In the xy-plane, the graph of the quadratic function f(x) = -x^2 + bx + c, where b and c a… — SAT

In the $xy$ -plane, the graph of the quadratic function $f(x) = -x^2 + bx + c$ , where $b$ and $c$ are constants, has its vertex at $(4, 9)$ . What is the positive difference between the two $x$ -intercepts of the graph?

Answer: 6

Answer

The correct answer is 6.

The quadratic function can be represented in vertex form as

f(x) = -(x - 4)^2 + 9

since the leading coefficient is

-1

and the vertex is at

(4, 9)

. Setting the function equal to zero to find the

x

-intercepts gives

(x - 4)^2 = 9

, which yields

x = 7

and

x = 1

. The positive difference between these intercepts is

7 - 1 = 6

Step-by-Step Solution

Write the quadratic function in vertex form using the given vertex

(4, 9)

and the leading coefficient.

f(x) = -(x - 4)^2 + 9

The vertex form of a quadratic function is

f(x) = a(x - h)^2 + k

, where

(h, k)

is the vertex. Since the coefficient of

x^2

-1

, we have

a = -1

h = 4

, and

k = 9

Set

f(x) = 0

to find the

x

-intercepts.

(x - 4)^2 = 9

The

x

-intercepts of a graph are the points where

f(x) = 0

Solve for

x

x = 7

and

x = 1

Taking the square root of both sides gives

x - 4 = \pm 3

, which results in

x = 7

and

x = 1

Find the positive difference between the two

x

-intercepts.

Subtract the smaller

x

-intercept from the larger

x

-intercept:

7 - 1 = 6

Key Concept

Finding the intercepts of a quadratic function using its vertex form

Alternative Method

Alternatively, expand the vertex form

f(x) = -(x - 4)^2 + 9

to get

f(x) = -(x^2 - 8x + 16) + 9 = -x^2 + 8x - 7

. Factoring this expression gives

f(x) = -(x - 7)(x - 1)

. The roots are

x = 7

and

x = 1

, and their difference is

7 - 1 = 6

Estimated Time:1m 30s

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Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Alternative Method

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