Question

the derivative of the function f is given by f(x)=-2x + 4 for all x, and f(-1)=5. which of the following is an equation for the line tangent to the graph of f at x=-1?
a y=-2x + 3
b y=-2x + 4
c y=6x + 5
d y=6x + 11

Explanation:

Step1: Find the slope of the tangent line

The slope of the tangent line to the graph of $y = f(x)$ at $x = a$ is given by $f^{\prime}(a)$. Here, $a=-1$ and $f^{\prime}(x)=-2x + 4$. Substitute $x=-1$ into $f^{\prime}(x)$:
$f^{\prime}(-1)=-2\times(-1)+4=2 + 4=6$.

Step2: Use the point - slope form of a line

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. We know that $x_1=-1$ and $y_1 = f(-1)=5$, and $m = 6$.
Substitute these values into the point - slope form: $y-5=6(x + 1)$.

Step3: Simplify the equation

Expand the right - hand side: $y-5=6x+6$.
Add 5 to both sides to get the equation in slope - intercept form $y=6x + 11$.

Answer:

D. $y = 6x+11$