Question

let $f(-3)=6$ and $f(-3)= - 2$. then the equation of the tangent line to the graph of $y = f(x)$ at $x=-3$ is $y=$

Explanation:

Step1: Recall the point - slope form

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

Step2: Identify the point and slope

Given $f(-3)=6$, so the point $(x_1,y_1)=(-3,6)$. Given $f'(-3)=-2$, and the slope of the tangent line to $y = f(x)$ at $x = a$ is $m=f'(a)$, so $m=-2$.

Step3: Substitute into the point - slope form

Substitute $x_1=-3$, $y_1 = 6$ and $m=-2$ into $y - y_1=m(x - x_1)$. We get $y-6=-2(x + 3)$.

Step4: Simplify the equation

Expand the right - hand side: $y-6=-2x-6$. Then add 6 to both sides to get $y=-2x$.

Answer:

$y=-2x$