In the xy-plane, the graph of the quadratic function f, defined by f(x) = -(x - 4)^2 + c w… — SAT

In the $xy$ -plane, the graph of the quadratic function $f$ , defined by $f(x) = -(x - 4)^2 + c$ where $c$ is a constant, intersects the $x$ -axis at two points. If the distance between these two points is $10$ , what is the value of $c$ ?

Answer: 25

Answer

The quadratic function is defined by

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

, which is in the vertex form

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

. The vertex of this parabola is at

(4, c)

, and the vertical line

x = 4

is its axis of symmetry. The distance between the two

x

-intercepts is

10

. By symmetry, the intercepts must be located

5

units to the left and right of the axis of symmetry, placing them at

x = 4 - 5 = -1

and

x = 4 + 5 = 9

. Since the graph intersects the

x

-axis at these points, substituting either

x

-coordinate into the function gives a

y

-value of

0

. Substituting

x = 9

yields

0 = -(9 - 4)^2 + c

, which simplifies to

0 = -25 + c

, so

c = 25

Step-by-Step Solution

Identify the axis of symmetry of the quadratic function.

The axis of symmetry is

x = 4

The function

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

is written in vertex form,

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

, where the vertex is

(h, k)

and the axis of symmetry is

x = h

Find the

x

-coordinates of the

x

-intercepts using the distance between them.

The

x

-intercepts are at

x = -1

and

x = 9

Since the parabola is symmetric about the line

x = 4

and the distance between the two intercepts is

10

, each intercept is

\frac{10}{2} = 5

units away from the axis of symmetry. Thus, the intercepts are at

4 - 5 = -1

and

4 + 5 = 9

Substitute one of the

x

-intercepts into the function to solve for the constant

c

c = 25

Since

(9, 0)

is on the graph,

f(9) = 0

. Substituting this gives

0 = -(9 - 4)^2 + c \implies 0 = -25 + c \implies c = 25

Key Concept

Using symmetry and the vertex form of a quadratic function to determine constants from key graphical features such as

x

-intercepts.

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Answer

Step-by-Step Solution

Key Concept

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