Basic Functions - Quadratic Functions and Their Graphs | AI tutor The No.1 Homework Finishing Free App

"When the quadratic function $f(x) = x^2 + ax + b$ satisfies $2f(x) = (x+1) f'(x) + 6$, find the values of constants $a, b$."

'(1)\\[\n\egin{aligned}\ny^{\\prime} & =-3 x^{2}+12 x-9 \\\\\n& =-3\\left(x^{2}-4 x+3\\right) \\\\\n& =-3(x-1)(x-3)\n\\end{aligned}\n\\]\nWhen \ y^{\\prime}=0 \, \ x=1,3 \\nThe table for the increase and decrease of \ y \ is as shown on the right. Therefore, the graph is as in Fig(1)\n\n(2) \\( y^{\\prime}=x^{2}+2 x+1=(x+1)^{2} \\)When \ y^{\\prime}=0 \, \ x=-1 \The table for the increase and decrease of \ y \ is as shown on the right. Therefore, it always increases monotonically. Therefore, the graph is as in Fig(2)\n\n\\n\egin{\overlineray}{c||c|c|c|c|c}\n\\hline x & \\cdots & 1 & \\cdots & 3 & \\cdots \\\\\n\\hline y^{\\prime} & - & 0 & + & 0 & - \\\\\n\\hline y & \\searrow & \\text{min -2} & \\nearrow & \\text{max 2} & \\searrow \n\\hline\n\\end{\overlineray}\n\'

'Graph the region represented by the inequality y > x^{2}-3 x.'

'Find the coefficients a, b, and c of f(x) so that the graph has a local maximum at the point (x=1).'

"What is the condition for f(x)=6x^2-6x+3a, where f'(x)=6x^2-6x+3a, to have two distinct real number solutions?"

'For the quadratic function of x, find all the quadratic functions that intersect the graph of y=x^2 at two points orthogonally. Two graphs are said to be orthogonal at a point if they share that point and their tangents are orthogonal to each other.'

'x^{2}+y^{2}=10\n(1) y=-x+2\nx^{2}-2 x-3=0\n(x+1)(x-3)=0\nTherefore, when x=-1 or x=3, y=3 or y=-1\nTherefore, circle (A) and line (1) intersect at two points (-1,3),(3,-1).'

'54 (1) f(x) = t^2 + at + b - 1'

'When the point P(x, y) moves on the unit circle, find the maximum value of 15x^2+10xy-9y^2 and the coordinates of point P that give the maximum value.'

'Maximum value at x=1,4 is \\frac{4}{3}; minimum value at x=0 is -\\frac{16}{3}'

"Let's review the maximum and minimum values of a quadratic function, as well as equations involving trigonometric functions! Let's revisit Example 72 in Mathematics I! Recall how we found the maximum and minimum values of a quadratic function. First, complete the square and plot the graph. To plot the graph of the quadratic function y=4t^{2}+4t+6, complete the square on the right side to put it in standard form."

'For the two parabolas y=x^{2} (1) y=-x^{2}+x-a (2), answer the following questions.'

"Let's review equations involving quadratic functions, maximums, minimums, and trigonometric functions!"

"(2) If y' = 12x - 3x^2 = -3x(x-4) and y' = 0, then the table of increasing and decreasing of y at x = 0, 4 is as follows. Therefore, at x = 4 it takes a maximum value of 32 and at x = 0 it takes a minimum value of 0."

'Find the average rate of change when the value of x changes as follows in the function f(x)=x^{2}+2x-1.'

'The region represented by the inequality y>x^{2} is above the parabola y=x^{2}. Using this as a reference, illustrate the regions represented by the following inequalities:'

'Plot the region represented by the following inequalities. (1) \\left\\{\egin{\overlineray}{l}x-3 y-9<0 \\\\ 2 x+3 y-6>0\\end{\overlineray}\\right. (2) \\left\\{\egin{\overlineray}{l}x^{2}+y^{2} \\leqq 9 \\\\ x-y<2\\end{\overlineray}\\right. (3) $1<x^{2}+y^{2} \\leqq 4$'

'Find the equation of the tangent line to the graph of the function y=x^{2}-x drawn from point C(1,-1).'

'Find the area enclosed by the following parabolas and the x-axis.\n(1) y=1-x^{2}\n(2) y=x^{2}+x-2\n(3) y=2x^{2}+x-1'

'Graph of a cubic function and its intersection points with the x-axis'

'Find the equations of the two tangents at the points (-1,1) and (2,4) on the parabola y=x^{2}, and calculate the area enclosed by this parabola.'

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'Find the equation of the tangent line at the point (-1,0) on the circle.'

'Find the maximum and minimum of a cubic function'

'Find the equation of the tangent line at point P(1, -2) on the circle x^2 + y^2 = 5.'

'Let r be greater than 0. Find the range of values of r when the parabola y=x^{2}-1 and the circle x^{2}+y^{2}=r^{2} have 4 common points.'

'What is the equation of the tangent line at the point (t, t^{2}+1) on the parabola C: y=x^{2}+1? There are two tangent lines to C, what are their equations?'

'When x = 3, the maximum value is 648, and when x = 1, the minimum value is 72.'

'At x = -1, there is a local maximum of 5, at x = 3, there is a local minimum of -27. At x = 4, there is a local maximum of 32, at x = 0, there is a local minimum of 0. No extremum, no extremum.'

'What are the strategies to pass the exam? Please provide specific suggestions.'

'When the point \\((x, y)\\) moves on the set determined by \\((x^{2}+y^{2})^{2}-\\left(3 x^{2}-y^{2}\\right) y=0, x \\geqq 0, y \\geqq 0\\) on the coordinate plane, find the maximum value of \x^{2}+y^{2}\ and the values of \x, y\ that give this maximum value. [Chiba University]'

'Find the coordinates of the points where the graph of \\( f(x)=-\\frac{1}{2}x^{2}+2(x \\leqq 0) \\) and its inverse function \\( f^{-1}(x) \\) intersect.'

'Let the parabola y² = 4x be C. \n(1) Find the equation of the normal to the parabola C with slope m. \n(2) How many normals can be drawn from the point (a, 0) on the x-axis to the parabola C? Given that a ≠ 0.'

'Answer which type of conic section is represented by the following quadratic equations:'

'Find the quadratic function whose vertex is at the point (2, -3) and the length of the segment cut from the x-axis is 6.'

'Graph of a Quadratic Function and x'

'Find the maximum and minimum values of the function. (3) y=x^2-4x+2 (-2 < x ≤ 4)'

'Plot the graphs of the following functions:'

'Find the quadratic function that passes through the given three points.'

'Determine the number of intersection points between the parabola y=x^{2}-2x+2k-4 and the x-axis by considering different cases based on the value of k.'

'Find a quadratic function that satisfies the following conditions:\n(1) The vertex of the graph is at the point (1,3), and it passes through the point (0,5).\n(2) The axis of the graph is the line x=-1, and it passes through the points (-2,9) and (1,3).\n(3) It reaches a minimum value of -1 when x=-3, and at x=1, y=31.'

'How is the parabola y=-x^{2}+3x-1 parallel shifted to obtain the parabola y=-x^{2}-5x+2.'

'Find the equation of a parabola that overlaps with the parabola y=-2x^2+3 when shifted 2 units along the x-axis and -1 units along the y-axis.'

'Prove that the graph of the function $y = mx^2 - 4(m+1)x + m+3$ always has a point of intersection with the x-axis regardless of the value of the constant $m$.'

'Find the maximum and minimum values of the function f(x) = -x^2 + 2ax (0 ≤ x ≤ 4) where a is a constant.'

'Find the maximum and minimum values of the function. (2) y=2x^2-4x+3 (x ≥ 2)'

'Given line segments of length a and b, draw a line segment with the positive root as the length satisfying the quadratic equation x^2 - ax - b^2 = 0.'

'Intersection points of parabola and straight line'

'For the function f(x)=x^{2}-2 a x-a+6, find the range of values for the constant a such that f(x) ≥ 0 always holds for -1 ≤ x ≤ 1.'

'Determine the values of the constants a and b so that the graph of the quadratic function y = ax^{2} + bx - 1 passes through the points (1, 0) and (-2, -15).'

'When the function f(x)=ax^{2}-2ax+a+b has a maximum value of 3 and a minimum value of -5, find the values of the constants a and b.'

'Find the number of intersection points and the coordinates of those points where the graphs of the following two functions share the x-axis: (1) y = 9x^2 - 6x + 1 (2) y = x^2 - x + 1 (3) y = 3x^2 - 8x - 1'

'Basic Example 86 Intersection Points of a Parabola and a Line\nThere is a parabola y=x^{2}-3 x+3 and a line y=2 x-a.\n(1) Find the coordinates of the intersection points of the two graphs when a=1.\n(2) Determine the value of the constant a so that the two graphs have only one intersection point.\n(3) Determine the range of values of the constant a so that the two graphs have no intersection points.'

'Find the equations of the parabolas that satisfy the following conditions.'

'Find the maximum and minimum values of the following function. Let PR (a) be a constant. For the function f(x)=x^2-10x+a where a ≤ x ≤ a+1:\n\n(1) Find the maximum value.\n(2) Find the minimum value.'

'Find the quadratic function with a vertex at (2,-3) and a line segment of length 6 cut from the x-axis.'

