Find an equation of the tangent line to the parabola 𝑓(𝑥) = x^2+2x+2 a point (2,2)

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

### Step 1: Verify that the point lies on the curve
Substitute \( x = 2 \) into the function \( f(x) \) to check if \( f(2) = 2 \).

\[
f(2) = (2)^2 + 2(2) + 2 = 4 + 4 + 2 = 10
\]

But the given point is \( (2, 2) \), and we calculated \( f(2) = 10 \), not 2. This means the point \( (2, 2) \) is **not** on the parabola \( f(x) = x^2 + 2x + 2 \), so there is no tangent line at this point.

### Alternative Approach
Since there was likely a miscommunication with the given point, I suggest verifying the point or function again. If you want to work through this problem with a different point on the curve, feel free to provide more details.

Do you have any questions or want to clarify the problem? 

Here are 5 related questions:
1. How do you find the slope of a tangent line to a curve at a given point?
2. What happens if the point does not lie on the curve?
3. How can derivatives help in determining tangent lines?
4. Can a point outside the curve have a unique tangent line?
5. How can we generalize the formula for the equation of a tangent line?

**Tip:** When verifying whether a point is on a curve, always substitute the \(x\)-coordinate into the equation and check if the \(y\)-value matches.

Find an equation of the tangent line to the parabola 𝑓(𝑥) = x^2+2x+2 at the point (2,2).

Explore how to find the equation of the tangent line to the parabola f(x) = x^2 + 2x + 2. This solution checks whether the point (2,2) lies on the curve, applies differentiation to find the slope, and reviews the tangent line concept for quadratic functions. Suitable for high school 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