In the xy-plane, a cubic polynomial function f crosses the x-axis at (-3, 0), (1, 0), and … — SAT

In the $xy$ -plane, a cubic polynomial function $f$ crosses the $x$ -axis at $(-3, 0)$ , $(1, 0)$ , and $(4, 0)$ . Given that $f(2) = -10$ , what is the $y$ -coordinate of the $y$ -intercept of the graph of $f$ ?

A
-12
B
-4
12Answer
D
60

Answer

The correct answer is 12. Since the cubic function has

x

-intercepts at

(-3, 0)

(1, 0)

, and

(4, 0)

, it can be represented in factored form as

f(x) = a(x + 3)(x - 1)(x - 4)

for some constant

a

. Substituting the given coordinate

(2, -10)

into the function yields

-10 = a(2 + 3)(2 - 1)(2 - 4)

, which simplifies to

-10 = -10a

, meaning

a = 1

. The fully determined function is

f(x) = (x + 3)(x - 1)(x - 4)

. The

y

-intercept occurs when

x = 0

. Evaluating

f(0)

gives

(3)(-1)(-4) = 12

Step-by-Step Solution

Write the general factored form of the cubic function using its intercepts.

f(x) = a(x + 3)(x - 1)(x - 4)

, where

a

is a constant.

Since the graph of

f

has

x

-intercepts at

(-3, 0)

(1, 0)

, and

(4, 0)

, the values

-3

1

, and

4

are roots of the polynomial. Therefore,

(x + 3)

(x - 1)

, and

(x - 4)

are factors of the polynomial.

Substitute the point

(2, -10)

into the equation to solve for the constant coefficient

a

a = 1

Plugging in the given values yields

-10 = a(2 + 3)(2 - 1)(2 - 4)

, which simplifies to

-10 = a(5)(1)(-2)

, or

-10 = -10a

. Dividing both sides by

-10

gives

a = 1

Calculate the

y

-intercept of the function by evaluating it at

x = 0

f(0) = 12

The

y

-intercept occurs where the graph crosses the

y

-axis, which corresponds to

x = 0

. Substituting

x = 0

into the function

f(x) = (x + 3)(x - 1)(x - 4)

gives

f(0) = (3)(-1)(-4) = 12

Key Concept

Polynomial Factors and Graphs

Estimated Time:1m 30s

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