Let's analyze the given function [tex]\( f(x) = (x+4)^2 \)[/tex] to draw some conclusions about its properties.
### Vertex:
The function [tex]\( f(x) = (x+4)^2 \)[/tex] is a parabola in the standard form [tex]\( (x-h)^2 + k \)[/tex], where the vertex [tex]\((h, k)\)[/tex] can be identified as:
[tex]\[ h = -4, \][/tex]
[tex]\[ k = 0. \][/tex]
Thus, the vertex of [tex]\( f(x) = (x+4)^2 \)[/tex] is [tex]\((-4, 0)\)[/tex].
### [tex]\(x\)[/tex]-intercepts:
To find the [tex]\(x\)[/tex]-intercepts, we need to solve for [tex]\(x\)[/tex] when [tex]\( f(x) = 0 \)[/tex]:
[tex]\[ (x+4)^2 = 0. \][/tex]
Solving this equation:
[tex]\[ x + 4 = 0 \][/tex]
[tex]\[ x = -4. \][/tex]
Therefore, the function has one [tex]\(x\)[/tex]-intercept at [tex]\((-4, 0)\)[/tex].
### [tex]\(y\)[/tex]-intercept:
To find the [tex]\(y\)[/tex]-intercept, we evaluate [tex]\( f(0) \)[/tex]:
[tex]\[ f(0) = (0+4)^2 = 16. \][/tex]
Thus, the [tex]\(y\)[/tex]-intercept of the function is [tex]\((0, 16)\)[/tex].
### Range:
The function [tex]\( f(x) = (x+4)^2 \)[/tex] represents a parabola that opens upwards. Therefore, the lowest point on the graph is the vertex, and the function's values increase from there. Therefore, the range of [tex]\( f(x) \)[/tex] is:
[tex]\[ [0, \infty). \][/tex]
### Comparing with the other function represented by a graph:
Without specific information about the other function directly from the graph, we cannot make precise determinations about the properties such as the vertex, intercepts, and range.
However, the conclusions we can make about [tex]\( f(x) = (x+4)^2 \)[/tex] alone allow us to address the provided options:
1. They have the same vertex.
This is only true if the other function shares the vertex [tex]\((-4, 0)\)[/tex].
2. They have one [tex]\( x \)[/tex]-intercept that is the same.
This is only true if the other function also has an [tex]\( x \)[/tex]-intercept at [tex]\((-4, 0)\)[/tex].
3. They have the same [tex]\( y \)[/tex]-intercept.
This can be concluded if the other function has a [tex]\( y \)[/tex]-intercept at [tex]\((0, 16)\)[/tex].
4. They have the same range.
Given that [tex]\( f(x) = (x+4)^2 \)[/tex] has a range of [tex]\([0, \infty)\)[/tex], this statement is true if the other function also shares the same range [tex]\([0, \infty)\)[/tex].
Given only the analysis of the function [tex]\( f(x) = (x+4)^2 \)[/tex], and the options provided, Zander can conclude:
- They have the same range, if the graph of the other function is also a parabola opening upwards starting from a value of 0 to infinity or if it eventually encompasses all non-negative values.
In summary, the statement that can be made with certainty about [tex]\( f(x) = (x+4)^2 \)[/tex] without additional information about the graph of the other function is:
[tex]\[ \text{"They have the same range."} \][/tex]
