Question

question 20 of 45
the graph of the function $f(x) = -9(x + 1)^2 + 81$ has its vertex at point $(-1, 81)$ and intersects the $x$-axis at points $(-4, 0)$ and $(a, 0)$. what is the value of $a$
a 2
b 3
c 72
d 81

Explanation:

Step1: Set \( f(x) = 0 \)

We know that the function is \( f(x)=-9(x + 1)^2+81 \). To find the \( x \)-intercepts, we set \( f(x) = 0 \), so:
\[
0=-9(x + 1)^2+81
\]

Step2: Solve for \( (x + 1)^2 \)

First, we can rearrange the equation. Add \( 9(x + 1)^2 \) to both sides:
\[
9(x + 1)^2=81
\]
Then divide both sides by 9:
\[
(x + 1)^2=\frac{81}{9}=9
\]

Step3: Solve for \( x \)

Take the square root of both sides. Remember that \( \sqrt{(x + 1)^2}=\pm(x + 1) \), so:
\[
x + 1=\pm\sqrt{9}=\pm3
\]
This gives us two equations:

1. \( x + 1 = 3 \), solving for \( x \) we get \( x=3 - 1=2 \)
2. \( x + 1=-3 \), solving for \( x \) we get \( x=-3 - 1=-4 \)

We already know one \( x \)-intercept is at \( x = - 4 \) (from the point \( (-4,0) \)), so the other \( x \)-intercept is at \( x = 2 \). So \( a = 2 \).

Answer:

A. 2