Answer :
Let's analyze each of the given function options to determine which one meets the specified criteria: [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].
1. Option 1: [tex]\( f(x) = -6(x+1)(x+5) \)[/tex]
- Finding the [tex]\(x\)[/tex]-intercepts:
Set [tex]\( f(x) = 0 \)[/tex]:
[tex]\[ -6(x+1)(x+5) = 0 \][/tex]
This equation is zero when either [tex]\( x+1 = 0 \)[/tex] or [tex]\( x+5 = 0 \)[/tex]:
[tex]\[ x = -1 \quad \text{or} \quad x = -5 \][/tex]
Hence, the [tex]\(x\)[/tex]-intercepts are at [tex]\( (-1,0) \)[/tex] and [tex]\( (-5,0) \)[/tex].
- Finding the [tex]\(y\)[/tex]-intercept:
Set [tex]\( x = 0 \)[/tex]:
[tex]\[ f(0) = -6(0+1)(0+5) = -6(1)(5) = -30 \][/tex]
Hence, the [tex]\(y\)[/tex]-intercept is at [tex]\( (0, -30) \)[/tex].
Both the [tex]\(x\)[/tex]-intercepts and the [tex]\(y\)[/tex]-intercept match the given criteria.
2. Option 2: [tex]\( f(x) = -6(x-1)(x-5) \)[/tex]
- Finding the [tex]\(x\)[/tex]-intercepts:
Set [tex]\( f(x) = 0 \)[/tex]:
[tex]\[ -6(x-1)(x-5) = 0 \][/tex]
This equation is zero when either [tex]\( x-1 = 0 \)[/tex] or [tex]\( x-5 = 0 \)[/tex]:
[tex]\[ x = 1 \quad \text{or} \quad x = 5 \][/tex]
Hence, the [tex]\(x\)[/tex]-intercepts are at [tex]\( (1,0) \)[/tex] and [tex]\( (5,0) \)[/tex], which does not match the given criteria.
Since the [tex]\(x\)[/tex]-intercepts do not match, there is no need to check the [tex]\(y\)[/tex]-intercept.
3. Option 3: [tex]\( f(x) = -5(x+1)(x+5) \)[/tex]
- Finding the [tex]\(x\)[/tex]-intercepts:
Set [tex]\( f(x) = 0 \)[/tex]:
[tex]\[ -5(x+1)(x+5) = 0 \][/tex]
This equation is zero when either [tex]\( x+1 = 0 \)[/tex] or [tex]\( x+5 = 0 \)[/tex]:
[tex]\[ x = -1 \quad \text{or} \quad x = -5 \][/tex]
Hence, the [tex]\(x\)[/tex]-intercepts are at [tex]\( (-1,0) \)[/tex] and [tex]\( (-5,0) \)[/tex].
- Finding the [tex]\(y\)[/tex]-intercept:
Set [tex]\( x = 0 \)[/tex]:
[tex]\[ f(0) = -5(0+1)(0+5) = -5(1)(5) = -25 \][/tex]
Hence, the [tex]\(y\)[/tex]-intercept is at [tex]\( (0, -25) \)[/tex], which does not match the given criteria.
4. Option 4: [tex]\( f(x) = -5(x-1)(x-5) \)[/tex]
- Finding the [tex]\(x\)[/tex]-intercepts:
Set [tex]\( f(x) = 0 \)[/tex]:
[tex]\[ -5(x-1)(x-5) = 0 \][/tex]
This equation is zero when either [tex]\( x-1 = 0 \)[/tex] or [tex]\( x-5 = 0 \)[/tex]:
[tex]\[ x = 1 \quad \text{or} \quad x = 5 \][/tex]
Hence, the [tex]\(x\)[/tex]-intercepts are at [tex]\( (1,0) \)[/tex] and [tex]\( (5,0) \)[/tex], which does not match the given criteria.
Since the [tex]\(x\)[/tex]-intercepts do not match, there is no need to check the [tex]\(y\)[/tex]-intercept.
