# 1.How is the maximum or minimum point of the original quadratic function related to the maximum or minimum point of its reciprocal function?

58,090 results

1. ### math

1.How is the maximum or minimum point of the original quadratic function related to the maximum or minimum point of its reciprocal function? 2.How can you use the leading coefficient of the original quadratic function to tell you whether its reciprocal will have a maximum or ...
2. ### Algebra- Maximum and minimum.

I don't understand how to find the maximum and minimum. Determine whether the given quadratic function has a minimum value or maximum value. Then find the coordinates of the minimum or maximum point. 15.f(x) = -x2 + 2x - 4 20.f(x) = 2x2 - 4x 19.f(x) = x2 - 2x + 1 30.f(x) = -...

1.Give a linear function with its zero at x=a, what is the equation of the vertical asymptote of its reciprocal function? 2.Give a quadratic function with its zeros at x=a and x=b, what are the equations of the vertical asymptotes of its reciprocal function? 3.Once you know ...
4. ### ALGEBRA

Determine whether the quadratic function has a minimum or maximum value.Then find the coordinates of the minimum or maximum point. f(x)=2x^2-4x
5. ### algebra

determine the given quadratic function has a minimum valueor maximum vale. Then find the coordinates of the minimum or maximum point. f(x)=x^2=2x-9
6. ### Maximum and Minimum

Its been awhile since I have done compleating the square, can you please help me? Determine whether the given quadratic function has a minimum value or maximum value. Then find the coordinates of the minimum or maximum point. 19.f(x) = x2 - 2x + 1 And how you would complete ...
7. ### math

Determine whether the given quadratic function has a minimum value or maximum value. Then find the coordinates of the minimum or maximum point.f(x) = x2 + 2x - 9
8. ### Algebra

Find the vertex, the line of symmetry, the maximum or minimum of the quadratic function, and graph the function. f(x)=x^2-4x-3 What is the vertex? (Type an ordered pair) What is the equation of the line of symmetry? x= What is the maximum/minimum of f(x)? Is the value, f(2)= -...
9. ### Algebra2

Complete parts a – c for each quadratic function: a. Find the y-intercept, the equation of the axis of symmetry and the x- coordinate of the vertex. b. Make a table of values that includes the vertex. c. Use this information to graph the function. 1. f(x) = 2x^2 2. f(x) = x^...
10. ### calculus

there are no examples of this type of problem in my book so if you could help walk me through it - that would be extremely helpful. thanks ahead of time. Find the extreme values of the function on the interval and where they occur. 4) F(x)=³√(x); -3</=x</=64 A. ...
11. ### Calculus

Give an example of a function with a critical point that is neither a maximum nor a minimum. Specify the relevant point(s). This is what I have and this is what I got wrong... A function with a critical point that is neither a maximum nor a minimum would be f(x)=x^2 if F(x) = ...
12. ### Algebra 2

The vertices of a feasible region are A(1,2), B(5,2), C(1,40. Write a function that satisfies each equation. a) A is the maximum and B is the minimum. b) C is the maximum and B is the minimum. c) B is the maximum and A is the minimum.
13. ### Algebra 2

The vertices of a feasible region are A(1,2), B(5,2), C(1,4). Write a function that satisfies each equation. a) A is the maximum and B is the minimum. b) C is the maximum and B is the minimum. c) B is the maximum and A is the minimum.
14. ### precalculus

Determine the maximum and minimum number of turning points for the function h(x) = -2x^4 - 8x^3 + 5x -6. Maximum:3 Minimum:1 Is this a valid reason: A quartic polynomial function has a 3 Turning points. The turning point is always 1 less than the degree. For example degree 4=3...
15. ### Math

Determine the maximum and minimum number of turning points for the function h(x) = -2x^4 - 8x^3 + 5x -6. Maximum:3 Minimum:1 Is this a valid reason: A quartic polynomial function has a 3 Turning points. The turning point is always 1 less than the degree. For example degree 4=3...
16. ### Math

Can someone help me with these questions? 1.What are the maximum and minimum values for Y=28(1.21)^x on the interval 0<x<12? a) 275.8 and 0 b) 275.8 and 28 c) 249.7 and 28 d) 188.4 and 28 2. For what value of x does the maximum or minimum of f(x)= x^2 +8x + 12 occur? Is ...
17. ### Calculus (pleas help!!!)

