Answer :
To determine which linear function represents the line given by the point-slope equation \( y + 1 = -3(x - 5) \), we will follow these steps:
1. Distribute the constant on the right-hand side:
Start with the given equation:
[tex]\[ y + 1 = -3(x - 5) \][/tex]
Distribute the \(-3\) through \( (x - 5) \):
[tex]\[ y + 1 = -3x + 15 \][/tex]
2. Isolate \( y \) to convert the equation into slope-intercept form (\(y = mx + b\)):
Subtract 1 from both sides to isolate \( y \):
[tex]\[ y = -3x + 15 - 1 \][/tex]
Simplify the equation:
[tex]\[ y = -3x + 14 \][/tex]
3. Identify the linear function that matches the simplified equation \( y = -3x + 14 \):
Compare this with the given options:
- \( f(x) = -3x - 6 \)
- \( f(x) = -3x - 4 \)
- \( f(x) = -3x + 16 \)
- \( f(x) = -3x + 14 \)
The correct function that matches \( y = -3x + 14 \) is:
[tex]\[ f(x) = -3x + 14 \][/tex]
Thus, the linear function that represents the given point-slope equation is:
[tex]\[ \boxed{f(x) = -3x + 14} \][/tex]
1. Distribute the constant on the right-hand side:
Start with the given equation:
[tex]\[ y + 1 = -3(x - 5) \][/tex]
Distribute the \(-3\) through \( (x - 5) \):
[tex]\[ y + 1 = -3x + 15 \][/tex]
2. Isolate \( y \) to convert the equation into slope-intercept form (\(y = mx + b\)):
Subtract 1 from both sides to isolate \( y \):
[tex]\[ y = -3x + 15 - 1 \][/tex]
Simplify the equation:
[tex]\[ y = -3x + 14 \][/tex]
3. Identify the linear function that matches the simplified equation \( y = -3x + 14 \):
Compare this with the given options:
- \( f(x) = -3x - 6 \)
- \( f(x) = -3x - 4 \)
- \( f(x) = -3x + 16 \)
- \( f(x) = -3x + 14 \)
The correct function that matches \( y = -3x + 14 \) is:
[tex]\[ f(x) = -3x + 14 \][/tex]
Thus, the linear function that represents the given point-slope equation is:
[tex]\[ \boxed{f(x) = -3x + 14} \][/tex]
