Find an equation of the tangent line to the graph of f at the given point.
f(x) = x3 + 8,    (−2, 0)

To find the equation of the tangent line to the graph of \( f(x) = x^3 + 8 \) at the point \((-2, 0)\), we need to follow these steps:

### 1. **Find the derivative of \( f(x) \):**

The derivative \( f'(x) \) will give us the slope of the tangent line at any point on the curve.

\[
f(x) = x^3 + 8
\]

Differentiate with respect to \( x \):

\[
f'(x) = 3x^2
\]

### 2. **Find the slope at \( x = -2 \):**

To get the slope of the tangent line at the point \( (-2, 0) \), substitute \( x = -2 \) into \( f'(x) \):

\[
f'(-2) = 3(-2)^2 = 3 \times 4 = 12
\]

So, the slope of the tangent line at \( (-2, 0) \) is 12.

### 3. **Use the point-slope form of the equation of a line:**

The point-slope form of a line is:

\[
y - y_1 = m(x - x_1)
\]

where \( m \) is the slope, and \( (x_1, y_1) \) is the point on the line. We know:
- \( m = 12 \)
- \( (x_1, y_1) = (-2, 0) \)

Substitute these values into the point-slope form:

\[
y - 0 = 12(x - (-2))
\]

Simplify:

\[
y = 12(x + 2)
\]

### Final Answer:
The equation of the tangent line is:

\[
y = 12(x + 2)
\]

Would you like further details or have any questions?

Here are five related questions:
1. How do you find the derivative of a more complex function?
2. How does the point-slope form differ from other forms of linear equations?
3. What is the significance of the derivative at a point on a curve?
4. How do we confirm that \( (-2, 0) \) lies on the graph of \( f(x) \)?
5. What would happen if we needed the tangent line at a different point?

**Tip:** When finding tangent lines, always verify that the point lies on the curve by substituting the \( x \)-value back into the original function!

Find an equation of the tangent line to the graph of f at the given point. f(x) = x^3 + 8, (−2, 0)

This problem demonstrates how to find the equation of the tangent line to the graph of f(x) = x^3 + 8 at the point (-2, 0). Learn how to compute the derivative of a cubic function, determine the slope at a specific point, and use the point-slope formula to write the equation of the tangent line. Ideal for students in Grades 11-12 studying calculus.

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