Find the absolute maximum and absolute minimum values of the function f(x)=(x−2)(x−5)^3+11 on each of the indicated intervals. Enter -1000 for any absolute extrema that does not exist. (A) Interval = [1,4] Absolute maximum = Absolute minimum = (B) Interval = [1,8...
18. ### Calculus (pleas help!!!)

Find the absolute maximum and absolute minimum values of the function f(x)=(x−2)(x−5)^3+11 on each of the indicated intervals. Enter -1000 for any absolute extrema that does not exist. (A) Interval = [1,4] Absolute maximum = Absolute minimum = (B) Interval = [1,8...
19. ### Please check my Calculus

1. Which of the following describes the behavior of f(x)=x^3-x A. Relative maximum: x=0 B. Relative maximum: x=(1/sqrt(3)); Relative minimum: x=(-1/sqrt(3)) C. Relative maximum: x=(-1/sqrt(3)); Relative minimum: x=(1/sqrt(3)) D. Relative minimum: x=0 E. none of these I got C. ...
20. ### calculus

Find the absolute extrema of the function (if any exist) on each interval. (If an answer does not exist, enter DNE.) f(x) = square root of (25 − x^2) (a) [−5, 5] minimum (x, y) = (smaller x-value) (x, y) = (larger x-value) maximum (x, y) = (b) [−5, 0) minimum...
21. ### Algebra2

Complete parts a – c for each quadratic function. a. Find the y-intercept, the equation of the axis of symmetry and the x-coordinate of the vertex. b. Make a table of values that includes the vertex. c. Use this information to graph the function. 1. f(x) = 2x2 2. f(x) = x2...
22. ### Calculus

Use analytical methods to find the exact global maximum and minimum values of the function f(x)=8x-ln(4x) for x >0. If a global maximum or minimum does not exist, enter the word NONE. For the global maximum at x=none, But for the Global minimum at x=? Is the Global minimum ...
23. ### Pre-calculus

Find the critical point of the function y=-2(x-1)^2-3. Then, determine whether the point represents a maximum, a minimum, or a point of inflection.
24. ### Pre-calculus

Find the critical point of the function y=-2(x-1)^2-3. Then, determine whether the point represents a maximum, a minimum, or a point of inflection.
25. ### Algebra 2

What is the relative maximum and minimum of the function? f(x) = 2x^3 + x^2 -11x A - The realative maximum is at (-1.53,8.3) and the realative minimum is at (1.2,-12.01) B - The realative maximum is at (-1.53,12.01)and the realative minimum is at (1.2, -8.3) C - The realative ...
26. ### MATHS

The quadratic function f(x) has roots at x=4 and x=-2 and has value of 2 at x=0. Does f(x) have a minimum or maximum? Find the value of the /minimum/maximum value of f(x).
27. ### Caluclus

Find the absolute maximum and absolute minimum of f on the interval (-1,2]: f(x)=(-x^3+x^2+3x+1)/(x+1) A. Maximum: (1, -2); Minimum: (-1, 2) B. Maximum: (1, -2); Minimum: None C. Maximum: None; Minimum: None D. Maximum: None; Minimum: (-1, 2) E. None of these I got to my ...
28. ### maths

Given p(x)=x^4+ax^3+bx^2+cx+d,such that x=0 is the only real root of p'(x)=0.If p(-1)<p(1),then in the interval [-1,1],which is maximum and minimum of p(-1) and p(1)?: a)p(-1) is minimum and p(1) is maximum. b)p(-1) is not minimum and p(1) is maximum. c)neither p(-1) is ...
29. ### maths

Given p(x)=x^4+ax^3+bx^2+cx+d,such that x=0 is the only real root of p'(x)=0.If p(-1)<p(1),then in the interval [-1,1],which is maximum and minimum of p(-1) and p(1)?: a)p(-1) is minimum and p(1) is maximum. b)p(-1) is not minimum and p(1) is maximum. c)neither p(-1) is ...
30. ### Calculus

Find the absolute maximum and absolute minimum values of the function f(x)=x^3+6x^2-63x+4 on each of the indicated variables. Enter DNE for does not exist. (A) Interval = [-8,0] Absolute maximum = Absolute minimum = (B) Interval = [-5,4] Absolute maximum = Absolute minimum = (...
31. ### calculus

suppose f(x) is a function such that f(3)=5, f'(3)=0, and f''(3)=3. Is the point (3, 5) a relative maximum, minimum or inflection point?
32. ### math

