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 $x = 2$?
a $y = 2x - 1$
b $y = 4x - 1$
c $y = 4x - 8$
d $y = 5.9x - 4.8$

Explanation:

Step1: Recall tangent - line formula

The equation of the tangent line to the graph of $y = f(x)$ at the point $(x_0,y_0)$ is given by $y - y_0=f^{\prime}(x_0)(x - x_0)$. We are given that the point $(2,7)$ is on the graph of $f$, so $x_0 = 2$ and $y_0=7$. The slope of the tangent line to the graph of $f$ at $x = 2$ is $f^{\prime}(2)$. Since the graph of $f^{\prime}$ is not shown completely in the problem - statement, but we know that the slope of the tangent line of $y = f(x)$ at $x = 2$ can be used to find the equation of the tangent line. The slope - intercept form of a line is $y=mx + b$, where $m$ is the slope and $b$ is the y - intercept. We know that the line passes through the point $(2,7)$.

Step2: Substitute values into point - slope form

Using the point - slope form $y - y_0=m(x - x_0)$ with $x_0 = 2$, $y_0 = 7$. We check each option by substituting $x = 2$ into the equations of the lines.
For option A: When $x = 2$, $y=2\times2−1=4 - 1=3
eq7$.
For option B: When $x = 2$, $y = 4\times2-1=8 - 1 = 7$. The slope of the line $y = 4x-1$ is $m = 4$. The point - slope form of a line passing through $(2,7)$ with slope $m = 4$ is $y - 7=4(x - 2)$. Expanding, we get $y-7=4x-8$, or $y = 4x - 1$.
For option C: When $x = 2$, $y=4\times2−8=8 - 8=0
eq7$.
For option D: When $x = 2$, $y=5.9\times2−4.8=11.8 - 4.8 = 7$. But we can also note that the general form of the tangent line is $y - y_0=f^{\prime}(x_0)(x - x_0)$. If we assume the slope of the tangent line at $x = 2$ is found from the derivative information (and since the most straightforward slope - value check works for option B), and we know that the point $(2,7)$ must satisfy the equation of the tangent line. The slope of the line $y = 4x-1$ is consistent with the fact that if we use the point - slope formula $y - 7=m(x - 2)$ and solve for $m$ when the line passes through $(2,7)$ and has the form $y=mx + b$.

Answer:

B. $y = 4x - 1$