'For the graph of a quadratic function y=a x^{2}+b x+c as shown on the right, determine the positive, zero, or negative nature of the following values: (1) a, (2) b, (3) c, (4) b^{2}-4 a c, (5) a+b+c, (6) a-b+c'

"Company A sells chocolates. The number of chocolates sold, denoted as y (where y is an integer greater than or equal to 1), is related to the selling price p yen per chocolate as follows:\ny = 10 - p\n(1) Find the values of the selling price p and the number of chocolates sold y that maximize Company A's revenue. Revenue is defined as the product of selling price and quantity sold.\n(2) The total cost c(y) of selling y chocolates is given by c(y) = y^2. Determine the values of the selling price p and the number of chocolates sold y that maximize Company A's profit (revenue minus total cost).\n(3) In (2), if the total cost c(y) changes to c(y) = y^2 + 20y - 20, find the values of the selling price p and the number of chocolates sold y that maximize Company A's profit."

'Select two functions from the following (1)~(4) that have a maximum value at x = 2, and find the maximum and minimum values of those functions.'

'Determination of Quadratic Functions (2)'

'For the graph C of the quadratic function y = x^2 - 4x + 3 and the point A(0, -1), find the following: (1) Move the graph C parallel to the x-axis so that it passes through point A'

'Determine the value of the constant a such that the parabola y=x^{2}-a x+a+1 is tangent to the x-axis. Also, find the coordinates of the point of tangency.'

'Plot the graph of a quadratic function y=ax^{2}+bx+c using computer graph plotting software. In this software, inputting values for coefficients a, b, and c in the A, B, C on the screen will display the graph corresponding to those values.'

'Find the quadratic function that satisfies the following conditions for its graph:'

'Translate the parabola y=x^{2}-3x-1 by shifting it to pass through the points (1,-1) and (2,0), find the vertex of the parabola.'

'Graph of a quadratic function with absolute value'

'Determine the value of the constant c so that the function has a maximum value of 7. Also, find the minimum value at that point.'

'After a certain parabola is translated parallel to the x-axis by 1 unit and parallel to the y-axis by -2 units, and then reflected about the x-axis, it becomes the parabola y=-x^2-3x+3. Find the original equation of the parabola.'

'This is a problem to find the maximum and minimum of a quadratic function. Please find the vertex of the quadratic function provided below and determine the maximum or minimum based on that value.'

'Quadratic functions and graphs\n\nGraph of quadratic functions\nGraph of $\\ Delta y=a(x-p)^{2}+q(a \\ neq 0)$: Vertex $(p, q)$, axis is the line $x=p$\nIf $a>0$, the parabola is concave down; If $a<0$, the parabola is concave up\n\nGraph of $D y=a x^{2}+b x+c(a \\ neq 0)$: Completing the square\n\\[ y=a\\left(x+\\frac{b}{2 a}\\right)^{2}-\\frac{b^{2}-4 a c}{4 a} \\]\nVertex $\\left(-\\frac{b}{2 a},-\\frac{b^{2}-4 a c}{4 a}\\right)$, axis is the line $x=-\\frac{b}{2 a}$\nIf $a>0$, the parabola is concave down; If $a<0$, the parabola is concave up'

'Find the length of the segment cut off by the graph of the following quadratic functions from the x-axis.'

'Find the range of values for the constant $a$ such that the 2nd degree equation $x^{2}-2(a-1)x+(a-2)^{2}=0$ has two distinct real number solutions $\\alpha, \eta$ satisfying $0<\\alpha<1<\x08eta<2$.'

'Find the quadratic functions that satisfy the following conditions: (1) The vertex of the graph is at (1,3) and it passes through the point (-1,4). (2) The axis of the graph is the line x=4 and it passes through the points (2,1) and (5,-2). (3) It has a maximum value of 10 at x=3 and y=-6 when x=-1.'

'Explain how to move the parabola y=-2x^2+3 parallel to the x-axis by -2 units and parallel to the y-axis by 1 unit, and find the equation of the resulting parabola.'

'A parabola y=2 x^{2}+a x+b is translated 2 units along the x-axis and -3 units along the y-axis, coinciding with the parabola y=2 x^{2}. Find the values of the constants a and b.'

'Translate the given text into multiple languages.'

'60 (1) \\ (x= \\pm 2 \\) has a maximum value of 8, a minimum value of -4 at \\ (x=0 \\)\n(2) \\ (x=2 \\) has a minimum value of 3, with no maximum value\n(3) \\ (x=2 \\) has a minimum value of -2, with no maximum value\n(4) \\ (x=0 \\) has a maximum value of 1, with no minimum value'

'When 64 (1) \ a<2 \, for \ x=4 \ the maximum value is \ -24 a+53 \; when \ a=2 \, for \ x=0,4 \ the maximum value is 5; when \ a>2 \, for \ x=0 \ the maximum value is 5; (2) when \ a<0 \, for \ x=0 \ the minimum value is 5; for \ 0 \\leqq a \\leqq 4 \, for \ x=a \ the minimum value is \ -3 a^{2}+5 \; when \ a>4 \, for \ x=4 \ the minimum value is \ -24 a+53 \'

'Find the equation of the parabola obtained by translating the parabola y=x^2-4x parallel to the x-axis by 2 units and parallel to the y-axis by -1 unit.'

'There are three forms of a quadratic function: standard form, general form, and factored form. Please explain the characteristics and situations where each form should be used.'

'Find the range of values for the constant $a$ such that the two distinct real roots of the equation $a x^{2}+(a+7) x+2 a-7=0$ both lie within the interval $-3<x<3$.'

'Find the number of points of intersection between the graphs of the following two quadratic functions and the x-axis.'

'Find the maximum and minimum values of the following function: (4) y=-x^2-6x+1 (0 ≤ x < 2)'

'Find the maximum and minimum values of the function. (1) y=3x^2-4 (-2 ≤ x ≤ 2)'

'Graph the following quadratic functions. Also, find their vertices and axis.'

'Maximum and minimum of a quadratic function'

'Chapter 4 Quadratic Functions: Solving 132 Quadratic Functions'

'When the parabola y=x^{2}-(k+2)x+2k intersects the x-axis and cuts a line segment of length 3, find the value of the constant k.'

'Chapter 4 Quadratic Functions: Graph of a 112th-degree function'

'Plot the graph of a quadratic function y=ax^2+bx+c using the method of completing the square.'

'Find the value of the constant a when the vertex of the parabola y=x^{2}+a x-2 lies on the line y=2 x-1.'

'When a ball is launched vertically upwards from the ground, and the height after t seconds is y meters, y becomes a quadratic function of t. If the height of the ball reaches a maximum of 176.4 meters after 6 seconds, how can y be expressed as a function of t?'

'Find the value of the constant a when the vertex of the parabola y=x^2+ax-2 lies on the line y=2x-1.'

'Find the maximum and minimum values of 2x^{2}+3y^{2} when x^{2}+2(y-2)^{2}=18. Also, determine the values of x and y at that time.'

'Find the maximum and minimum values for the following quadratic functions:'

'Calculate the number of points of intersection between the graph of the following quadratic functions and the x-axis.'

'For the cases where a is a constant, find the maximum and minimum values of the function f(x)=3 x^{2}-6 a x+2 within the range 0 ≤ x ≤ 2.'

'Determine whether the following example is a function: "For the square root of x, x = 16, therefore y = 4 and y = -4, so y is not a function of x"'

'Let the graph of the quadratic function $y=-2x^{2}+ax+b$ pass through the point $(3,-8)$. Find the value of the constant $a$ when the graph touches the $x$-axis.'

'Find the equation of the parabola obtained by symmetrically moving the parabola y=-2x^{2}+4x-4 with respect to the x-axis, and then moving it parallelly 8 units in the x-axis direction and 4 units in the y-axis direction.'

'How to determine a quadratic function that satisfies the given conditions'

'Find the maximum and minimum values, if any, of the following quadratic functions.'

'Find the coordinates of intersection points between the following parabolas and lines.'

'Maximum and minimum of a quadratic function'

'When x^{2}+y^{2}=4, find the maximum and minimum values of 2 y+x^{2}. Also, determine the values of x and y at that time.'

'For the graph of the quadratic function $y=x^{2}+2(k-1) x+k^{2}-3$, answer the following question: (1) Find the range of values for the constant $k$ when it does not have any points in common with the x-axis.'

'Plot the graphs of the following equations: (1) y=2x^2-4x-1 (2) y=-x^2-2x+4 (3) y=-x^2+4x-3'

'Find a quadratic function where the coefficient of the quadratic term is -1, the graph passes through the point (1,1), and the vertex lies on the line y=x.'

'Find the maximum and minimum values for the following functions:'

'Find the coordinates of the points where the graph of the quadratic function y = x^2 - 6x + 6 intersects the x-axis.'

'Plot the graph of the following quadratic function.'

'The graph of the quadratic function y = mx^2 + 3x + m is always above the x-axis'

'Explain the relationship between the graph of the following functions and the x-axis.'

'Find a quadratic function that satisfies the following conditions.'

