Question

find an equation of the tangent line to the graph of $y = g(x)$ at $x = 5$ if $g(5) = -4$ and $g(5) = 2$. (enter your answer as an equation in terms of $y$ and $x$.)

Explanation:

Step1: Recall the point - slope form of a line

The point - slope form of a line is given by $y - y_1=m(x - x_1)$, where $(x_1,y_1)$ is a point on the line and $m$ is the slope of the line.

For the tangent line to the graph of $y = g(x)$ at $x = 5$:

• The point on the tangent line: When $x = 5$, $y=g(5)$. We know that $g(5)=- 4$, so the point $(x_1,y_1)=(5,-4)$.
• The slope of the tangent line: The slope of the tangent line to the graph of $y = g(x)$ at $x = a$ is given by $g^{\prime}(a)$. Here, $a = 5$, and $g^{\prime}(5)=2$, so the slope $m = 2$.

Step2: Substitute into the point - slope form

Substitute $x_1 = 5$, $y_1=-4$, and $m = 2$ into the point - slope form $y - y_1=m(x - x_1)$:
\[

$$\begin{align*} y-(-4)&=2(x - 5)\\ y + 4&=2x-10 \end{align*}$$

\]
Then, we can rewrite it in the slope - intercept form (or just simplify) by subtracting 4 from both sides:
\[
y=2x-10 - 4\\
y=2x-14
\]
Or we can leave it in the form $y + 4=2(x - 5)$ and expand it to the general form $2x-y-14 = 0$, but the most common form for a line equation (in terms of $y$ and $x$) is the slope - intercept form or the point - slope expanded form. The expanded form from point - slope is $y=2x - 14$.

Answer:

$y = 2x-14$ (or $y+4 = 2(x - 5)$ or $2x-y-14=0$, but the slope - intercept form $y = 2x-14$ is a standard answer)