Maximum or minimum values of the function f(x) = (1 – x)^2 e^x is: i need solution (a) Minimum value at x = –1; maximum value x = 1 (b) Minimum value at x = 1; maximum value x = –2 (c) Minimum value at x = 1; maximum value x = –1 (d) None of these
33. ### algebra

Find the line of symmetry, the maximum or minimum value of the quadratic function, and graph the function f(x)=2-x^2
34. ### algebra

Find the line of symmetry, the maximum or minimum value of the quadratic function, and graph the function f(x)=-2x^2+2x+9
35. ### algebra

find the vertex, the line of symmetry, the maximum or minimum value of the quadratic function, and graph the function f(x)=x^2-4x-5
36. ### calc

let f be function given by f(x)= Ln(x)/x for all x> 0. the dervative of f is given by f'(x)= (1 - Ln(x))/x squared. a) write equation for the line tangent to the graph of f at x=e squared b) Find the x-coordinate of the critical point of f. Determine wheter this point is a ...
37. ### Calculus

A differentiable function called f(x) achieves its maximum when x=0. Which of the following must then be true? 1. The function p(x) = xf(x) has a critical point when x = 0. 2. The function m(x) = (f(x))^2 has its maximum at x = 0. 3. The function j(x) = f(x^2) has its maximum ...
38. ### math

Indicate the correct answer for the function. If (x)= x^3 - 3x^2 + 3x + 7, then f(x) is: a) Maximum b) Minimum c) Neither maximum or minimum d) not possible
39. ### algebra(reiny)

Hey thanks for all your help but you kinda confused me on a few: 1)Determine whether f(x)=-5x^2-10x+6 has a maximum or minimum value and find that value A.minimum -1 B.maximum 11 C.maximum -1 D.minimum 11 and you said the function opens up, so there is a minimum. it occurs ...
40. ### calculus

I needed help with these FRQ in my APCalc course. Any help or walkthrough would be extremely helpful - thanks in advance. Let f be the function given by f(x)=3ln((x^2)+2)-2x with the domain [-2,4]. (a) Find the coordinate of each relative maximum point and each relative ...
41. ### math

I'm having some troubles with these. Thanks in advance. If a>0 find the minimum value. If a<0 find the maximum value. 1. y=-x²+2x+5 2. y=-2x²-3x+4 These are functions for parabolas, so f(x)=ax²+bx+c when a does not equal 0 In both your examples, the "a", which is the ...
42. ### ALGEBRA

Find the vertex, the line of symmetry, the maximum or minimum value of the quadratic function. f(x) = -2x^2 + 2x + 9 x coordinate is y coordinate is the equation of the line of symmetry is x = Maximum/minimum of f(x) is the value, f (1half) = 19 over 2 (fraction) is ? min or ...
43. ### math

Answer the following questions for the function f(x)=x(sqrtx2+25) defined on the interval [–5,4]. A. f(x) is concave down on the region to B. f(x) is concave up on the region to C. The inflection point for this function is at D. The minimum for this function occurs at E. The...
44. ### calculus

a function has a local maximum at x=-2 and x=6 and a local minimum at x=1. how do u find the concavity of this function and point of inflection??
45. ### Calculus

Find the values of x that give relative extrema for the function f(x)=3x^5-5x^3 A. Relative maximum: x= 1; Relative minimum: x=sqrt(5/3) B. Relative maximum: x=-1; Relative minimum: x=1 C. Relative maxima: x=+or- 1; Relative minimum: x=0 D. Relative maximum: x=0; Relative ...
46. ### calculus

a function has a local maximum at x=-2 and x=6 and a local minimum at x=1. how do u find the concavity of this function and point of inflection?? ***the only given info is the max and min points
47. ### math

Find the maximum and minimum value of the quadratic function g(x) = 2x^2 + 6x + 3

The derivative of a function is f'(x)=x(x+2)(x-5). Find the value of x at each point where f has a (a) local maximum, (b) local minimum, or (c) point of inflection
49. ### Math

The derivative of a function is f'(x)=x(x+2)(x-5). Find the value of x at each point where f has a (a) local maximum, (b) local minimum, or (c) point of inflection
50. ### Calculus