'Regarding the graph of the quadratic function y=x^{2}+2(k-1) x+k^{2}-3, answer the following questions:'

"Maxima and minima of a quadratic function...... the position of the axis is crucial. When reviewing the solution to example 83 again, it can be seen that there are different cases to consider, but the key point is where the axis (vertex) is located with respect to the domain. For example, for a concave down graph of a quadratic function, it can be divided as follows: 1. The axis (vertex) is within the domain. 2. The axis (vertex) is outside the domain. In the case of a concave up graph, it is as follows. That's right. The magnitude will be reversed. Whether concave up or concave down, various cases exist depending on whether the axis (vertex) is inside or outside the domain, and whether the axis (vertex) is to the left or right of the middle position of the domain. In example 83, although both the maximum and minimum values are considered simultaneously, what happens if we consider the maximum and minimum values separately?"

'Plot the graphs of the following two quadratic functions and find their vertex and axis.'

'(1) y=-(x-1)^{2}+1, \\quad y=-(x-2)^{2}+2 \\\\ \\left[y=-x^{2}+2 x, \\quad y=-x^{2}+4 x-2\\right]'

'(1) Since the axis is the straight line x=4, the quadratic function to be determined can be expressed as y=a(x-4)^{2}+q. Since the graph passes through the points (2,1),(5,-2), we have 1=a(2-4)^{2}+q, -2=a(5-4)^{2}+q. Organizing these equations, we get 4a + q = 1 and a + q = -2. Solve these simultaneous equations to find the quadratic function.'

'(1) Show that the parabola y = x^2 + ax + a-4 always has two distinct points of intersection with the x-axis, regardless of the value of the constant a.'

'Find the quadratic function that takes a maximum value of 1 at x=3 and y=-1 at x=5.'

'Determine the range of values for the constant a such that the parabola y=x^{2}-8ax-8a+24 intersects the positive part of the x-axis at two distinct points.'

'Find the maximum and minimum values of the following quadratic functions.'

'To investigate the relationship between the graph of a quadratic function and the x-axis, it is necessary to understand the following. Since the y-coordinate of the intersection points between the graph of a quadratic function and the x-axis is always 0, the x-coordinate of the intersection points where y = 0 is the value of x. In other words, solving the following problems is crucial.\n1. Find the x-coordinate of the intersection point between the graph of the quadratic function y=ax^2+bx+c and the x-axis.\n2. What is the term for a single intersection point? What is that point?'

'Find the number of intersection points of the graphs of the following quadratic functions with the x-axis.'

'When a ball is thrown vertically upwards from the ground and the height after t seconds is y meters, y becomes a quadratic function of t. If the height of the ball reaches a maximum of 176.4m after 6 seconds from the launch, how is y expressed as a function of t?'

'Transform the quadratic function y=ax^{2}+bx+c into the maximum and minimum form y=a(x-p)^{2}+q (completing the square). When a>0, the minimum value is at x=p, which is q. When a<0, the maximum value is at x=p, which is q.'

"Let's try to make a parabola!"

'Plot the graphs of the following functions.'

'Find the quadratic functions for the following parabolas: (1) Parabola with vertex at (2,-3) passing through the point (3,-1) (2) Parabola with axis at the line x=4 passing through the points (2,1) and (5,-2)'

'Draw the graph of a quadratic function y=ax^2+bx+c (2)'

'Please find the maximum value of the following quadratic function: f(x) = -2x^2 + 4x'

'The maximum value is 21 when x=-4, and the minimum value is -3 when x=0'

'Rewrite the following conditions (a), (b), (c) so that they all become equivalent to condition (d).'

'Let a and b be constants, and let F be the graph of the quadratic function y=x^{2}+ax+b. Choose two correct statements about F from the following (1) to (6).'

'By translating the parabola (2) parallel to the x-axis by -3 and the y-axis by -6, and then reflecting it with respect to the origin, it returns to parabola ①. During this movement, the vertex (2, -3) is moved by the parallel translation to point (2-3, -3-6) which is point (-1, -9), and further moved by the symmetric translation with respect to the origin to point (1, 9). Therefore, the equation of (1) is y=(x-1)^{2}+9 which is equivalent to y=x^{2}-2x+10, hence a=-2, b=10.'

'(3) Find a quadratic function that has a minimum at x=-2 and passes through the points (-1,2) and (0,11).'

'Find the equation of the parabola obtained by symmetrically moving the parabola y=-2x^2+4x-4 with respect to the x-axis and then parallelly shifting it by 8 units in the x-direction and 4 units in the y-direction.'

'Plot the graph of the following quadratic function and find its vertex and axis. (1) y=5 x^{2}+3 x+4'

'Find a quadratic function that satisfies the following conditions.'

'(2) For the parabola y=x^2+ax+a-4 and the x-coordinates of the points of intersection as α and β, find the range of constant a such that (α-β)^2<28 holds true.'

'Find the quadratic function passing through the following 3 points.'

'Find the quadratic function that passes through the following 3 points:\n(1) (-1,7),(0,-2),(1,-5)\n(2) (-1,0),(3,0),(1,8)'

'Let P be the vertex of the parabola y=x^{2}+2 x-1. Answer the following questions: (1) Find the coordinates of the point Q which is symmetrical to point P with respect to the x-axis. (2) Find the equation of the parabola that is symmetrical to this parabola with respect to the x-axis.'

'(1) y=2(x-2)^2-3[y=2x^2-8x+5] (2) y=(x-4)^2-3[y=x^2-8x+13]'

'Steps for determining a quadratic function'

'As the point P moves on the parabola y=x², perpendicular lines PQ and PR are drawn to the lines y=x-1, y=5x-7 respectively. Find the minimum value of the product PQ・PR. Also, determine the coordinates of point P at that moment.'

'Passing through point A(-1,0), let ℓ be the line with slope a. The parabola y = 1/2 x^2 intersects the line ℓ at two distinct points P and Q.'

'Find the equation of the tangent line drawn on the graph of the function y=x^{2}-x passing through the point C(1,-1).'

'Let 99a be a constant. For the parabola y=x^{2}+ax+3-a, when a varies over all real values, find the locus of the vertex.'

'Find the area enclosed by the tangent lines to the curve y=-x²+1 at the point (0,0) and the point (2,-2).'

'Given the parabola y=x^{2} and the circle x^{2}+(y-4)^{2}=r^{2}(r>0), find the range of r for which the parabola and the circle have 4 intersection points.'

'Calculate the average rate of change for the given basic problems 169'

'When positive real numbers x and y satisfy 9x^2 + 16y^2 = 144, find the maximum value of xy.'

'Find the equation of the tangent line with a slope of -1 on the parabola y=x^{2}-5 x+4.'

'Find the maximum value of y=-x^{2}+2x+3 and the corresponding value of x.'

'Find the condition for the real number a that the two parabolas y = -2(x-a)^2 + 3a and y = x^2 have two distinct common points. Find the maximum area enclosed by the two parabolas.'

'Finding the maximum and minimum area'

'After filling in the third row of the increase-decrease table, draw the graph. When you fill in the value of y corresponding to the x value in the first row, draw the graph according to the content. In this case, when drawing the graph of a quadratic function, it is recommended to first find the vertex so that y is at a maximum or minimum. Also, find the coordinates of the intersection with the y-axis (substitute x=0 into the expression for y).'

'Find the equation of the line passing through the point (2, -4) and tangent to the curve y=x^{2}-2 x.'

'Let a>0, consider the tangent line m at the point (a, a^{2}) on the parabola E: y=x^{2}. Find the area enclosed by E, m, and the y-axis, expressed in terms of a.'

'From the point (3,4), find the equation of the tangent line drawn to the curve y=-x^{2}+4x-3.'

'Find the minimum value for the following:\n31. (1) x=4 with minimum value of 8\n(2) x=2 with minimum value of 3'

'(1) At x=-\\frac{1}{2}, the minimum value is -\\frac{3}{16}. At x=0, the maximum value is 0. At x=2, the minimum value is -8. (2) At x=1, the minimum value is 0'

'Find the equation of the common tangent of the two parabolas C_{1}: y=x^{2}+1, C_{2}: y=-2 x^{2}+4 x-3.'

"Let's take a closer look at the content of basic example 214. It should be noted that in the actual answer, calculations should be done as in the example solution without using the following as a formula."

'The x-coordinates of the intersection points of the curve y=-2x^{2}+4x+6 and the x-axis are obtained by solving the equation -2x^{2}+4x+6=0, ie. x^{2}-2x-3=0, which gives (x+1)(x-3)=0, thus -1 ≤ x ≤ 3 and y ≥ 0, so the required area S is'

'There are two quadratic functions f(x)=-\\frac{1}{2}x^2+\\frac{3}{2} and g(x)=x^2+ax+3.'

"When the quadratic function $f(x)$ satisfies $f'(0)=1, f'(1)=2$, find the value of $f'(2)$."

'Find the equation of the line passing through the circle x^2 + y^2 = 8 and perpendicular to the line 7x + y = 0.'

'At the point P(a, a^2) on the parabola C: y = x^2, where a > 0, find the tangent line l1. If the tangent line l2 at a point Q on C different from point P is perpendicular to l1, find the equation of l2.'

"The second line is filled in using the graph of y'"

'Let a be a constant, and consider the function f(x)=-2x^{2}+6x+1 over the interval a ≤ x ≤ a+1. Answer the following questions.'

'Find the quadratic function when the graph of the practice meets the following conditions:'

'Find the maximum and minimum of the quadratic function y=ax^{2}+bx+c.'

'Find the minimum value.'

'Find the value of the constant a when the parabola y=x^{2}+3x+a and the line y=x+4 are tangent.'

