The graph of the quadratic function f in the xy-plane has its vertex at (4, 12). The funct… — SAT

The graph of the quadratic function $f$ in the $xy$ -plane has its vertex at $(4, 12)$ . The function is defined by $f(x) = -(x - c)(x - d)$ , where $c$ and $d$ are constants. What is the value of the product $cd$ ?

Answer: 4

Answer

The value of the product

cd

4

The

x

-coordinate of the vertex of a quadratic function of the form

f(x) = -(x-c)(x-d)

is the average of the

x

-intercepts

c

and

d

. Since the vertex is

(4, 12)

, we have

\frac{c+d}{2} = 4

, which means

c+d = 8

. Substituting the vertex

(4, 12)

into the function gives

12 = -(4-c)(4-d)

, which expands to

12 = -(16 - 4(c+d) + cd)

. Substituting

c+d = 8

gives

12 = -(16 - 32 + cd) = -(-16 + cd) = 16 - cd

. Solving for

cd

yields

cd = 4

Step-by-Step Solution

Find the sum of the constants

c

and

d

using the

x

-coordinate of the vertex.

c + d = 8

The graph of a quadratic function in the form

f(x) = -(x-c)(x-d)

has a vertical line of symmetry at the

x

-coordinate of its vertex, which is the midpoint of its

x

-intercepts

c

and

d

. Therefore,

\frac{c+d}{2} = 4

, which simplifies to

c + d = 8

Substitute the vertex coordinates

(4, 12)

into the function definition.

(4-c)(4-d) = -12

Since

(4, 12)

is the vertex, the point lies on the graph of

f

, meaning

f(4) = 12

. Substituting

x = 4

into the function gives

12 = -(4-c)(4-d)

, which simplifies to

(4-c)(4-d) = -12

Expand the expression

(4-c)(4-d)

and substitute

c+d = 8

to solve for

cd

cd = 4

Expanding

(4-c)(4-d) = -12

gives

16 - 4(c+d) + cd = -12

. Substituting

c+d = 8

yields

16 - 32 + cd = -12

, which simplifies to

-16 + cd = -12

. Adding

16

to both sides gives

cd = 4

Key Concept

Quadratic Functions and Graphs

Question

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