Let f(x) be a polynomial function such that f(-2)=5, f'(-2)=0, and f"(-2)=3. The point (-2,5) is a(n)____of the graph f. A. Relative maximum B. Relative minimum C. Intercept D. Point of Inflection E. None of these
51. ### Calculus

Let f(x) be a polynomial function such that f(-2)=5, f’(-2)=0 and f”(-2)=3. The point (-2, 5) is which of the following for the graph of f? A. relative maximum B. relative minimum C. intercept D. inflection point E. none of these
52. ### Absolute Extrema

Find the minimum and maximum values of the function f(x)= 2/3 sin pix on the interval [1,2] (domain). I got (1.5, -2/3) as the minimum and no maximum on the interval. Is it correct? Thank you.
53. ### algebra

find the maximum or minimum of the following quadratic function: y =x2 - 10x + 25
54. ### Calculus check

Let f(x) be a polynomial function such that f(3)=3, f'(3)=0 and f"(3)=-3. What is the point (3,3) on the graph y=f(x)? A. Relative maximum B. Relative minimum C. Intercept D. Inflection point E. None of these I got C
55. ### Math

Find the stationary points of function E(a,b) = a^4 - 2a^2 + 2b^2 and classify each of these as either a minimum, maximum or saddle point.
56. ### Minimum and maximum value

Let f(x) be a decreasing the function defined for x>or=0 and f(2a^2+a+1)<or= f(3a^2-4a+1) then find the minimum and maximum value of a plz help help help!!!show ur work
57. ### Calc

Let f be the function given by f(x)=2ln(x^2+3)-x with domain -3 is less than or equal to x which is less than or equal to 5 a) Find the x-coordinate of each relative maximum point and each relative minimum point of f. Justify your answer. b) Find the x-coordinate of each ...
58. ### Physics ( Calculus & Mecahnics )

A proton with mass m moves in one dimension.The potential-energy function is where and are positive constants. The proton is released from rest a x0 = a/b. (a) Show that can be written as U(x) = a --- [ ( x0/x )^2 - x0/x ] x0^2 Graph U(x). Calculate U(x0) and thereby locate ...
59. ### Math

Context: The function v(t) represents the velocity of a particle moving along a horizontal line at any time, t, greater than or equal to zero. If the velocity is positive, the particle moves to the right. If the velocity is negative, the particle is moving to the left. The ...
60. ### Calculus

Let f(x) be a polynomial function such that f(4)=-1, f’(4)=2 and f”(4)=0. If x<4, then f”(x)<0 and if x>4, then f”(x)>0. The point (-4, -1) is which of the following for the graph of f? A. relative maximum B. relative minimum C. critical value D. inflection...
61. ### math

Answer the following questions for the function f(x)=sin^2(x/5) defined on the interval .(-15.507923,3.82699075) Rememer that you can enter "pi" for as part of your answer. a.what is f(x) concave down on the region B. A global minimum for this function occurs at C. A local ...
62. ### math

Given an equation and after figuring out the equation's maximum or minimum. How would I explain in sentence how I know the point is a maximum or minimum?
63. ### AP CALC. AB

2. A cubic function is a polynomial of degree 3 and has the form y=mx^3+bx^2+cx+d; m≠0. What is the maximum quantity of local extreme values a given cubic function can have? a. 2 b. 1 c. 0 d. 3 Is it (a)?? 2. Let f(x)=xln(x). The minimum value attained by f is a) there ...
64. ### trigonometry

An object is attached by a string to the end of a spring. When the weight is released it starts oscillating vertically in a periodic way that can be modeled by a trigonometric function. The object's average height is −20 cm (measured from the top of the spring). It ...
65. ### Finite Mathematics

Find the maximum and minimum value of the objective function given. z = 21x + 12y maximum value z = minimum value z =
66. ### math

to find the cordinates of the maximum or minimum point of a quadratic relationship?
67. ### Math

When you're finding the sign of f' and f'' to find a local maximum/minimum, concavity and inflection point where do you substitute the tangent/critical points? In which function? f(x), f'(x) or f''(x)?
68. ### Calculus

Answer the following questions for the function f(x) = sin^2(x/3) defined on the interval [ -9.424778, 2.356194]. Rememer that you can enter pi for \pi as part of your answer. a.) f(x) is concave down on the interval . b.) A global minimum for this function occurs at . c.) A ...
69. ### Calculus

