To find the [tex]\( x \)[/tex]-intercept of the function [tex]\( f(x) = (x + 4)^2 \)[/tex], we follow these steps:
1. Understand the definition of the [tex]\( x \)[/tex]-intercept: The [tex]\( x \)[/tex]-intercept is the point where the graph of the function crosses the [tex]\( x \)[/tex]-axis. This means the [tex]\( y \)[/tex]-value at the [tex]\( x \)[/tex]-intercept is 0.
2. Set the function equal to zero: To find the [tex]\( x \)[/tex]-intercept, we need to solve the equation where the function [tex]\( f(x) \)[/tex] is equal to zero:
[tex]\[
(x + 4)^2 = 0
\][/tex]
3. Solve for [tex]\( x \)[/tex]:
[tex]\[
(x + 4)^2 = 0
\][/tex]
Taking the square root of both sides, we get:
[tex]\[
x + 4 = 0
\][/tex]
Solving for [tex]\( x \)[/tex], we find:
[tex]\[
x = -4
\][/tex]
4. Substitute [tex]\( x \)[/tex] back into the function to verify: To ensure our solution, we substitute [tex]\( -4 \)[/tex] back into the function:
[tex]\[
f(-4) = (-4 + 4)^2 = 0
\][/tex]
Indeed, the [tex]\( y \)[/tex]-value at [tex]\( x = -4 \)[/tex] is 0.
5. Record the intercept: Therefore, the [tex]\( x \)[/tex]-intercept of the function [tex]\( f(x) = (x + 4)^2 \)[/tex] is [tex]\( (-4, 0) \)[/tex].
Now, let's plot this intercept on the graph:
[tex]\[
\begin{tabular}{|l|l|}
\hline
$x$ & $y$ \\
\hline
-4 & 0 \\
\hline
& \\
\hline
& \\
\hline
& \\
\hline
\end{tabular}
\][/tex]
Click or tap on the graph at the point [tex]\((-4, 0)\)[/tex] to plot the [tex]\( x \)[/tex]-intercept.
