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For a function [tex]g[/tex], we are given that [tex]g(-6)=-8[/tex] and [tex]g^{\prime}(-6)=3[/tex].

What is the equation of the tangent line to the graph of [tex]g[/tex] at [tex]x=-6[/tex]?

Sagot :

GinnyAnsweridn
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.
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