A quadratic function f is defined by f(x) = a(x - 4)(x - k), where a and k are constants. … — SAT

A quadratic function $f$ is defined by $f(x) = a(x - 4)(x - k)$ , where $a$ and $k$ are constants. In the $xy$ -plane, the graph of $y = f(x)$ has its vertex at $(6, 12)$ . What is the value of $k$ ?

Answer: 8

Answer

The value of

k

8

The given quadratic function is in the factored form

f(x) = a(x - 4)(x - k)

, meaning the x-intercepts of its graph are

x = 4

and

x = k

. Due to the symmetry of a parabola, the x-coordinate of the vertex is the midpoint of the x-intercepts. Since the vertex is at

(6, 12)

, its x-coordinate is

6

. Setting the midpoint of the intercepts equal to 6 gives the equation

\frac{4 + k}{2} = 6

. Multiplying both sides by 2 gives

4 + k = 12

, and subtracting 4 gives the correct answer

k = 8

Step-by-Step Solution

Identify the x-intercepts of the function

f(x) = a(x - 4)(x - k)

The x-intercepts are at

x = 4

and

x = k

For any quadratic function written in the factored form

f(x) = a(x - r_1)(x - r_2)

, the values

r_1

and

r_2

correspond to the x-intercepts of the graph.

Relate the x-coordinate of the vertex to the x-intercepts using symmetry.

The axis of symmetry equation is

\frac{4 + k}{2} = 6

Because a parabola is symmetric, the x-coordinate of the vertex always lies exactly halfway between the two x-intercepts.

Solve the equation for

k

k = 8

Multiplying both sides of the equation

\frac{4 + k}{2} = 6

by 2 yields

4 + k = 12

. Subtracting 4 from both sides gives

k = 8

Key Concept

Symmetry of quadratic graphs and their vertices relative to their x-intercepts.

Alternative Method

Alternatively, substitute the vertex coordinates

(6, 12)

into the function:

12 = a(6 - 4)(6 - k) = 2a(6 - k)

. Since the vertex is the maximum point, the derivative

f'(x) = a(x - k) + a(x - 4)

must be equal to 0 at

x = 6

. This yields

a(6 - k) + a(6 - 4) = 0

. Since

a \neq 0

, dividing by

a

gives

(6 - k) + 2 = 0

, which simplifies to

8 - k = 0

, or

k = 8

Estimated Time:1m 30s

Question

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Alternative Method

Topics

More questions from this topic