Find the absolute maximum value and the absolute minimum value, if any, of the function. (If an answer does not exist, enter DNE.) f(x) = −x2 + 2x + 3 on [3, 6] maximum minimum

For each function, the point given is the maximum or minimum. Use the difference quotient to verify that the slope of the tangent at this point is zero. a) f(x) = 0.5x^2 + 6x + 7.5; (-6, -10.5) Difference quotient is f(a + h) - f(a)/h m= f(a+h) - f(a)/h = f(-6+h) - f(-6)/h = 0...
71. ### algebra

given the following quadratic equation, determine if it has a maximum or a minimum value. Then find the maximum or minimum value f(x) - -11x^2 - 2x - 7
72. ### Calculus (Urgent Help)

Okay, I need major help! Can someone tell me if these statements are true or false ASAP please. Thank you. 1. If ƒ′(x) < 0 when x < c then ƒ(x) is decreasing when x < c. True 2. The function ƒ(x) = x^3 – 3x + 2 is increasing on the interval -1 < x < 1...
73. ### Calculus

f(x) = sin^2(x/2) defined on the interval [ -5.683185, 1.270796]. Remember that you can enter pi for \pi as part of your answer. a.) f(x) is concave down on the interval . b.) A global minimum for this function occurs at . c.) A local maximum for this function which is not a ...
74. ### maths

a)Express y = 2 (x - 1) (x - 5) in the form y = ax^2 + bx + c. b) From the above quadratic equation calculate the following: I) x and y intercepts II) Coordinate of the turning point III) Is the graph a maximum or minimum turning point?
75. ### how to sketch a graph of..

Local minimum and local maximum imply that the function approaches negative and positive infinite at opposite sides of the graph. Local minimum (1,1) and local maximum (3,3) means the slope of the function is 0 at these points. Thank you so much. So 1)when the local minima is...
76. ### calculus

If g is a differentiable function such that g(x) is less than 0 for all real numbers x and if f'(x)=(x2-4)g(x), which of the following is true? f has a relative maximum at x=-2 and a relative minimum at x=2, f has a relative minimum at x=-2 and has a relative maximum at x=2, f...
77. ### Calculus

Find the absolute maximum and absolute minimum of f on the interval (-4, -1]: f(x)=(x^3+8x^2+19x+12)/(x+4) A. Maximum: None; Minimum: (-2, -1) B. Maximum: (-4, 3); Minimum (-1, 0) C. Maximum: (-4, 3); Minimum: (-2, -1) D. Maximum: None; Minimum: (-1, 0) E. none of these
78. ### physics

The potential energy of a particle on the x-axis is given by U= 8xe ^-x ^2/19 where x is 1 meter and U is in Joules. Can you explain for me how to find the point on the x-axis for which the potential is a maximum or minimum and is this answer a point of maximum potential ...
79. ### Economics

A manufacturing firm produces output using a single plant. The relevant cost function is TC=45,000+8Q squared and the demand function is Q=100-0.02P a)What is the minimum level of AC b)What is the firms profit when AC is at a minimum? c)What is the firms maximum profit
80. ### differentiability

If g is a differentiable function such that g(x) is less than 0 for all real numbers x and if f prime of x equals (x2-4)*g(x), which of the following is true? A. f has a relative maximum at x=-2 and a relative minimum at x=2, B. f has a relative minimum at x=-2 and a relative ...
81. ### Algebra 2

3. Identify the vertex and the axis of symmetry of the graph of the function y = 3(x + 2)2 – 3. (1 point) A- vertex: (2, –3); axis of symmetry: x = 2 B- vertex: (–2, –3); axis of symmetry: x = –2 C- vertex: (2, 3); axis of symmetry: x = 2 D- vertex: (–2, 3); axis ...
82. ### math

find the vertex, the line of symmetry and the maximum or minimum value of f(x). graph the function. f(x)=-(x+6)^2-2 The vertex is The line of symmetry is x= The minimum/maximum value of f(x) is? The value of f(-6) =-2 is maximum or minimum? and the graph goes up or down. Thank...
83. ### 12th Calculus

use the derivative of the function y=f(x)to find the points at which f has a local maximum local minimum point of inflection y'=(x-1)^2(x-2)(x-4)
84. ### calc

