A polynomial function f has the form f(x) = a(x - 2)(x + 3)(x - 5), where a is a constant.… — SAT

A polynomial function $f$ has the form $f(x) = a(x - 2)(x + 3)(x - 5)$ , where $a$ is a constant. In the $xy$ -plane, the graph of $y = f(x)$ has a $y$ -intercept of $(0, 60)$ . What is the value of $f(1)$ ?

Answer: 32

Answer

The

y

-intercept of the graph is given as

(0, 60)

, which means that when

x = 0

f(0) = 60

. Substituting

x = 0

into the function gives

f(0) = a(0 - 2)(0 + 3)(0 - 5) = 30a

. Equating this to 60 gives

30a = 60

, so

a = 2

. Therefore, the function is

f(x) = 2(x - 2)(x + 3)(x - 5)

. To find the value of

f(1)

, substitute

x = 1

into this expression:

f(1) = 2(1 - 2)(1 + 3)(1 - 5) = 2(-1)(4)(-4) = 32

Step-by-Step Solution

Identify the relation between the

y

-intercept and the function's value.

f(0) = 60

The

y

-intercept of a graph

y = f(x)

is the point where

x = 0

Substitute

x = 0

into the definition of

f(x)

and set it equal to 60.

a(0 - 2)(0 + 3)(0 - 5) = 60 \implies 30a = 60

This allows us to solve for the unknown constant coefficient

a

Solve the linear equation for

a

a = 2

Dividing both sides of the equation by 30 isolates

a

Evaluate the complete function

f(x) = 2(x - 2)(x + 3)(x - 5)

x = 1

f(1) = 2(1 - 2)(1 + 3)(1 - 5) = 2(-1)(4)(-4) = 32

This yields the requested value of

f(1)

Key Concept

Using the factors and a known point (such as the y-intercept) of a polynomial function to determine its algebraic expression and evaluate it.

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