Therefore, the correct function that meets all the given criteria for [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]
1. Option 1: [tex]\( f(x) = -6(x+1)(x+5) \)[/tex]
- Finding the [tex]\(x\)[/tex]-intercepts:
Set [tex]\( f(x) = 0 \)[/tex]:
[tex]\[ -6(x+1)(x+5) = 0 \][/tex]
This equation is zero when either [tex]\( x+1 = 0 \)[/tex] or [tex]\( x+5 = 0 \)[/tex]:
[tex]\[ x = -1 \quad \text{or} \quad x = -5 \][/tex]
Hence, the [tex]\(x\)[/tex]-intercepts are at [tex]\( (-1,0) \)[/tex] and [tex]\( (-5,0) \)[/tex].
- Finding the [tex]\(y\)[/tex]-intercept:
Set [tex]\( x = 0 \)[/tex]:
[tex]\[ f(0) = -6(0+1)(0+5) = -6(1)(5) = -30 \][/tex]
Hence, the [tex]\(y\)[/tex]-intercept is at [tex]\( (0, -30) \)[/tex].
Both the [tex]\(x\)[/tex]-intercepts and the [tex]\(y\)[/tex]-intercept match the given criteria.
2. Option 2: [tex]\( f(x) = -6(x-1)(x-5) \)[/tex]
- Finding the [tex]\(x\)[/tex]-intercepts:
Set [tex]\( f(x) = 0 \)[/tex]:
[tex]\[ -6(x-1)(x-5) = 0 \][/tex]
This equation is zero when either [tex]\( x-1 = 0 \)[/tex] or [tex]\( x-5 = 0 \)[/tex]:
[tex]\[ x = 1 \quad \text{or} \quad x = 5 \][/tex]
Hence, the [tex]\(x\)[/tex]-intercepts are at [tex]\( (1,0) \)[/tex] and [tex]\( (5,0) \)[/tex], which does not match the given criteria.
Since the [tex]\(x\)[/tex]-intercepts do not match, there is no need to check the [tex]\(y\)[/tex]-intercept.
3. Option 3: [tex]\( f(x) = -5(x+1)(x+5) \)[/tex]
- Finding the [tex]\(x\)[/tex]-intercepts:
Set [tex]\( f(x) = 0 \)[/tex]:
[tex]\[ -5(x+1)(x+5) = 0 \][/tex]
This equation is zero when either [tex]\( x+1 = 0 \)[/tex] or [tex]\( x+5 = 0 \)[/tex]:
[tex]\[ x = -1 \quad \text{or} \quad x = -5 \][/tex]
Hence, the [tex]\(x\)[/tex]-intercepts are at [tex]\( (-1,0) \)[/tex] and [tex]\( (-5,0) \)[/tex].
- Finding the [tex]\(y\)[/tex]-intercept:
Set [tex]\( x = 0 \)[/tex]:
[tex]\[ f(0) = -5(0+1)(0+5) = -5(1)(5) = -25 \][/tex]
Hence, the [tex]\(y\)[/tex]-intercept is at [tex]\( (0, -25) \)[/tex], which does not match the given criteria.
4. Option 4: [tex]\( f(x) = -5(x-1)(x-5) \)[/tex]
- Finding the [tex]\(x\)[/tex]-intercepts:
Set [tex]\( f(x) = 0 \)[/tex]:
[tex]\[ -5(x-1)(x-5) = 0 \][/tex]
This equation is zero when either [tex]\( x-1 = 0 \)[/tex] or [tex]\( x-5 = 0 \)[/tex]:
[tex]\[ x = 1 \quad \text{or} \quad x = 5 \][/tex]
Hence, the [tex]\(x\)[/tex]-intercepts are at [tex]\( (1,0) \)[/tex] and [tex]\( (5,0) \)[/tex], which does not match the given criteria.
Since the [tex]\(x\)[/tex]-intercepts do not match, there is no need to check the [tex]\(y\)[/tex]-intercept.
Therefore, the correct function that meets all the given criteria for [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]
