To find the equation of the tangent line to the graph of $y = f(x)$ at $x = -1$, we need to use the point-slope form of the equation of a line:

$y - y_1 = m(x - x_1)$

where:

• $(x_1, y_1)$ is a point on the line,
• $m$ is the slope of the line.

Step 1: Identify the point on the line

From the problem, we know:

$f(-1) = -7$

So, the point $(x_1, y_1)$ is $(-1, -7)$.

Step 2: Determine the slope of the tangent line

The slope of the tangent line at $x = -1$ is given by the derivative of the function at that point:

$f'(-1) = 6$

Step 3: Write the equation of the tangent line

Substituting $m = 6$, $x_1 = -1$, and $y_1 = -7$ into the point-slope form:

$y - (-7) = 6(x - (-1))$

Simplify this equation:

$y + 7 = 6(x + 1)$

Expanding further:

$y + 7 = 6x + 6$

$y = 6x + 6 - 7$

$y = 6x - 1$

Final Answer

The equation of the tangent line to the graph of $y = f(x)$ at $x = -1$ is:

$\boxed{y = 6x - 1}$

Would you like to see more details or have any questions?

Related Questions:

1. How do you find the derivative of a function at a specific point?
2. What is the significance of the derivative in determining the slope of a tangent line?
3. How do you rewrite a linear equation in standard form?
4. How can you verify that a line is tangent to a curve at a given point?
5. How does changing the slope affect the steepness of the tangent line?

Tip:

Always double-check the values of the function and its derivative at the given point to ensure accuracy in tangent line calculations!