To solve this problem, we need to find which of the given equations, when graphed, produces [tex]$x$[/tex]-intercepts at [tex]$(-1,0)$[/tex] and [tex]$(-5,0)$[/tex], and a [tex]$y$[/tex]-intercept at [tex]$(0,-30)$[/tex].
Let's break down each of the equations and determine the intercepts.
1. [tex]\( f(x) = -6(x+1)(x+5) \)[/tex]:
To find the [tex]\(x\)[/tex]-intercepts, set [tex]\( f(x) = 0 \)[/tex]:
[tex]\[ -6(x+1)(x+5) = 0 \][/tex]
[tex]\[ (x+1)(x+5) = 0 \][/tex]
This gives us two solutions:
[tex]\[ x + 1 = 0 \rightarrow x = -1 \][/tex]
[tex]\[ x + 5 = 0 \rightarrow x = -5 \][/tex]
Thus, the [tex]\(x\)[/tex]-intercepts are at [tex]\((-1, 0)\)[/tex] and [tex]\((-5, 0)\)[/tex], which matches the problem's requirements.
Next, let's find the [tex]\(y\)[/tex]-intercept by setting [tex]\(x = 0\)[/tex]:
[tex]\[ f(0) = -6(0+1)(0+5) \][/tex]
[tex]\[ f(0) = -6(1)(5) \][/tex]
[tex]\[ f(0) = -6 \cdot 5 \][/tex]
[tex]\[ f(0) = -30 \][/tex]
The [tex]\(y\)[/tex]-intercept is at [tex]\((0, -30)\)[/tex], which also matches the problem's requirements.
Therefore, the equation [tex]\( f(x) = -6(x+1)(x+5) \)[/tex] meets all the required conditions.
2. [tex]\( f(x) = -6(x-1)(x-5) \)[/tex]:
To find the [tex]\(x\)[/tex]-intercepts, set [tex]\( f(x) = 0 \)[/tex]:
[tex]\[ -6(x-1)(x-5) = 0 \][/tex]
[tex]\[ (x-1)(x-5) = 0 \][/tex]
This gives us two solutions:
[tex]\[ x - 1 = 0 \rightarrow x = 1 \][/tex]
[tex]\[ x - 5 = 0 \rightarrow x = 5 \][/tex]
Thus, the [tex]\(x\)[/tex]-intercepts are at [tex]\((1, 0)\)[/tex] and [tex]\((5, 0)\)[/tex], which do not match the problem's requirements.
3. [tex]\( f(x) = -5(x+1)(x+5) \)[/tex]:
To find the [tex]\(x\)[/tex]-intercepts, set [tex]\( f(x) = 0 \)[/tex]:
[tex]\[ -5(x+1)(x+5) = 0 \][/tex]
[tex]\[ (x+1)(x+5) = 0 \][/tex]
This gives us two solutions:
[tex]\[ x + 1 = 0 \rightarrow x = -1 \][/tex]
[tex]\[ x + 5 = 0 \rightarrow x = -5 \][/tex]
Thus, the [tex]\(x\)[/tex]-intercepts are at [tex]\((-1, 0)\)[/tex] and [tex]\((-5, 0)\)[/tex], which match the problem's requirements.
Next, let's find the [tex]\(y\)[/tex]-intercept by setting [tex]\(x = 0\)[/tex]:
[tex]\[ f(0) = -5(0+1)(0+5) \][/tex]
[tex]\[ f(0) = -5(1)(5) \][/tex]
[tex]\[ f(0) = -5 \cdot 5 \][/tex]
[tex]\[ f(0) = -25 \][/tex]
The [tex]\(y\)[/tex]-intercept is at [tex]\((0, -25)\)[/tex], which does not match the problem's requirements.
4. [tex]\( f(x) = -5(x-1)(x-5) \)[/tex]:
To find the [tex]\(x\)[/tex]-intercepts, set [tex]\( f(x) = 0 \)[/tex]:
[tex]\[ -5(x-1)(x-5) = 0 \][/tex]
[tex]\[ (x-1)(x-5) = 0 \][/tex]
This gives us two solutions:
[tex]\[ x - 1 = 0 \rightarrow x = 1 \][/tex]
[tex]\[ x - 5 = 0 \rightarrow x = 5 \][/tex]
Thus, the [tex]\(x\)[/tex]-intercepts are at [tex]\((1, 0)\)[/tex] and [tex]\((5, 0)\)[/tex], which do not match the problem's requirements.
Based on the intercepts derived, the correct equation that has [tex]\(x\)[/tex]-intercepts at [tex]\((-1,0)\)[/tex] and [tex]\((-5,0)\)[/tex] and a [tex]\(y\)[/tex]-intercept at [tex]\((0,-30)\)[/tex] is:
[tex]\[ f(x) = -6(x+1)(x+5) \][/tex]
