To determine which linear function represents the line given by the point-slope equation [tex]\( y + 1 = -3(x - 5) \)[/tex], we need to first convert this equation into the slope-intercept form [tex]\( y = mx + b \)[/tex], where [tex]\( m \)[/tex] is the slope and [tex]\( b \)[/tex] is the y-intercept.
Here’s a detailed step-by-step process to convert the given equation into slope-intercept form:
1. Start with the given point-slope equation:
[tex]\[
y + 1 = -3(x - 5)
\][/tex]
2. Distribute -3 on the right-hand side:
[tex]\[
y + 1 = -3 \cdot x + (-3) \cdot (-5)
\][/tex]
[tex]\[
y + 1 = -3x + 15
\][/tex]
3. Isolate [tex]\( y \)[/tex] by subtracting 1 from both sides:
[tex]\[
y = -3x + 15 - 1
\][/tex]
[tex]\[
y = -3x + 14
\][/tex]
The slope-intercept form of the equation is therefore:
[tex]\[
y = -3x + 14
\][/tex]
Now that we have the linear function in slope-intercept form, we can match it with one of the given options:
- [tex]\(f(x) = -3x - 6\)[/tex]
- [tex]\(f(x) = -3x - 4\)[/tex]
- [tex]\(f(x) = -3x + 16\)[/tex]
- [tex]\(f(x) = -3x + 14\)[/tex]
The function that matches [tex]\( y = -3x + 14 \)[/tex] is:
[tex]\[
f(x) = -3x + 14
\][/tex]
Therefore, the correct linear function representing the equation [tex]\( y + 1 = -3(x - 5) \)[/tex] is [tex]\( \boxed{4} \)[/tex].
