The function f is defined by f(x) = (x - 4)(x - 2)(x + k), where k is a constant. If the y… — SAT

The function $f$ is defined by $f(x) = (x - 4)(x - 2)(x + k)$ , where $k$ is a constant. If the $y$ -intercept of the graph of $y = f(x)$ in the $xy$ -plane is $(0, 24)$ , what is the value of $k$ ?

Answer: 3

Answer

The

y

-intercept of the graph of

y = f(x)

is the point where

x = 0

. Since the

y

-intercept is

(0, 24)

, it follows that

f(0) = 24

. Substituting

0

for

x

in the equation for

f(x)

gives

f(0) = (0 - 4)(0 - 2)(0 + k) = (-4)(-2)(k) = 8k

. Setting this equal to the

y

-value of the intercept yields

8k = 24

. Dividing both sides of the equation by

8

gives

k = 3

Step-by-Step Solution

Use the definition of the

y

-intercept to find the value of

f(0)

f(0) = 24

The

y

-intercept of a graph is the point where the graph crosses the

y

-axis, corresponding to

x = 0

. Given the point

(0, 24)

f(0)

must equal

24

Evaluate the polynomial at

x = 0

in terms of

k

f(0) = 8k

Substituting

0

for

x

f(x) = (x - 4)(x - 2)(x + k)

gives

f(0) = (-4)(-2)(k)

, which simplifies to

8k

Set the evaluated expression equal to the

y

-intercept value and solve for

k

k = 3

Equating

8k

24

and dividing both sides by

8

yields

k = 3

Key Concept

Evaluating a factored polynomial function at

x = 0

determines its

y

-intercept. Using a given

y

-intercept allows solving for unknown coefficients or constants within the factors of the polynomial.

Question

Answer

Step-by-Step Solution

Key Concept

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