'Find the quadratic function passing through the points (1,0), (3,0), and (4,2).'

'Find the quadratic function that satisfies the following conditions: Passing through the points (1,8), (-2,2), (-3,4).'

'(2) Translate the parabola $y=x^{2}-3 x+4$ so that it passes through the point (2,4) and its vertex lies on the line $y=2 x+1$.'

'Determine the range of the constant m so that the graph of the quadratic function y=-x^{2}+(m-10)x-m-14 satisfies the following conditions: (1) Intersects with both the positive and negative parts of the x-axis. (2) Shares points only with the negative part of the x-axis.'

'Explain how to solve the maximum and minimum problems of a quadratic function and describe its characteristics.'

'How does the number of points of intersection between the graph of the quadratic function y=x^{2}-2x+2k-4 and the x-axis change with different values of the constant k?'

'Find the quadratic function that satisfies the following conditions: passing through the points (-1,16), (4,-14), (5,-8).'

'Practice Determine the values of the constants a and b for the function f(x) = a x^2 + 4ax + b, with the domain -1 <= x <= 2, when the maximum value is 5 and the minimum value is 1.'

'Find the equation of a quadratic function y = ax^2 + bx + c. The graph passes through the points (-1, 22), (5, 22), and (1, -2).'

'For the following quadratic functions, if there exist maximum or minimum values, please find them. (1) y=x^{2}-2 x-3 (2) y=-2 x^{2}+3 x-5 (3) y=-2 x^{2}+6 x+1 (4) y=3 x^{2}-5 x+8'

'Let a be a constant, and f(x) = x^2 - 2ax + a + 2. Find the range of values for a such that f(x) > 0 holds true for all x values in the range 0 ≤ x ≤ 3.'

'Find the range of values of a when the graph of the function y=ax^2+4x+2 has 2 different points of intersection with the x-axis. Also, determine the value of a when the graph intersects the x-axis at just one point.'

'Determine the number of intersection points between the graph of the quadratic function y=-x^{2} and the line y=-2x+k. Where k is a constant.'

'Does the graph of the following parabolic functions have a common point with the x-axis? If so, find the coordinates:\n(1) y = -3x^2 + 6x - 3\n(2) y = 2x^2 - 3x + 4\n(3) y = -x^2 + 4x - 2'

'Use graphing software on a computer to display the graph of a quadratic function y=ax^{2}+bx+c. In this software, input the values of coefficients a, b, and c at positions A, AB, and C on the screen, and the graph corresponding to those values will be displayed. Now, after inputting the values at A, B, and C, a graph similar to the one on the right was displayed. (1) Choose an appropriate combination from 11 to 8 in the table on the right as the values input at positions A, B, and C. (2) To display a curve symmetrical to the origin of the graph currently displayed, what values should be input at A, B, and C? Choose the appropriate combination from 1 to 8 in table (1).'

'Please explain how to effectively utilize charts and analysis to solve example problems.'

'Find the values of constants a and b when the quadratic function y=3x^2-(3a-6)x+b takes a minimum value of -2 at x=1.'

'Find the equation of the parabola obtained by shifting the parabola $y=-2 x^{2}+4 x-4$ 3 units to the left along the $x$-axis and 1 unit up along the $y$-axis.'

'Find the quadratic function when the following conditions are met.'

'Transform the graph of the quadratic function y=ax^{2}+bx+c into the form a(x-p)^{2}+q by completing the square.'

'Determine the range of values for the constant m such that the graph of the quadratic function y=-x^{2}+(m-10)x-m-14 satisfies the following conditions: (1) Intersects the positive and negative parts of the x-axis. (2) Only shares a point with the negative part of the x-axis.'

'Usually, when two parabolas have two points in common, find the equation of the line passing through them.'

'By using graphing software, you can observe how the graph changes based on the value of a.'

'Chapter 3: Quadratic Functions\nSection 8: Functions and Graphs\nProblem\nDraw the graphs of the following functions:\n(a) $y = x^2$\n(b) $y = -x^2$'

'Find the length of the segment cut by the graph of the quadratic function y=-3x^{2}-4x+2 from the x-axis.'

'Determine the range of the constant m so that the graph of the quadratic function y=x^{2}-m x+m^{2}-3 m meets the following conditions:'

'Find the quadratic function that passes through the points (1,1), (3,5), and (4,4).'

'The desired solution is in the combined range of \ -1<x<-1+2 \\sqrt{3} \\n(2) \\( x^{2}-6 x-7=(x+1)(x-7) \\) so\n\ x^{2}-6 x-7 \\geqq 0 \ the solution is \ x \\leqq-1,7 \\leqq x \\n\ x^{2}-6 x-7<0 \ the solution is \ \\quad-1<x<7 \'

'Mathematics II\n(1) Determine the value of the constant a so that the graph of the function y=x^{2}+ax+a is tangent to the line y=x+1. Find the coordinates of the point of tangency.\n(2) Let k be a constant. Investigate the number of intersection points between the graph of the function y=x^{2}-2kx and the line y=2x-k^{2}.'

'Since the graph passes through the point (2,4), 2(2-p)^{2}+2 p-4=4.\nSimplify to get p^{2}-3 p=0, so p=0,3.\nWhen p=0, y=2 x^{2}-4.\nWhen p=3, y=2(x-3)^{2}+2.'

'How does the number of intersection points between the graph of the quadratic function y=x^{2}-2 x+2 k-4 and the x-axis change with the value of the constant k?'

'For the quadratic function y=ax^2+bx+c, passing through points (-1,0) and (3,8), and tangent to the line y=2x+6, what are the values of a, b, and c?'

'Do these two parabolas have intersection points? If so, find their coordinates.'

'Range of existence of solutions of the equation'

"Let's consider the position of the points of intersection between a parabola and the x-axis. Draw a parabolic graph that satisfies the following conditions.\n\nConditions:\n1. The graph intersects the positive part of the x-axis at two different points.\n2. The position of the axis is positive.\n3. f(0) > 0.\n\nExplain what characteristics a graph that satisfies these conditions would have."

'Practice: In the following graph of quadratic functions, how much were the graphs within the square brackets parallel shifted? Also, draw each graph, and find their axes and vertices.'

'How is the graph of the quadratic function y=2x^{2}+6x+7 shifted to obtain the graph of the quadratic function y=2x^{2}-4x+1?'

'The shape of the graph of the function f(x) changes depending on the value of a, but explain for which values of a f(x) does not have a maximum value.'

'Find the value of the constant k when the length of the segment cut by the parabola y=x^{2}-(k+2)x+2k from the x-axis is 4.'

'For [5], [6], [7], y is minimum at x=2.'

'Mathematics I\nDo the following parabolas and lines have common points? If so, find the coordinates.\n(1) \\ left\\ { \\ begin {\overlineray} {l} y=x^{2}-2 x+3\\ y=x+6 \\ end {\overlineray} \\ right. \n(2) \\ left\\ { \\ begin {\overlineray} {l} y=x^{2}-4 x\\ y=2 x-9 \\ end {\overlineray} \\ right. \n(3) \\ left\\ { \\ begin {\overlineray} {l} y=-x^{2}+4 x-3\\ y=2 x \\ end {\overlineray} \\ right. '

'When the coordinates of A and B are sought, it is as follows.'

'The graph of the quadratic function y=x^{2}+ax-a+3 has points of intersection with the x-axis, but does not have any points of intersection with the line y=4x-5. Here, a is a constant.\n(1) Determine the range of values for a.\n(2) Let m be the minimum value of the quadratic function y=x^{2}+ax-a+3, find the range of values for m. [Hokkaido Information University]'

'Find the values of a, b, and c for the quadratic function y=a x^{2}+b x+c passing through points (-1,0) and (3,8) and tangent to the line y=2 x+6.'

'Do the graphs of these two quadratic functions intersect with the x-axis? If so, find the coordinates of the intersection points.'

'Practice: Find the maximum and minimum values of the following functions.'

'Find the x-coordinates of the intersection points of the graph of the function y=x^{2}-2ax+a^{2}-3 and the x-axis.'

'Find the coordinates of the points of intersection of the following quadratic functions with the x-axis.'

'Find the values of constants a, b when the quadratic function y=x^{2}+ax+b takes a maximum value of 1 in the range 0 ≤ x ≤ 3 and a maximum value of 9 in the range 0 ≤ x ≤ 6.'

'Determine the sign (positive, 0, negative) of the following values for the graph of a quadratic function as shown on the right:'

'Find the conditions for the graph of the quadratic function y = ax^2 + bx - 1 to pass through the points (1,0) and (-2,-15).'

'Find the number of intersection points between the graphs of the functions y = x^2 - 4 and y = a(x + 1)^2.'

'Find the quadratic function represented by graphs (1) and (2) of graph C of the function y=x^{2}-4 x+3 and point A(0,-1), where (1) is a graph obtained by translating C parallel to the x-axis passing through point A and (2) is a graph obtained by translating C parallel to the y-axis passing through point A.'

