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To determine which function represents a parabola with zeros at [tex]\( x = -2 \)[/tex] and [tex]\( x = 4 \)[/tex], and a y-intercept at [tex]\( (0, -16) \)[/tex], we need to verify these conditions for each given function.
### Checking Zero Points
1. Function A: [tex]\( f(x) = 2x^2 + 4x - 16 \)[/tex]
- Calculate [tex]\( f(-2) \)[/tex]:
[tex]\[ f(-2) = 2(-2)^2 + 4(-2) - 16 = 2(4) - 8 - 16 = 8 - 8 - 16 = -16 \neq 0 \][/tex]
- Calculate [tex]\( f(4) \)[/tex]:
[tex]\[ f(4) = 2(4)^2 + 4(4) - 16 = 2(16) + 16 - 16 = 32 + 16 - 16 = 32 \neq 0 \][/tex]
- Thus, Function A does not have zeros at [tex]\( x = -2 \)[/tex] and [tex]\( x = 4 \)[/tex].
2. Function B: [tex]\( f(x) = x^2 - 2x - 8 \)[/tex]
- Calculate [tex]\( f(-2) \)[/tex]:
[tex]\[ f(-2) = (-2)^2 - 2(-2) - 8 = 4 + 4 - 8 = 0 \][/tex]
- Calculate [tex]\( f(4) \)[/tex]:
[tex]\[ f(4) = (4)^2 - 2(4) - 8 = 16 - 8 - 8 = 0 \][/tex]
- Thus, Function B has zeros at [tex]\( x = -2 \)[/tex] and [tex]\( x = 4 \)[/tex].
3. Function C: [tex]\( f(x) = x^2 + 2x - 8 \)[/tex]
- Calculate [tex]\( f(-2) \)[/tex]:
[tex]\[ f(-2) = (-2)^2 + 2(-2) - 8 = 4 - 4 - 8 = -8 \neq 0 \][/tex]
- Calculate [tex]\( f(4) \)[/tex]:
[tex]\[ f(4) = (4)^2 + 2(4) - 8 = 16 + 8 - 8 = 16 \neq 0 \][/tex]
- Thus, Function C does not have zeros at [tex]\( x = -2 \)[/tex] and [tex]\( x = 4 \)[/tex].
4. Function D: [tex]\( f(x) = 2x^2 - 4x - 16 \)[/tex]
- Calculate [tex]\( f(-2) \)[/tex]:
[tex]\[ f(-2) = 2(-2)^2 - 4(-2) - 16 = 2(4) + 8 - 16 = 8 + 8 - 16 = 0 \][/tex]
- Calculate [tex]\( f(4) \)[/tex]:
[tex]\[ f(4) = 2(4)^2 - 4(4) - 16 = 2(16) - 16 - 16 = 32 - 32 = 0 \][/tex]
- Thus, Function D has zeros at [tex]\( x = -2 \)[/tex] and [tex]\( x = 4 \)[/tex].
### Checking the Y-Intercept
To find the y-intercept, we substitute [tex]\( x = 0 \)[/tex] into each function and see if the result equals -16:
1. Function A: [tex]\( f(x) = 2x^2 + 4x - 16 \)[/tex]
[tex]\[ f(0) = 2(0)^2 + 4(0) - 16 = -16 \][/tex]
- Function A has a y-intercept at [tex]\( (0, -16) \)[/tex].
2. Function B: [tex]\( f(x) = x^2 - 2x - 8 \)[/tex]
[tex]\[ f(0) = (0)^2 - 2(0) - 8 = -8 \][/tex]
- Function B does not have a y-intercept at [tex]\( (0, -16) \)[/tex].
3. Function C: [tex]\( f(x) = x^2 + 2x - 8 \)[/tex]
[tex]\[ f(0) = (0)^2 + 2(0) - 8 = -8 \][/tex]
- Function C does not have a y-intercept at [tex]\( (0, -16) \)[/tex].
4. Function D: [tex]\( f(x) = 2x^2 - 4x - 16 \)[/tex]
[tex]\[ f(0) = 2(0)^2 - 4(0) - 16 = -16 \][/tex]
- Function D has a y-intercept at [tex]\( (0, -16) \)[/tex].
### Conclusion
Among the functions, only Function D: [tex]\( f(x) = 2x^2 - 4x - 16 \)[/tex] has the zeros [tex]\( x = -2 \)[/tex] and [tex]\( x = 4 \)[/tex], and the y-intercept at [tex]\( (0, -16) \)[/tex].
