Question

find an equation for the tangent line to the graph of the given function at (-4,7). f(x)=x^{2}-9 find an equation for the tangent line to the graph of f(x)=x^{2}-9 at (-4,7). y = (type an expression using x as the variable.)

Explanation:

Step1: Find the derivative of the function

The derivative of $f(x)=x^{2}-9$ using the power - rule $(x^n)' = nx^{n - 1}$ is $f'(x)=2x$.

Step2: Calculate the slope of the tangent line

Substitute $x=-4$ into $f'(x)$. So, $m = f'(-4)=2\times(-4)=-8$.

Step3: Use the point - slope form of a line

The point - slope form is $y - y_1=m(x - x_1)$, where $(x_1,y_1)=(-4,7)$ and $m=-8$.
$y - 7=-8(x + 4)$.

Step4: Simplify the equation

$y-7=-8x-32$.
$y=-8x - 32 + 7$.
$y=-8x-25$.

Answer:

$y=-8x - 25$