To identify which function has a vertex at the origin, we need to evaluate each function at [tex]\( x = 0 \)[/tex] and determine if [tex]\( f(0) = 0 \)[/tex]. This is because a vertex at the origin means that the function passes through the point [tex]\((0, 0)\)[/tex].
Here are the three steps we will use for each function:
1. Substitute [tex]\( x = 0 \)[/tex] into the function.
2. Calculate the value of the function at [tex]\( x = 0 \)[/tex].
3. Check if the result is equal to 0.
Given Functions:
1. [tex]\( f(x) = (x + 4)^2 \)[/tex]
2. [tex]\( f(x) = x(x - 4) \)[/tex]
3. [tex]\( f(x) = (x - 4)(x + 4) \)[/tex]
4. [tex]\( f(x) = -x^2 \)[/tex]
### Evaluation:
1. For [tex]\( f(x) = (x + 4)^2 \)[/tex]:
Substitute [tex]\( x = 0 \)[/tex]:
[tex]\[ f(0) = (0 + 4)^2 = 4^2 = 16 \][/tex]
Since [tex]\( f(0) \neq 0 \)[/tex], this function does not have a vertex at the origin.
2. For [tex]\( f(x) = x(x - 4) \)[/tex]:
Substitute [tex]\( x = 0 \)[/tex]:
[tex]\[ f(0) = 0 \cdot (0 - 4) = 0 \][/tex]
Since [tex]\( f(0) = 0 \)[/tex], this function has a vertex at the origin.
3. For [tex]\( f(x) = (x - 4)(x + 4) \)[/tex]:
Substitute [tex]\( x = 0 \)[/tex]:
[tex]\[ f(0) = (0 - 4)(0 + 4) = -4 \cdot 4 = -16 \][/tex]
Since [tex]\( f(0) \neq 0 \)[/tex], this function does not have a vertex at the origin.
4. For [tex]\( f(x) = -x^2 \)[/tex]:
Substitute [tex]\( x = 0 \)[/tex]:
[tex]\[ f(0) = -(0^2) = 0 \][/tex]
Since [tex]\( f(0) = 0 \)[/tex], this function has a vertex at the origin.
### Conclusion:
The functions that have a vertex at the origin are:
- [tex]\( f(x) = x(x - 4) \)[/tex]
- [tex]\( f(x) = -x^2 \)[/tex]