'Chapter 3 Quadratic Functions\n[1] $a+\\frac{1}{2}<5$ that is when $a<\\frac{9}{2}$\nFrom figure [1], maximum occurs at $x=a$.\nThe maximum value is $f(a)=a^{2}-9 a$\n[2] $a+\\frac{1}{2}=5$ meaning $a=\\frac{9}{2}$\n[1] $x=a+\\frac{1}{2}$\nFrom figure [2], maximum occurs at $x=\\frac{9}{2}, \\frac{11}{2}$.\nThe maximum value is $f\\left(\\frac{9}{2}\\right)=f\\left(\\frac{11}{2}\\right)=-\\frac{81}{4}$\n[3] $5<a+\\frac{1}{2}$ meaning $\\frac{9}{2}<a$\nFrom figure [3], maximum occurs at $x=a+1$. The maximum value is\n$f(a+1) =(a+1)^{2}-10(a+1)+a =a^{2}-7 a-9$'

'After a certain parabola was translated parallel to the x-axis by 1 unit and parallel to the y-axis by -2 units, then symmetrically moved with respect to the x-axis, it became the parabola y=-x^{2}-3x+3. Find the equation of the original parabola.'

'Find the maximum and minimum values of the following quadratic functions:'

'Find the values of the constants a and b when the quadratic function y = x^2 + a x + b takes a maximum value of 1 in the range 0 ≤ x ≤ 3 and a maximum value of 9 in the range 0 ≤ x ≤ 6.'

'The maximum and minimum values of the quadratic function f(x)=-x^{2}+2x for a ≤ x ≤ a+2 are functions of a, denoted by F(a) and G(a) respectively. Plot the graphs of functions F(a) and G(a).'

'Given that x and y satisfy x^2 + 2y^2 = 1, find the maximum and minimum values of 2x + 3y^2.'

"The graph of a quadratic function is a parabola, and the values of a, b, c determine whether it opens downwards or upwards, the degree of openness, the position of the axis and vertex, etc. Here, let's think about how the graph changes when we vary the values of a, b, and c."

'Find the quadratic function that passes through the following three points.'

'Find the quadratic function that satisfies the following conditions: (1) The vertex is at point (1,3), and it passes through point (0,5).'

'Find the quadratic function passing through (-1,7), (0,-2), (1,-5).'

'In the range of -1 <= x <= 2, for the function f(x) = ax^2 - 2ax + a + b, when the maximum value is 3 and the minimum value is -61, find the values of constants a and b.'

'By using the graph of a quadratic function and its relationship with the x-axis, find the real solutions of the following equation.'

'Chapter 3 Quadratic Functions'

'When $5 < \\alpha$, from figure [6], it can be observed that $x=a$ results in the minimum value. The minimum value is $f(a)=a^{2}-9 a$'

'Find a quadratic function that satisfies the following conditions: (2) The axis of the graph is the line x=4, and it passes through the points (2,1) and (5,-2).'

'Basic Example 84: Find the number of intersection points between the parabola y=x^{2}-2 x+2 k-4 and the x-axis by considering different cases of the constant k.'

'Find the quadratic function with vertex at point (2, -3) and a length of 6 for the segment cut from the x-axis.'

'Points to note when completing the square'

'Determine the value of the constant a, so that the parabola y=x^{2}-ax+a+1 is tangent to the x-axis. Also, find the coordinates of the point of tangency.'

'For -4 <= x <= 0, from figure (1), it can be observed that the maximum value is obtained at x=-1 with f(-1)=3, and the minimum value is obtained at x=-4 with f(-4)=-15.'

'Find the maximum and minimum values of the following quadratic functions:\n(1) y=2 x^{2}+4 x+1\n(2) y=-x^{2}+2 x+3'

'Find a quadratic function that satisfies the following conditions: (2) The axis of the graph is the line x=-1 and it passes through the points (-2,9) and (1,3).'

'Find the maximum and minimum values of the following quadratic functions: (1) y=x^{2}-2 x-3 (2) y=-2 x^{2}+x (3) y=3 x^{2}+4 x-1 (4) y=-2 x^{2}+3 x-5'

'Prove that the length of the line segment cut from the x-axis by the graph of the quadratic function y=x^{2}-2ax+a^{2}-3 is constant regardless of the value of the constant a.'

'Describe the graph of the following quadratic functions.'

'Find the quadratic function that satisfies the following conditions: At x=-3, it takes the minimum value of -1, and at x=1, y=31.'

'Find the length of the line segment cut by the graph of the quadratic function y=-x^{2}+3x+3 from the x-axis.'

'Chapter 3 Quadratic Functions\n(2) Since the vertex of the parabola to be determined lies on the line y=-x+2, the coordinates of the vertex are represented as (p,-p+2). Therefore, the required equation is y=\\frac{1}{2}(x-p)^{2}-p+2. Since the parabola passes through the point (1,5), we have 5=\\frac{1}{2}(1-p)^{2}-p+2, which simplifies to p^{2}-4 p-5=0. Therefore, (p+1)(p-5)=0, hence p=-1,5. When p=-1, (1) becomes y=\\frac{1}{2}(x+1)^{2}+3 (or y=\\frac{1}{2} x^{2}+x+\\frac{7}{2}). When p=5, (1) becomes y=\\frac{1}{2}(x-5)^{2}-3 (or y=\\frac{1}{2} x^{2}-5 x+\\frac{19}{2}).'

'Let x, y be real numbers. Find the minimum value of 6 x^{2}+6 x y+3 y^{2}-6 x-4 y+3 and the corresponding values of x, y.'

'Find the quadratic function that passes through the following 3 points.'

'How should the parabola y = 3x^2-6x+5 be translated to overlap with the parabola y = 3x^2+9x?'

'Determine the number of intersection points of the graph of the following quadratic functions with the x-axis.'

'Find the equation of a parabola that overlaps with the parabola y=-2 x^{2}+3 after being translated 2 units along the x-axis and -1 unit along the y-axis.'

'Plot the graphs of the following two quadratic functions and find their axes and vertices. (1) y=x^{2}+4 x+3 (2) y=-2 x^{2}+6 x-1'

'Find a quadratic function that satisfies the following conditions: It reaches a maximum value of 10 at x=3, and y=-6 when x=-1.'

'In problems involving the determination of quadratic functions, choosing the correct form - standard form, general form, or factored form - is crucial as it can affect the complexity of calculations and the likelihood of errors. It is important to consider when to use each form.'

'By what amount should the parabola y=-x^2+3x-1 be translated in order to obtain the parabola y=-x^2-5x+2.'

'Find the coordinates of the two intersection points of the parabolas y = x^2 - x + 1 and y = -x^2 - x + 3.'

'Find a quadratic function that satisfies the following conditions: (1) The vertex of the graph is at the point (1,3) and it passes through the point (-1,4). (2) The axis of the graph is the line x=4, and it passes through the points (2,1) and (5,-2). (3) It has a maximum value of 10 at x=3, and at x=-1 the function value is -6.'

'Basic Example 832 Function Graph and Intersection Points with the x-axis'

'Determine the maximum and minimum of a quadratic function'

'Let the graph G of y=2x^{2}-4x+5 be shifted parallel to the y-axis by k to obtain the graph H. When the graph H intersects the x-axis at two different points, and it intersects at two different points within the range of 2≦ x≦6, the range of possible values for k such that the graph intersects the x-axis at two different points is ≦ k< .'

'Find the equation of the parabola after shifting the parabola y = -x^2 + x - 2 parallel to the x-axis by -3 and parallel to the y-axis by 1.'

'When shifting the parabola y=x^2-3x-1 parallelly to pass through points (1,-1) and (2,0), find the vertex of the parabola.'

'Find the quadratic function that passes through the following 3 points.'

'The parabola y=2x^2-4x+1 was parallel shifted by two units from the parabola y=2x^2+6x+7.'

'Find the quadratic function that satisfies the following conditions: (1) The vertex of the graph is at the point (1,3), and it passes through the point (-1,4).'

'Find the length of the line segment cut by the graph of the following quadratic functions from the x-axis. (1) y=4 x^{2}-7 x-11 (2) y=-4 x^{2}+4 a x-a^{2}+9'

"Regarding 'a', the sign and absolute value of the coefficient 'a' depend on whether the parabola is concave downwards or upwards, as well as the degree of openness of the parabola."

'A certain parabola was parallelly translated by 1 unit along the x-axis and by -2 units along the y-axis, and then symmetrically translated with respect to the x-axis, resulting in the equation of the parabola becoming y=-x^2-3x+3. Determine the original equation of the parabola.'

'Let PR be a constant. For the function f(x)=3 x^{2}-6 a x+5 (0 ≤ x ≤ 4), find:\n(1) The maximum value.\n(2) The minimum value.\nf(x)=3 x^{2}-6 a x+5=3(x-a)^{2}-3 a^{2}+5\nThe graph of this function is a concave parabola, with the axis being the line x=a.'

'When x and y satisfy 2x^{2}+y^{2}-4y-5=0, find the maximum and minimum values of x^{2}+2y.'

'Find the maximum and minimum values of the function f(x) under the following conditions.\n(2) -2<x≤1\n(3) 0≤x≤3\nFind the optimal x based on the diagram and each condition.'

'Let a be a constant. Find the minimum value of the function f(x) = x^2 - 2x + 2 in the range a ≤ x ≤ a + 2.'

'Investigate the number and nature of solutions of the function f(x) under the following conditions:'

'Find the maximum and minimum values of the function f(x)=-x^2+2ax+a+b.'

'(1) After entering 1 in A and values in B and C, a graph like Figure 1 was displayed. Which of the following is the most appropriate combination of b and c values?'

'Shape represented by ax^2 + by^2 + cx + dy + e = 0'

'Find the equation of the tangent line drawn from the given point to the following parabola.'

