Find an equation of the tangent line to the curve at the given point: f(x) = -2x²+1 at (2, -7)

To find the equation of the tangent line to the curve \( f(x) = -2x^2 + 1 \) at the point \( (2, -7) \), we need to follow these steps:

1. **Find the derivative of \( f(x) \)**: This represents the slope of the tangent line at any point on the curve.
2. **Evaluate the derivative at the given point**: This gives the slope of the tangent line at \( x = 2 \).
3. **Use the point-slope form of the equation of a line**: With the slope and a point on the tangent line, we can find the equation.

### Step 1: Find the Derivative

The function given is:
\[
f(x) = -2x^2 + 1
\]

The derivative \( f'(x) \) is found using the power rule:
\[
f'(x) = \frac{d}{dx}(-2x^2 + 1) = -4x
\]

### Step 2: Evaluate the Derivative at \( x = 2 \)

Now, plug \( x = 2 \) into the derivative to find the slope of the tangent line at this point:
\[
f'(2) = -4(2) = -8
\]

The slope of the tangent line at \( x = 2 \) is \(-8\).

### Step 3: Use the Point-Slope Form

The point-slope form of a line equation is:
\[
y - y_1 = m(x - x_1)
\]

Here, \( m = -8 \), \( x_1 = 2 \), and \( y_1 = -7 \). Plugging in these values gives:
\[
y - (-7) = -8(x - 2)
\]

Simplify the equation:
\[
y + 7 = -8x + 16
\]

Subtract 7 from both sides:
\[
y = -8x + 9
\]

### Conclusion

The equation of the tangent line to the curve \( f(x) = -2x^2 + 1 \) at the point \( (2, -7) \) is:
\[
\boxed{y = -8x + 9}
\]

Would you like more details on any step, or do you have any questions?

Here are some related questions you might find useful:
1. How do you find the derivative of a polynomial function?
2. What is the power rule in calculus?
3. How do you apply the point-slope form to find the equation of a line?
4. What is the significance of the slope of the tangent line at a point on a curve?
5. How do you verify that a point lies on the tangent line to a curve?

**Tip:** Remember that the slope of the tangent line represents the instantaneous rate of change of the function at a specific point.

Learn how to find the equation of the tangent line to the curve f(x) = -2x² + 1 at the point (2, -7) using calculus. This problem involves calculating derivatives, applying the point-slope form of a line equation, and understanding tangent lines in calculus. Suitable for students in Grades 11-12.

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