In the xy-plane, the graph of the polynomial function g, defined by g(x) = 3(x - k)(x + 2)… — SAT

In the $xy$ -plane, the graph of the polynomial function $g$ , defined by $g(x) = 3(x - k)(x + 2)^2$ , has a $y$ -intercept at $(0, -24)$ , where $k$ is a constant. What is the value of $k$ ?

Answer: 2

Answer

The correct answer is 2.

Evaluating the function at

x = 0

yields

g(0) = 3(-k)(2)^2 = -12k

. Setting this equal to the given

y

-intercept value of

-24

gives

-12k = -24

. Solving for

k

gives

k = 2

Step-by-Step Solution

Determine the value of the function at the

y

-intercept

g(0) = -24

The

y

-intercept of a graph in the

xy

-plane is the point where

x = 0

. Since the

y

-intercept is

(0, -24)

, the function value when

x = 0

must be

-24

Substitute

x = 0

into the function definition

g(0) = 3(0 - k)(0 + 2)^2

To evaluate the expression at

x = 0

, we substitute

0

for every instance of

x

in the equation.

Simplify the algebraic expression

g(0) = -12k

Simplifying the terms:

(0+2)^2 = 4

, and

3(0-k) = -3k

. Multiplying these gives

-3k \times 4 = -12k

Solve for the constant

k

k = 2

Equating the simplified expression to the known

y

-value at the intercept gives

-12k = -24

. Dividing both sides by

-12

yields

k = 2

Key Concept

Using the y-intercept of a polynomial function to solve for an unknown constant coefficient.

Question

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Key Concept

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