Question

question
find the y - coordinate of the y - intercept of the polynomial function defined below:
$f(x)=-(x - 4)^2(x^2 - 1)$

Explanation:

Step1: Recall the definition of y-intercept

The y-intercept of a function \( y = f(x) \) is the point where \( x = 0 \). So we need to find \( f(0) \) to get the y - coordinate of the y - intercept.

Step2: Substitute \( x = 0 \) into the function \( f(x)=-(x - 4)^{2}(x^{2}-1) \)

First, substitute \( x = 0 \) into each part of the function:

• For \( (x - 4)^{2} \), when \( x = 0 \), we have \( (0 - 4)^{2}=(-4)^{2}=16 \)
• For \( (x^{2}-1) \), when \( x = 0 \), we have \( (0^{2}-1)=- 1 \)

Then, substitute these values back into the function \( f(x) \):
\( f(0)=-(0 - 4)^{2}(0^{2}-1)=-(16)\times(-1) \)

Step3: Simplify the expression

\( -(16)\times(-1)=16 \)

Answer:

The y - coordinate of the y - intercept is \( 16 \)