'Coordinates of intersection points of a quadratic curve and a straight line'

'Find the equation of the tangent line at the point $(-1, \\frac{1}{5})$ on the ellipse $\\frac{(x+4)^{2}}{25}+\\frac{(y+3)^{2}}{16}=1$.'

'Find the equations of the tangent and normal to the point P on the following curves.\n(1) \\( y^{2}=4 p x(p \\neq 0), \\mathrm{P}\\left(x_{1}, y_{1}\\right) \\)\n(2) \\( x^{2}-y^{2}=1, \\mathrm{P}(2, \\sqrt{3}) \\)\n(3) \ x=\\cos 2 \\theta, y=\\sin \\theta+1, \\mathrm{P} \ corresponds to \ \\theta=\\frac{\\pi}{6} \'

'32 Curves and Regions'

'(2) From $\\sin ^{2} \\theta+\\cos ^{2} \\theta=1$, we have $\\sin ^{2} \\theta=1-\\cos ^{2} \\theta$'

'Find the equations of the following conic sections. Assume p≠0, a>0, b>0. (1) A parabola obtained by translating the parabola y^{2}=4 p x, with the directrix as the line x=-1 and the focus at the point (3,4). (2) A hyperbola obtained by translating the hyperbola x^{2}/a^{2}-y^{2}/b^{2}=1, with the asymptotes being the lines y=x+1 and y=-x+1, and passing through the point (3,3).'

'Find the x-coordinate of the intersection point of parabola C₁ and the x-axis.'

'Equation passing through the intersection of two curves'

'When the function \\( f(x)=x^{2}-2 a|x|+a^{2}-1 \\) is given, find the area \ S \ enclosed by the graph of \\( y=f(x) \\) and the x-axis. Here, \ a \ is a positive constant.'

'Let f(x) = x^{2} - 2a|x| + a^{2} - 1, find the area S enclosed by the graph of y = f(x) and the x-axis. Here, a is a positive constant.'

'Find the vertex of the parabola y=x^{2}+px+p(|p| ≠ 2).'

'Find the locus of the vertex of the parabola y = x^{2}+px+p(p ≠ 2).'

'Given y=(x^{2}+x-1)(-2x-1)+5x+8, when x=(-1 \\pm \\sqrt{5})/2, y^{\\prime}=0. Therefore, from x^{2}+x-1=0, we can prove that when x=(-1-\\sqrt{5})/2, y=(11-5 \\sqrt{5})/2. Similarly, show that when x=(-1+\\sqrt{5})/2, y=5(-1+\\sqrt{5})/2+8.'

'The distance between point P on the parabola y = -x^2 + x + 2 and a point on the line y = -2x + 6 takes the minimum value when the coordinates of P are α.'

'Let the vertex of the parabola $y=x^{2}+(2 t-10) x-4 t+16$ be $P$. Find the trajectory of vertex $P$ as $t$ varies over non-negative values.'

"(1) y'=4x+1\n(2) y'=3x^2-5\n(3) y'=-24x^2+24x-6\n(4) y'=6x^5-16x^3+8x\n(5) y'=27x^2+6x-8"

'When a parabola and a circle have four distinct common points, it means that the quadratic equation (1) has two different real solutions in the range 1<y<3. Therefore, the range of values of a that satisfy conditions [1] to [3] simultaneously is to be determined. Let f(y)=2y^{2}-7y-a+6.\n[1] Let D be the discriminant of equation (1), it is required that D>0, which implies 8a+1>0, leading to a>-−1/8.\n[2] About the axis, it always holds that 1<\\frac{7}{4}<3.\n[3] From f(3)=3-a>0, it follows that a<3, and from f(1)=1-a>0, it follows that a<1. The common range of (3) to (5) is determined to be -\\frac{1}{8}<\\alpha<1.'

'At x = \\frac{-1+\\sqrt{5}}{2}, the maximum value is \\frac{11+5 \\sqrt{5}}{2}, and at x = 2, the minimum value is -7.'

'Let a and b be real numbers. Illustrate the region of points (a, b) such that the curve y = ax^2 + bx + 1 does not share any points with the positive part of the x-axis.'

''

'74 Parabolic y=3 x ^{2}+\\frac{2}{3}'

'Let m be a constant. The parabola y=f(x) passes through the origin, and the slope of the tangent at point (x, f(x)) is 2x + m. Let S be the area of the region enclosed by the parabolas y=f(x) and y=-x^{2}+4x+5. Find the minimum value of S.'

'In both cases, x = 0 is a minimum.'

'Find the x-coordinate of the intersection points between curve C and the x-axis. Also, find the x-coordinate of the intersection points between curve C and the curve y=-x^{2}+2x.'

'Find the portion of the parabola y=2x^{2}-3x where x<0 and 2<x'

'Given (1) maximum value is -2 at x = 5/2, (2) maximum value is 3 at x = 5 and minimum value is 0 at x = 1, (3) maximum value is 0 at x = 1/3 and 27, minimum value is -4 at x = 3.'

'Let D be the region represented by the system of inequalities x²+(y-4)²≥4 and y≥-2/3x². Find the line with y-intercept between 0 and 2 that is contained in D and has the maximum slope.'

''

'110 (2) Parabola y=3x^2-6x+2'

'101 (2) \ y=-x \\pm \\sqrt{2} \'

'Plot the region represented by the following inequalities.'

'When real numbers x, y satisfy x²+y²=2, find the maximum and minimum values of 2x+y. Also, determine the values of x and y in that case.'

'Please verify the symmetry of the following function: f(x) = x^2 + 3x + 2'

'(2) Let m be a constant. Find the number of intersection points between the parabola y^2=-8x and the line x+my=2.'

'The slope of the normal to the parabola C at the point (x₁, y₁) on C (x₁ ≠ 0, y₁ ≠ 0) is -1 / (2 / y₁) = -y₁ / 2. Let y₁ / 2 = m, then y₁² = 4x₁. Thus, x₁ = m². Therefore, the equation of the required normal is y = mx - m³ - 2m.'

'Let the parabola $y^{2}=4 x$ be denoted as curve C.\n(1) Find the equation of the normal to curve $C$ with slope $m$.\n(2) How many normals can be drawn from the point $(a, 0)$ on the x-axis to curve $C$? Where $a \neq 0$.'

'(3) has a local minimum at x=-2 of -1/3 and a local maximum at x=0 of 1'

'(3) The domain is 1-4 x^{2} \\geqq 0 with \\quad-\\frac{1}{2} \\leqq x \\leqq \\frac{1}{2}'

'(1) x=-a ± √(a²+1)'

'Practice whether the following pair of quadratic curves and lines have any intersection points. If they do intersect, determine whether it is a point of intersection or tangency, and find the coordinates of that point.'

'For f(x) = 2 sqrt{x} and the interval [1,4], find the value of c that satisfies the conditions of the Mean Value Theorem.'

'Find the equation of the tangent line drawn from the given point to the following quadratic curve: (a) \ x^{2}-4y^{2}=4 \, point \\( (2,3) \\)'

'When real numbers x, y satisfy the two inequalities x^2 + 9y^2 ≤ 9 and y ≥ x, find the maximum and minimum values of x + 3y.'

'Determine the curve drawn by the vertex of a parabola.'

'The functions f(x) and g(x) are continuous on the interval [a, b], with the maximum value of f(x) greater than the maximum value of g(x) and the minimum value of f(x) less than the minimum value of g(x). Show that the equation f(x) = g(x) has a solution in the range a ≤ x ≤ b.'

'Using the formulas of basic concepts from the previous page, find the equation of the tangent line at the given points on the following quadratic curves.'

'Please solve basic problems involving equations of tangent and normal lines. For example, find the equation of the tangent line at point (1, 1) on the curve y = x^2.'

'Find the values of the constants a and b when the range of the function y = √(2x + 4) (a ≤ x ≤ b) is 1 ≤ y ≤ 3.'

'Maximum value is 18 when x=4, y=0'

'Find the equation of the tangent line drawn from the given point to the following parabola: (i) \ y^{2}=8x \, point \\( (3,5) \\)'

'Find the equations of the lines that are tangent to both curves y=-x^{2}, y=\\frac{1}{x}.'

'Find the area of the curve represented by x=2t+t^2, y=t+2t^2 (-2 ≤ t ≤ 0) and enclosed by the y-axis, by parameter t.'

'Find the equation of the tangent line drawn from point A(1,4) to the hyperbola 4 x^{2}-y^{2}=4. Also, find the coordinates of the point of tangency.'

'Find the foci of the hyperbola $\\frac{x²}{a²} - \\frac{y²}{b²} = 1 (a > 0, b > 0)$.'

"Chapter 4 Applications of Differentiation\n117\nSince 2x^2+x+1=2(x+\\frac{1}{4})^2+\\frac{7}{8)>0, let y'=0, then x=1\nThe table of increase and decrease of y is as follows.\n\nQuestion: Find the point where y takes a maximum value of 1."

'Maximum value of 1 at x=0, minimum value of 17/9 at x=4'

'Find the range of values of the constant k for which the parabola y=x^2+k intersects the ellipse x^2+4y^2=4 at four distinct points.'

