To find the equation of the tangent line to the graph of the function [tex]\( h \)[/tex] at the point where [tex]\( x = -2 \)[/tex], we follow these steps:
1. Identify the point on the function:
The problem states that [tex]\( h(-2) = -5 \)[/tex]. This means that the point of tangency on the graph is [tex]\( (-2, -5) \)[/tex].
2. Determine the slope of the tangent line:
The slope of the tangent line at [tex]\( x = -2 \)[/tex] is given by the derivative of the function at that point. The problem tells us that [tex]\( h'(-2) = -9 \)[/tex]. Therefore, the slope of the tangent line is [tex]\( -9 \)[/tex].
3. Use the point-slope form of the line:
The point-slope form of the equation of a line is:
[tex]\[
y - y_1 = m(x - x_1)
\][/tex]
where [tex]\((x_1, y_1)\)[/tex] is a point on the line and [tex]\( m \)[/tex] is the slope. Plugging in the given point [tex]\((-2, -5)\)[/tex] and the slope [tex]\(-9\)[/tex], we have:
[tex]\[
y - (-5) = -9(x - (-2))
\][/tex]
Simplify this equation:
[tex]\[
y + 5 = -9(x + 2)
\][/tex]
4. Expand and simplify the equation:
Now, distribute the slope on the right-hand side:
[tex]\[
y + 5 = -9x - 18
\][/tex]
5. Isolate [tex]\( y \)[/tex] to obtain the equation in slope-intercept form:
Subtract 5 from both sides to isolate [tex]\( y \)[/tex]:
[tex]\[
y = -9x - 18 - 5
\][/tex]
[tex]\[
y = -9x - 23
\][/tex]
Therefore, the equation of the tangent line to the graph of [tex]\( h \)[/tex] at [tex]\( x = -2 \)[/tex] is:
[tex]\[
y = -9x - 23
\][/tex]
