https://file.aiquickdraw.com/mathbot/file/1727054706614apbk84pt.jpg

We are given the function \( f(x) = x^2 - 4 \), and we need to find the equation of the tangent line at \( x = -1 \).

### Steps:

1. **Find the derivative**: 
   The derivative of \( f(x) = x^2 - 4 \) will give us the slope of the tangent line at any point. The derivative is:
   \[
   f'(x) = 2x
   \]

2. **Evaluate the slope at \( x = -1 \)**: 
   Substitute \( x = -1 \) into \( f'(x) \) to find the slope at that point:
   \[
   f'(-1) = 2(-1) = -2
   \]
   So, the slope of the tangent line at \( x = -1 \) is \( -2 \).

3. **Find the point on the curve at \( x = -1 \)**:
   Substitute \( x = -1 \) into the original function \( f(x) \) to find the y-coordinate:
   \[
   f(-1) = (-1)^2 - 4 = 1 - 4 = -3
   \]
   So, the point on the curve is \( (-1, -3) \).

4. **Equation of the tangent line**:
   Now, using the point-slope form of a line equation \( y - y_1 = m(x - x_1) \), where \( m \) is the slope and \( (x_1, y_1) \) is the point, we have:
   \[
   y - (-3) = -2(x - (-1))
   \]
   Simplifying:
   \[
   y + 3 = -2(x + 1)
   \]
   \[
   y + 3 = -2x - 2
   \]
   \[
   y = -2x - 5
   \]
   Therefore, the equation of the tangent line at \( x = -1 \) is:
   \[
   y = -2x - 5
   \]

Let me know if you would like further details or have any questions.

### Related questions:
1. How do you find the tangent line of a function at any arbitrary point?
2. What does the derivative represent geometrically in the context of this problem?
3. Can the tangent line ever intersect the curve at more than one point?
4. How do you find the normal line to a curve at a given point?
5. How does the concavity of the function affect the shape of the tangent line?

### Tip:
Always remember that the derivative at a point gives the slope of the tangent line to the curve at that point!

For the function f(x) = x^2 - 4, find the equation of the tangent line at x = -1.

Learn how to find the equation of the tangent line to the quadratic function f(x) = x^2 - 4 at x = -1. This step-by-step guide explains how to differentiate the function and use the point-slope form to find the tangent line. Suitable for high school calculus 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