Therefore, the correct function is:
[tex]\[ \boxed{2} \][/tex]
### Checking Zero Points
1. Function A: [tex]\( f(x) = 2x^2 + 4x - 16 \)[/tex]
- Calculate [tex]\( f(-2) \)[/tex]:
[tex]\[ f(-2) = 2(-2)^2 + 4(-2) - 16 = 2(4) - 8 - 16 = 8 - 8 - 16 = -16 \neq 0 \][/tex]
- Calculate [tex]\( f(4) \)[/tex]:
[tex]\[ f(4) = 2(4)^2 + 4(4) - 16 = 2(16) + 16 - 16 = 32 + 16 - 16 = 32 \neq 0 \][/tex]
- Thus, Function A does not have zeros at [tex]\( x = -2 \)[/tex] and [tex]\( x = 4 \)[/tex].
2. Function B: [tex]\( f(x) = x^2 - 2x - 8 \)[/tex]
- Calculate [tex]\( f(-2) \)[/tex]:
[tex]\[ f(-2) = (-2)^2 - 2(-2) - 8 = 4 + 4 - 8 = 0 \][/tex]
- Calculate [tex]\( f(4) \)[/tex]:
[tex]\[ f(4) = (4)^2 - 2(4) - 8 = 16 - 8 - 8 = 0 \][/tex]
- Thus, Function B has zeros at [tex]\( x = -2 \)[/tex] and [tex]\( x = 4 \)[/tex].
3. Function C: [tex]\( f(x) = x^2 + 2x - 8 \)[/tex]
- Calculate [tex]\( f(-2) \)[/tex]:
[tex]\[ f(-2) = (-2)^2 + 2(-2) - 8 = 4 - 4 - 8 = -8 \neq 0 \][/tex]
- Calculate [tex]\( f(4) \)[/tex]:
[tex]\[ f(4) = (4)^2 + 2(4) - 8 = 16 + 8 - 8 = 16 \neq 0 \][/tex]
- Thus, Function C does not have zeros at [tex]\( x = -2 \)[/tex] and [tex]\( x = 4 \)[/tex].
4. Function D: [tex]\( f(x) = 2x^2 - 4x - 16 \)[/tex]
- Calculate [tex]\( f(-2) \)[/tex]:
[tex]\[ f(-2) = 2(-2)^2 - 4(-2) - 16 = 2(4) + 8 - 16 = 8 + 8 - 16 = 0 \][/tex]
- Calculate [tex]\( f(4) \)[/tex]:
[tex]\[ f(4) = 2(4)^2 - 4(4) - 16 = 2(16) - 16 - 16 = 32 - 32 = 0 \][/tex]
- Thus, Function D has zeros at [tex]\( x = -2 \)[/tex] and [tex]\( x = 4 \)[/tex].
### Checking the Y-Intercept
To find the y-intercept, we substitute [tex]\( x = 0 \)[/tex] into each function and see if the result equals -16:
1. Function A: [tex]\( f(x) = 2x^2 + 4x - 16 \)[/tex]
[tex]\[ f(0) = 2(0)^2 + 4(0) - 16 = -16 \][/tex]
- Function A has a y-intercept at [tex]\( (0, -16) \)[/tex].
2. Function B: [tex]\( f(x) = x^2 - 2x - 8 \)[/tex]
[tex]\[ f(0) = (0)^2 - 2(0) - 8 = -8 \][/tex]
- Function B does not have a y-intercept at [tex]\( (0, -16) \)[/tex].
3. Function C: [tex]\( f(x) = x^2 + 2x - 8 \)[/tex]
[tex]\[ f(0) = (0)^2 + 2(0) - 8 = -8 \][/tex]
- Function C does not have a y-intercept at [tex]\( (0, -16) \)[/tex].
4. Function D: [tex]\( f(x) = 2x^2 - 4x - 16 \)[/tex]
[tex]\[ f(0) = 2(0)^2 - 4(0) - 16 = -16 \][/tex]
- Function D has a y-intercept at [tex]\( (0, -16) \)[/tex].
### Conclusion
Among the functions, only Function D: [tex]\( f(x) = 2x^2 - 4x - 16 \)[/tex] has the zeros [tex]\( x = -2 \)[/tex] and [tex]\( x = 4 \)[/tex], and the y-intercept at [tex]\( (0, -16) \)[/tex].
Therefore, the correct function is:
[tex]\[ \boxed{2} \][/tex]
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