To find the equation of the tangent line to the graph of the function [tex]\( g \)[/tex] at [tex]\( x = -6 \)[/tex], we can use the point-slope form of the equation for a line. The general form of the tangent line equation at a point [tex]\( (a, g(a)) \)[/tex] is given by:
[tex]\[ y - g(a) = g'(a) \cdot (x - a) \][/tex]
Given the information:
- The function value at [tex]\( x = -6 \)[/tex], [tex]\( g(-6) = -8 \)[/tex]
- The derivative at [tex]\( x = -6 \)[/tex], [tex]\( g'(-6) = 3 \)[/tex]
We can substitute [tex]\( a = -6 \)[/tex], [tex]\( g(a) = -8 \)[/tex], and [tex]\( g'(a) = 3 \)[/tex] into the point-slope equation:
[tex]\[ y - (-8) = 3 \cdot (x - (-6)) \][/tex]
Simplifying the equation step by step:
1. First, rewrite the left-hand side:
[tex]\[ y + 8 = 3 \cdot (x + 6) \][/tex]
2. Next, distribute the 3 on the right-hand side:
[tex]\[ y + 8 = 3x + 18 \][/tex]
3. Finally, isolate [tex]\( y \)[/tex] to get it into the slope-intercept form [tex]\( y = mx + b \)[/tex]:
[tex]\[ y = 3x + 18 - 8 \][/tex]
4. Simplify the constants:
[tex]\[ y = 3x + 10 \][/tex]
Therefore, the equation of the tangent line to the graph of [tex]\( g \)[/tex] at [tex]\( x = -6 \)[/tex] is:
[tex]\[ y = 3x + 10 \][/tex]
In this equation, the slope of the tangent line is 3, and the y-intercept is 10.