'Tangent of a Second Degree Curve\nThe equation of the tangent at a point \\( \\left(\oldsymbol{x}_{1}, \oldsymbol{y}_{1}\\right) \\) on the curve \ p \\neq 0, a>0, \\quad b>0 \\n1 Parabola\n\\[ \egin{array}{lll} y^{2}=4 p x \\\\ x^{2}=4 p y \\end{array} \\quad \\longrightarrow \\quad \egin{array}{l} y_{1} y=2 p\\left(x+x_{1}\\right) \\\\ x_{1} x=2 p\\left(y+y_{1}\\right) \\end{array} \\]\n2 Ellipse \ \\frac{x^{2}}{a^{2}}+\\frac{y^{2}}{b^{2}}=1 \\quad \\longrightarrow \\quad \\frac{x_{1} x}{a^{2}}+\\frac{y_{1} y}{b^{2}}=1 \\n3 Hyperbola \ \\quad \\frac{x^{2}}{a^{2}}-\\frac{y^{2}}{b^{2}}= \\pm 1 \\quad \\longrightarrow \\quad \\frac{x_{1} x}{a^{2}}-\\frac{y_{1} y}{b^{2}}= \\pm 1 \ (same sign sequence)'

'Explain the definition and properties of a parabola.'

'Find the number of intersection points between the following 2 curves and a line.'

'Find the equation of the tangent line when the point (x₁, y₁) lies on the curve of the parabola y²=4px.'

'Second degree curve and eccentricity e\nEllipses, hyperbolas, and parabolas can also be defined as the locus of points where the ratio of the distance from a fixed point F and a fixed line ℓ is constant, just like the parabola. That is, when a perpendicular PH is drawn from point P to ℓ, such that PF:PH=e:1 (e is a positive constant), the locus of points P satisfying this condition is a second degree curve with F as one focus, ℓ as the directrix, and e as the eccentricity. In this case, the second degree curve is classified according to the value of e as follows.\nWhen 0<e<1 Ellipse When e=1 Parabola When e>1 Hyperbola'

'Find the equation of the tangent lines drawn from the given points to the following quadratic curves.'

'Solve by considering the geometric meaning of the equation'

'Find the equation of the tangent line at the given point on the following curves.'

'What kind of shape does the following equation represent?'

'Consider the following equation:\n2. Equation: y^2 + x - 4y + 8 = 0'

'Draw the graphs of the following two quadratic functions and find their axes and vertices.'

'Determine the value of the constant a for which the graph of the quadratic function y=x^{2}+a x+a is tangent to the line y=x+1. Also, find the coordinates of the point of tangency.'

'The parabola y=4x^{2}+ax+b passes through the point (1,1) and touches the x-axis. Find all the combinations of a and b that satisfy this condition.'

'Mathematics I\n(2)\nThis function reaches its minimum value when x = -1, and that value is c - 5.\nThe condition for the minimum value to be -10 is c - 5 = -10.\nSolving this, we get c = -5.'

'Find the maximum and minimum values of the following quadratic functions.'

'Do the graphs of these two quadratic functions have any points in common? If so, find the coordinates of those points.'

'Determine the value of the constant c such that the minimum value of the function y=-x^2+6x+c (1≤x≤4) is 1. Also, find the maximum value at that point.'

'In mathematics, exercise 87, book 66, page 143, let f(x) = x^2-ax+a+8. The minimum value of f(x) for 0 ≤ x ≤ 5 should be greater than or equal to 0. Transforming f(x) gives'

'Given the above, when a<-\\frac{2}{3}, m(a)=2 a^{2}+3 a+2; when -\\frac{2}{3} \\leqq a \\leqq \\frac{2}{3}, m(a)=-\\frac{a^{2}}{4}+1; when \\frac{2}{3}<a, m(a)=2 a^{2}-3 a+2'

'Find the quadratic function that satisfies the following conditions: (1) The vertex is on the x-axis and passes through the points (0,4), (-4,36). (2) It is a parallel translation of the parabola y=2x^2 passing through point (2,3) and with the vertex on the line y=6x-5.'

'When the quadratic equation $x^{2}-ax+2a=0$ has two distinct real solutions, and only one of them is within the range $-1<x<1$, find the range of values for the constant $a$.'

'From the exercise problems of Chapter 3 on quadratic functions, please find the maximum and minimum values of the quadratic function graph.'

'Important example 93 Coefficient is a trigonometric equation (2) Quadratic equation in \ x \ \\( x^{2}-(\\cos \\theta) x+\\cos \\theta=0\\left(0^{\\circ} \\leqq \\theta \\leqq 180^{\\circ}\\right) \\) has two distinct real number solutions, the range of values of \ \\theta \ such that both solutions are contained in the interval \ -1<x<2 \. [Akita University] <Example 75'

'Find the quadratic functions that have curves with symmetrical movements about the following: (1) x-axis (2) y-axis (3) origin.'

'From the graph, it can be seen that when x=1, the minimum value of y=3(x-1)^{2}+2 is 2, and there is no maximum value.'

'When a < 0, the minimum is at x = 0 on the right graph. The minimum value is f(0) = 5. For 0 ≤ a ≤ 4, the minimum is at x = a on the right graph. The minimum value is f(a) = -3a^2 + 5.'

'Find the parabola y=2(x-p)^{2}-p+3.'

'The given function is a quadratic function, so a ≠ 0.'

'Considering a circle with diameter AC, let the radius be x, then the following points can be made.'

'Answer the following question 45.'

'Example 46 The length of the segment that a parabola cuts from the x-axis is a constant k. Let the parabola y=x^{2}-kx-18 cut a segment of length l from the x-axis.'

'Find the quadratic function y=a(x-p)^2+1.'

'(1) Graph of y=2x^2+3.\\n(2) Graph of y=2(x-1)^2.\\n(3) Graph of y=2(x+1)^2-4.'

'By symmetrically moving a point P(x, y) on the parabola C_{1} with respect to the line y=1 to a new point P^{\\prime}(x^{\\prime}, y^{\\prime}), where x^{\\prime}=x and \\frac{y+y^{\\prime}}{2}=1. Therefore, x=x^{\\prime}, y=2-y^{\\prime}, substituting this into y=ax^{2}+bx+4 yields 2-y^{\\prime}=ax^{\\prime 2}+bx^{\\prime}+4, so the equation of the parabola C_{2} is 2-y=ax^{2}+bx+4. Similarly, by finding the equation of the parabola C_{3} as 2-y=a(2-x)^{2}+b(2-x)+4. Since C_{2} and C_{3} pass through the points (-2,-10) and (3,-2) respectively, a system of equations for a and b is obtained, and solving it gives the values of a and b.'

'Find the coordinates of the intersection points of the following parabola and line.'

'When the axis x = a is within the range 0 ≤ x ≤ 2, in other words, when 0 ≤ a ≤ 2, it is clear from the graph on the right that x = a yields the minimum value. The minimum value is\n \\[ f(a) = -a^{2} - 4a\\]'

'Find the maximum and minimum values of x^{2}+y^{2}-6x, and the corresponding values of x and y when real numbers x, y satisfy 2x^{2}+y^{2}=8.'

'How are the graphs of these two quadratic functions shifted from the function y=2x^{2}?'

'Find a quadratic function that satisfies the following conditions.'

'When real numbers x, y satisfy x^2 + (y-1)^2 = 5, find the maximum and minimum values of 2x-y, and the values of x, y at that time.'

'Find the coordinates of the vertex for the equation y=x^{2}-3x+6 [y=(x-3/2)^{2}+15/4].'

'Practice whether the following parabolas and lines have points of intersection. If they do, find their coordinates.'

'Find the point of intersection of the following system of equations.'

'Determine the value of the constant c that satisfies the following conditions: (1) The maximum value of the function y=x^{2}-10x+c (3 ≤ x ≤ 8) is 10. (2) The minimum value of the function y=-x^{2}+4x+c (-1 ≤ x ≤ 1) is -10. First, complete the square and convert to standard form. Although an accurate graph is not drawn, it should be described in a way that shows the position of the axis and the relationship with the endpoints of the interval.'

'Find the equation of the parabola obtained by moving the parabola y=x^{2}-4 x 2 units in the x-axis direction and -1 unit in the y-axis direction.'

'Find a quadratic function that satisfies the following conditions.'

'Example: 1. Find the coordinates of the vertex and axis of the following quadratic function: y = 2x^2 - 4x + 1 2. Find the maximum and minimum values of the following quadratic function: y = -3x^2 + 6x - 2'

'What is the equation of the parabola obtained by translating the parabola y=-2 x^{2}+4 x-4 in the x-direction by -3 and in the y-direction by 1?'

For the quadratic function $y=x^{2}-2 k x+k^{2}-k+3$, answer the following question: (2) When it is tangent to the $x$ axis, find the value of the constant $k$ and the coordinates of the tangent point.

The graph passes through the points (1,3), (2,5), and (3,9). The quadratic function you are looking for is $y = a x^{2} + b x + c$.

Basics 76 | Determining a quadratic function from maximum and minimum conditions

Consider the parabola $y = x^{2} + 2x - 1 \cdots \cdots$ (1) with vertex $\mathrm{P}$. Answer the following questions: (1) Find the coordinates of the point $\mathrm{Q}$ which is symmetric to $\mathrm{P}$ with respect to the $x$-axis. (2) Find the equation of the parabola which is symmetric to this parabola with respect to the $x$-axis.

