To determine which equation, when graphed, has [tex]\(x\)[/tex]-intercepts at [tex]\((8, 0)\)[/tex] and [tex]\((-2, 0)\)[/tex] and a [tex]\(y\)[/tex]-intercept at [tex]\((0, -48)\)[/tex], we can examine each given function step-by-step.
Based on the standard form of a quadratic equation with [tex]\(x\)[/tex]-intercepts [tex]\(r_1\)[/tex] and [tex]\(r_2\)[/tex], we can write it as [tex]\(f(x) = k(x - r_1)(x - r_2)\)[/tex]. Given [tex]\(x\)[/tex]-intercepts [tex]\((8, 0)\)[/tex] and [tex]\((-2, 0)\)[/tex], we construct the quadratic equation accordingly.
### Checking [tex]\(f(x) = -3(x - 8)(x + 2)\)[/tex]:
1. X-intercepts: Set [tex]\(f(x) = 0\)[/tex].
- For [tex]\(x = 8\)[/tex]:
[tex]\[
f(8) = -3(8 - 8)(8 + 2) = -3(0)(10) = 0
\][/tex]
- For [tex]\(x = -2\)[/tex]:
[tex]\[
f(-2) = -3(-2 - 8)(-2 + 2) = -3(-10)(0) = 0
\][/tex]
Both intercepts match the required [tex]\(x\)[/tex]-intercepts.
2. Y-intercept: Set [tex]\(x = 0\)[/tex].
- For [tex]\(x = 0\)[/tex]:
[tex]\[
f(0) = -3(0 - 8)(0 + 2) = -3(-8)(2) = -3(-16) = 48
\][/tex]
Since the [tex]\(y\)[/tex]-intercept given is [tex]\(-48\)[/tex], this function does not match all criteria.
### Checking [tex]\(f(x) = -3(x + 8)(x - 2)\)[/tex]:
1. X-intercepts: Set [tex]\(f(x) = 0\)[/tex].
- For [tex]\(x = -8\)[/tex]:
[tex]\[
f(-8) = -3(-8 + 8)(-8 - 2) = -3(0)(-10) = 0
\][/tex]
- For [tex]\(x = 2\)[/tex]:
[tex]\[
f(2) = -3(2 + 8)(2 - 2) = -3(10)(0) = 0
\][/tex]
These intercepts do not match the required [tex]\(x\)[/tex]-intercepts.
### Checking [tex]\(f(x) = 3(x - 8)(x + 2)\)[/tex]:
1. X-intercepts: Set [tex]\(f(x) = 0\)[/tex].
- For [tex]\(x = 8\)[/tex]:
[tex]\[
f(8) = 3(8 - 8)(8 + 2) = 3(0)(10) = 0
\][/tex]
- For [tex]\(x = -2\)[/tex]:
[tex]\[
f(-2) = 3(-2 - 8)(-2 + 2) = 3(-10)(0) = 0
\][/tex]
Both intercepts match the required [tex]\(x\)[/tex]-intercepts.
2. Y-intercept: Set [tex]\(x = 0\)[/tex].
- For [tex]\(x = 0\)[/tex]:
[tex]\[
f(0) = 3(0 - 8)(0 + 2) = 3(-8)(2) = 3(-16) = -48
\][/tex]
This function matches all criteria.
### Checking [tex]\(f(x) = 3(x + 8)(x - 2)\)[/tex]:
1. X-intercepts: Set [tex]\(f(x) = 0\)[/tex].
- For [tex]\(x = -8\)[/tex]:
[tex]\[
f(-8) = 3(-8 + 8)(-8 - 2) = 3(0)(-10) = 0
\][/tex]
- For [tex]\(x = 2\)[/tex]:
[tex]\[
f(2) = 3(2 + 8)(2 - 2) = 3(10)(0) = 0
\][/tex]
These intercepts do not match the required [tex]\(x\)[/tex]-intercepts.
### Conclusion:
Given the requirements, the equation that fulfills all conditions is:
[tex]\[
f(x) = 3(x - 8)(x + 2)
\][/tex]