The derivative of a function f is f'(x) = x^2 - 8/x: Then at x = 2, f has A. an infection point B. a relative maximum C. a vertical tangent D. a discontinuity E. a relative minimum how do you test for all of these?
85. ### Math

Choose the three true statements about the graph of the quadratic function y = x2 − 3x − 4. Options A) The graph is a parabola with a minimum point. B) The graph is a parabola with a maximum point. C) The point (2, 2) lies on the graph. D) The point (1, 6) lies on ...
86. ### Please check my Calculus

1. Find all points of inflection: f(x)=1/12x^4-2x^2+15 A. (2, 0) B. (2, 0), (-2, 0) C. (0, 15) D. (2, 25/3), (-2, 25/3) E. none of these I got D. I found the second derivative and equaled it to 0 and solved for x. I plugged the x values in to get my points. 2. Find the ...
87. ### algebra

Find the maximum or minimum of the following quadratic function: y = x2 - x - 56. A. -56 B. 56 C. -225/4 D. 225/4 d
88. ### Algebra

min/max point minimum or maximum f(x)=2x^2+8x-2 a=2 x=(-b)/(2a)=(-8)/(2)(2)=-4 x=-4 y=2(-4)^2+8(-4)-2 2(16)-32-2 32-32-2 y=-2 (-4,-2) axis of symmetry is x=-4 range[-2,oo] minimum value -2
89. ### John

min/max point minimum or maximum f(x)=2x^2+8x-2 a=2 x=(-b)/(2a)=(-8)/(2)(2)=-4 x=-4 y=2(-4)^2+8(-4)-2 2(16)-32-2 32-32-2 y=-2 (-4,-2) axis of symmetry is x=-4 range[-2,oo] minimum value -2
90. ### Pre-Calc

A quadratic function has its vertex at the point (-5,-2). The function passes through the point (-8,-10). Find the quadratic and linear coefficients and the constant term of the function.
91. ### math

A quadratic function has its vertex at the point (-9,3). The function passes through the point (2,-1). Find the quadratic and linear coefficients and the constant term of the function.
92. ### Algebra

Consider the quadratic function f(x) = – x^2 + 10x – 26. Determine whether there is a maximum or minimum value and find that value.

The function f(x)=4x+9x^-1 has one local minimum and one local maximum. Question: This function has a local minimum at x=______ with a value _______ and a local maximum at x=________ with value _______?
94. ### calculus

5.In a certain state the maximum speed permitted on freeways is 65 km/h and the minimum speed is 40 km/h. The fine for violating these limits is Rs.15 for every kilometer per hour above the maximum speed or below the minimum speed. Express the amount of the fine as a function ...
95. ### Calculus

Consider a differentiable function f having domain all positive real numbers, and for which it is known that f'(x)=(4-x)x^-3 for x>0. A. Find the x-coordinate of the critical point of f. Determine whether the point is a relative maximum, a relative minimum, or neither for ...
96. ### AP Calculus

Consider the function f(x) = 1/4x^4 - 5/3x^3 - 3/2x^2 + 15x - 3. A. Find the relative extrema for f(x); be sure to label each as a maximum or minimum. You do not need to find function values; just find the x-values. B. Determine the interval(s) where f(x) is increasing (if any...
97. ### Math: Calculus

Answer the following questions for the function f(x)=sin(x/4)^2 defined on the interval [-12.266371, 2.241593]. Rememer that you can enter pi for as part of your answer. a.)f(x) is concave down on the interval ____. b.) A global minimum for this function occurs at ____. c.) A ...
98. ### Math

Football Lesson 5.1 1. y= -0.035x^2 + 1.4x + 1, where x is the distance (in yards) and y is the height (in yards) a. Does the graph open up or down? b. Does the graph have a Maximum value or a minimum value? c. graph the quadratic function. (i'll graph it just write it in the ...
99. ### calculus

Try to find the value of a such that the function f(x)=asinx+ (1/3) sin3x gets its extremum points at the point x= (π / 3). Decide if it is maximum or minimum, and find it.
100. ### Cosine Math Question

Graph a cosine function whose amplitude is 4, period is π , midline is y=−3 , and y-intercept is (0, 1) . The first point must be on the midline and the second point must be a maximum or minimum value on the graph closest to the first point. I appreciate the help thanks