Draw the graph of the quadratic function oldsymbol{y}=a x^{2}+b x+c (2)

For the graph of the quadratic function \( y=x^{2}+2(k-1)x+k^{2}-3 \), answer the following question: (2) When it is tangent to the x-axis, find the value of the constant $k$ and the coordinates of the tangent point.

Find a quadratic function that passes through the points (1,3), (2,5), and (3,9).

We have learned about the graph of the quadratic function $y=a x^{2}$. Here, we will learn about the graph of $y=a x^{2}+b x+c$, which is related to $y=a x^{2}$. For this purpose, understanding the graph of the quadratic function in the form \( y=a(x-p)^{2}+q \) is important, so let's take a closer look.

Find the quadratic functions that satisfy the following conditions. (1) A curve obtained by translating the parabola $y=-x^{2}-2x$ that passes through the points (-1, -2, 2, 1). (2) When translated by 2 units in the $x$-direction and -3 units in the $y$-direction, it passes through the points (1, 2), (2, -2), and (3, -4).

The relationship between the graph of a quadratic function and the x-axis Basic 90 The number of intersection points between the graph of a quadratic function and the axis

Find the coordinates of the intersection points of the quadratic function $y=x^{2}-6x+6$ with the x-axis.

Given real numbers a and b, find the values of the constants a and b when the vertices of the parabolas represented by the quadratic functions y=4x^{2}-8x+5 and y=-2(x+a)^{2}+b coincide.

Draw the graph of the quadratic function $y=ax^{2}+bx+c$ (1)

Find the quadratic functions that meet the following conditions: (1) The curve obtained by translating the parabola $y=-2 x^{2}$ such that its vertex is at the point (-3,1). (2) The curve obtained by translating the parabola $y=x^{2}+2 x-3$ such that it passes through the points \( (-1,6),(3,2) \).

Find the quadratic functions that satisfy the following conditions. (1) Takes a maximum value of 1 at $x=3$, and $y=-1$ when $x=5$. (2) Takes a minimum value at $x=-2$, and the graph passes through the points \( (-1,2) \) and \( (0,11) \).

Standard 73 | Determine coefficients from the maximum value of the quadratic function

Basic 72 | Maximum and Minimum Values of a Quadratic Function (2)

Find the quadratic function that has a minimum at oldsymbol{x}=1 and passes through the points \( (0,3) \) and \( (3,6) \).

Draw the graph of the quadratic function \( oldsymbol{y}=a(x-p)^{2}+q \)

Quadratic function y=α(x-p)²+q maximum and minimum values ① When the domain is not restricted (when the domain is the entire set of real numbers) There are two cases depending on the sign of a. When a>0 The minimum value of q is obtained at x=p There is no maximum value When a<0 The maximum value of q is obtained at x=p There is no minimum value

Find the number of intersection points between the graphs of the following quadratic functions and the $x$-axis. (1) $y=3 x^{2}+x-2$ (2) $y=-5 x^{2}+3 x-1$ (3) $y=2 x^{2}-16 x+32$

Determine the value of the constant $c$ such that the maximum value of the function \( f(x) = x^2 - 10x + c (3 \leqq x \leqq 8)\) is 10.

If the vertex of the parabola $y = x^{2} + a x - 2$ lies on the line $y = 2 x - 1$, find the value of the constant $a$.

For the quadratic function \( y=(x-3)^{2}+1 \) 1) $1 \leqq x \leqq 4$ 2) $2 \leqq x \leqq 5$ 3) $4 \leqq x \leqq 5$ 4) $2 \leqq x$

Find the quadratic function that has its vertex at the point (2,1) and passes through the point (4,9).

Find the coordinates of the points where the graph of each quadratic equation intersects the x-axis. (1) y=x^2-6x-4 (2) y=-4x^2+4x-1

Let's organize the steps for drawing the graph of the quadratic function $y=ax^{2}+bx+c$. Follow these steps: (1) Complete the square. (2) Write it in the form \( y=a(x-p)^{2}+q \). (3) Read the features of the graph. (4) If $a > 0$ it is concave up (opens upward), if $a < 0$ it is concave down (opens downward). (5) Determine the vertex coordinates \( (p,q) \). (6) The axis is the line $x=p$. (7) Actually draw the graph of $y=x^{2}-6x+10$.

Graph of the quadratic function and its relationship with the x-axis Basics 91 Conditions for the graph of the quadratic function to share points with the axis

Determine the maximum and minimum values, if they exist, for the following quadratic functions: (1) $y=x^{2}-6 x+3$ (2) $y=-2 x^{2}+8 x-3$

Basic 71 | Maximum and Minimum Values of Quadratic Functions (1)

For the function $y=x^{2}+2 x-1$, find the maximum and minimum values (if they exist) within the following ranges. (1) $-3 \leqq x \leqq 0$ (2) $-2<x<1$ (3) $0 \leqq x \leqq 2$

Find the maximum and minimum values of the following functions if they exist. (1) \( y=x^{2}-2x-3(-4 \leqq x \leqq 0) \) (2) \( y=2x^{2}-4x-6 \quad(0 \leqq x \leqq 3) \) (3) \( y=-x^{2}-4x+1(0 \leqq x \leqq 2) \) (4) \( y=x^{2}-4x+3(0<x<3) \)

Draw the graphs of the following quadratic functions and find their vertices and axes of symmetry. (1) \( y=3(x+1)^{2}-2 \) (2) \( y=-rac{1}{2}(x-1)^{2}+2 \)

Find the number of intersection points between the graphs of the following quadratic functions and the x-axis. (1) $y=x^{2}-x-3$ (2) $y=4 x^{2}+12 x+9$ (3) $y=-x^{2}+2 x-3$

Find the quadratic function that passes through the following points: (1) (-1,7),(0,-2),(1,-5) (2) (-1,0),(3,0),(1,8)

Find the maximum and minimum values of the function \( f(x)=(x-2)^{2} \) over the domain $0 \leqq x \leqq a$. Assume $a$ is a positive constant. (1) $0<a<2$ (2) $2 \leqq a<4$ (3) $a=4$ (4) $4<a$

For the quadratic function \( f(x) = -x^2 + 6x - 7 \), the maximum value in the interval $a \leq x \leq a+1$ is \( M(a) \).

Find the maximum and minimum values of the function \( f(x)=-(x-3)^{2} \) with the domain $1 \leqq x \leqq a$, for the following cases. Here, $a$ is a constant that satisfies $a>1$. (1) $1<a<3$ (2) $3 \leqq a<5$ (3) $a=5$ (4) $5<a$

Determine a quadratic function from the conditions of passing through 3 points

Development 81 | Maximum and minimum of quadratic functions and word problems (2)

Basics 75 | Determining quadratic functions from conditions such as vertices

4. The graph of the quadratic function $y=6 x^{2}+11 x-10$ is translated $a$ units in the $x$-axis direction and $b$ units in the $y$-axis direction to obtain the graph $F$. When $F$ passes through the origin (0,0), answer the following questions: (1) Express $b$ in terms of $a$. (2) When the quadratic function \( f(x) \) representing $F$ takes the same value at $x=-2$ and $x=3$, find the value of $a$ and the maximum and minimum values of \( f(x) \) in the interval $-2 \leqq x \leqq 3$.

Regarding the graph of the quadratic function \( y=x^{2}+2(k-1) x+k^{2}-3 \), answer the following questions. (1) When it does not intersect with the $x$ axis, find the range of the constant $k$.

Find a quadratic function such that its graph forms a parabola as follows. (1) A parabola with a vertex at point (-1,3) and passing through point (1,7) (2) A parabola with its axis as the line $x=1$ and passing through the points (3,-6) and (0,-3)

Find the coordinates of the intersection points between the graph of the following quadratic functions and the x-axis. (1) y=x^2+7x-18 (2) y=3x^2+8x+2 (3) y=x^2-6x+2 (4) y=-6x^2-5x+6 (5) y=9x^2-24x+16

Find the equation of the tangent line to the parabola $y^{2}=8 x$ at the point \( (2,4) \).

Find the equation of the tangent line at the point \( (\sqrt{2}, 1) \) on the ellipse rac{x^{2}}{4}+rac{y^{2}}{2}=1 .

111 x=p t^{2}, y=2 p t \quad(p ≠ 0)

Express the coordinates of the vertex of the parabola $y=x^{2}-2tx+2$ (A) in terms of t.

109 Parabola $y=x^{2}-5 x+3$

Example 1. The graph of the equation $9 x^{2}+4 y^{2}+54 x-16 y+61=0$.

Polar equation of a conic section (1)

Find the equation of the parabola that satisfies the following conditions: (1) Vertex at the origin, focus on the y-axis, and the directrix passes through the point (3,2) (2) Vertex at the origin, focus on the x-axis, and passes through the point (-2, √6)

Polar equation of a conic section (2)

Find the equation of the tangent line at the point \( \left(-1, rac{1}{5} ight) \) on the ellipse \( rac{(x+4)^{2}}{25}+rac{(y+3)^{2}}{16}=1 \).

The equation of the tangent line at a point on a quadratic curve

Find the equations of the parabolas that satisfy the following conditions. (1) Vertex at the origin, focus on the y-axis, and the directrix passes through the point (3,2). (2) Vertex at the origin, focus on the x-axis, and passes through the point (-2, √6).

Let's further investigate what type of shape is represented by the equation $a x^{2}+b y^{2}+c x+d y+e=0$.