Precalculus
Precalculus
9th Edition
ISBN: 9780321716835
Author: Michael Sullivan
Publisher: Addison Wesley
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Chapter 3.3, Problem 51AYU

In Problems 49-54, determine the quadratic function whose graph is given.

Chapter 3.3, Problem 51AYU, In Problems 49-54, determine the quadratic function whose graph is given.

Expert Solution & Answer
Check Mark
To determine

To calculate: The quadratic function using the given graphs.

Answer to Problem 51AYU

The equation of the given graph is f(x) = 2 x 2  - 4x - 1 .

Explanation of Solution

Given:

The given graph is

Precalculus, Chapter 3.3, Problem 51AYU

Formula Used:

The general form of a quadratic equation is f(x) = a (x - h) 2  + k, a  0 .

This general equation can also be written as f(x) = a x 2  + bx + c, a  0 , where

h = - b 2a and k = f( - b 2a )

If a > 0 , the graph opens upwards.

If a < 0 , the graph opens downwards.

The vertex of the above function is (h, k) = ( -b 2a , f( -b 2a ) ) .

The axis of symmetry will be x =  -b 2a .

We can find the y-intercept by equating the equation at x = 0 .

We can find the x-intercept by equation the equation at y = 0 .

1. If b 2  - 4ac = 0 then the vertex is the x-intercept .

2. If b 2  - 4ac < 0 then the graph has no x-intercept .

3. If b 2  - 4ac > 0 then the vertex is the x-intercept .

Calculation:

Let us consider the quadratic equation of the given graph to be

f(x) = a x 2  + bx + c, a  0    -----(1)

From the given graph, we can see that the y-intercept of the given function is (0, -1) .

We know that the y-intercept is the value of the function at x = 0 , thus, we have

f(0) = -1    -----(2)

At x = 0 , (1) becomes

f(0) = c    -----(3)

Thus, using (2) and (3), we get the value of c = -1 .

Thus, equation (1) becomes

f(x) = a x 2  + bx - 1    -----(4)

Now, we need to find a and b .

We know that a point in a graph is written as ( x, f(x) ) .

The given graph has 2 points (-1, 5) and (3, 5) .

Consider the point (-1, 5) .

Here, we have x= -1 and f(x) = 5 .

Therefore, substituting it in (4), we get

5 = a ( 1 ) 2  + b(1)  1  a  b = 6     -----(5)

Now, consider the point (3, 5) .

Here, we have x = 3 and f(x) = 5 .

Therefore, substituting it in (4), we get

5 = a ( 3 ) 2  + b(3)  1  9a + 3b = 6  3a + b = 2     -----(6)

In order to find the values of a and b , we have to solve the equations (5) and (6).

Therefore, we have

a - b = 6    -----(5) And

3a + b = 2    -----(6).

Now, on adding (5) and (6), we get

4a = 8  a = 2

On substituting the above value in (5), we get

2 - b = 6  2 - 6 = b  b = -4

Therefore, the equation of the given graph is

f(x) = 2 x 2  - 4x - 1

Chapter 3 Solutions

Precalculus

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