Given that

*f*(−1) = −7

and

*f* *'*(−1) = 6,

find an equation for the tangent line to the graph of

*y* = *f*(*x*)

at

*x* = −1.

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!

Given that f(-1) = -7 and f'(-1) = 6, find an equation for the tangent line to the graph of y = f(x) at x = -1.

Discover how to find the equation of the tangent line to the function y = f(x) at x = -1 using differentiation. This problem covers key calculus concepts, including the point-slope form and the derivative as the slope. Ideal for students in Grades 10-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation