To address the question, we need to analyze the function [tex]\( f(x) = (x + 4)^2 \)[/tex] and identify its key characteristics.
### 1. Vertex
The function [tex]\( f(x) = (x + 4)^2 \)[/tex] is in vertex form, which is [tex]\( (x - h)^2 + k \)[/tex]. Here, the function has been rewritten as [tex]\( (x + 4)^2 \)[/tex], which can be seen as [tex]\( (x - (-4))^2 + 0 \)[/tex]. Hence, the vertex of the function is at [tex]\( (-4, 0) \)[/tex].
### 2. [tex]\( x \)[/tex]-Intercept
To find the [tex]\( x \)[/tex]-intercepts of [tex]\( f(x) = (x + 4)^2 \)[/tex], we set [tex]\( f(x) = 0 \)[/tex].
[tex]\[
(x + 4)^2 = 0
\][/tex]
Solving for [tex]\( x \)[/tex]:
[tex]\[
x + 4 = 0 \implies x = -4
\][/tex]
Thus, the function has one [tex]\( x \)[/tex]-intercept at [tex]\( x = -4 \)[/tex].
### 3. [tex]\( y \)[/tex]-Intercept
To find the [tex]\( y \)[/tex]-intercept, we evaluate the function at [tex]\( x = 0 \)[/tex].
[tex]\[
f(0) = (0 + 4)^2 = 4^2 = 16
\][/tex]
Therefore, the [tex]\( y \)[/tex]-intercept is at [tex]\( y = 16 \)[/tex].
### 4. Range
Since [tex]\( f(x) = (x + 4)^2 \)[/tex] is a parabola that opens upwards (as indicated by the positive coefficient of [tex]\( x^2 \)[/tex]), the minimum value of the function occurs at the vertex.
The vertex is at [tex]\( (-4, 0) \)[/tex], so the minimum value of the function is [tex]\( 0 \)[/tex].
Thus, the range of [tex]\( f(x) \)[/tex] is [tex]\( [0, \infty) \)[/tex].
### Conclusion
Zander can conclude the following about the function [tex]\( f(x) = (x + 4)^2 \)[/tex]:
1. The vertex is [tex]\( (-4, 0) \)[/tex].
2. The function has one [tex]\( x \)[/tex]-intercept at [tex]\( x = -4 \)[/tex].
3. The [tex]\( y \)[/tex]-intercept is at [tex]\( y = 16 \)[/tex].
4. The range of the function is [tex]\( [0, \infty) \)[/tex].
By comparing these characteristics with those of the given graph, Zander can determine the similarities or differences between the two functions.
