A quadratic function g is defined by g(x) = 2x^2 - 12x + k, where k is a constant. In the … — SAT

A quadratic function $g$ is defined by $g(x) = 2x^2 - 12x + k$ , where $k$ is a constant. In the $xy$ -plane, the graph of $g$ has its vertex on the line $y = -5$ . What is the value of $k$ ?

A
-59
B
-23
C
-5
13Answer

Answer

The

x

-coordinate of the vertex for the quadratic function

g(x) = ax^2 + bx + c

is found using the formula

x = -\frac{b}{2a}

. Substituting

a = 2

and

b = -12

yields

x = -\frac{-12}{2(2)} = 3

. Evaluating

g(3)

gives the

y

-coordinate of the vertex:

g(3) = 2(3)^2 - 12(3) + k = -18 + k

. Since the vertex lies on the line

y = -5

, setting

-18 + k = -5

and solving for

k

gives

k = 13

Step-by-Step Solution

Find the

x

-coordinate of the vertex of the parabola.

The

x

-coordinate is

3

For a quadratic function

g(x) = ax^2 + bx + c

, the

x

-coordinate of the vertex is given by

x = -\frac{b}{2a}

. Here,

a = 2

and

b = -12

, so

x = -\frac{-12}{2(2)} = 3

Evaluate the function at

x = 3

to find the

y

-coordinate of the vertex in terms of

k

The

y

-coordinate is

-18 + k

Substituting

x = 3

into the function gives

g(3) = 2(3)^2 - 12(3) + k = 18 - 36 + k = -18 + k

Set the

y

-coordinate of the vertex equal to

-5

and solve for

k

k = 13

Since the vertex lies on the line

y = -5

, the

y

-coordinate of the vertex must be

-5

. Thus,

-18 + k = -5 \implies k = 13

Key Concept

Determining the vertex of a quadratic function and using its coordinates to solve for unknown parameters.

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