Question

the graph of $f$, the derivative of a function $f$, is shown above. the points $(2,7)$ and $(4,18.8)$ are on the graph of $f$. which of the following is an equation for the line tangent to the graph of $f$ at $x = 2$?

Explanation:

Step1: Recall tangent - line formula

The equation of a tangent line to the graph of a function $y = f(x)$ at the point $(x_0,y_0)$ is given by $y - y_0=f^{\prime}(x_0)(x - x_0)$.

Step2: Identify $x_0$, $y_0$ and $f^{\prime}(x_0)$

We are given that $x_0 = 2$. Since the point $(2,7)$ is on the graph of $f$, then $y_0=7$. To find the slope of the tangent line, we need to evaluate $f^{\prime}(2)$. Looking at the graph of $f^{\prime}$, we find the $y$-value of $f^{\prime}$ at $x = 2$. Let's assume from the graph that $f^{\prime}(2)=m$.

Step3: Write the tangent - line equation

Substitute $x_0 = 2$, $y_0 = 7$ and $m=f^{\prime}(2)$ into the point - slope form $y - y_0=m(x - x_0)$. We get $y-7=f^{\prime}(2)(x - 2)$.

Answer:

$y-7=f^{\prime}(2)(x - 2)$