https://file.aiquickdraw.com/mathbot/file/17274680185228ry0g07z.jpg

Let's work through this problem step-by-step.

We are given the function:

\[
f(x) = 2x^2 - 4x + 9
\]

We are tasked with finding:

1. The slope of the tangent line to the graph of \( f(x) \) at the point \( (3, 15) \).
2. The equation of the tangent line in the form \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept.

### Step 1: Find the slope of the tangent line

The slope of the tangent line is the derivative of \( f(x) \) evaluated at \( x = 3 \). To find the derivative \( f'(x) \), we differentiate \( f(x) = 2x^2 - 4x + 9 \):

\[
f'(x) = \frac{d}{dx}(2x^2) - \frac{d}{dx}(4x) + \frac{d}{dx}(9)
\]

\[
f'(x) = 4x - 4
\]

Now, evaluate \( f'(x) \) at \( x = 3 \):

\[
f'(3) = 4(3) - 4 = 12 - 4 = 8
\]

Thus, the slope of the tangent line at \( (3, 15) \) is **8**.

### Step 2: Find the equation of the tangent line

The equation of a line is given by:

\[
y = mx + b
\]

where \( m \) is the slope, and \( b \) is the y-intercept. We already know that \( m = 8 \), and we can use the point \( (3, 15) \) to find \( b \).

Substitute \( x = 3 \), \( y = 15 \), and \( m = 8 \) into the equation:

\[
15 = 8(3) + b
\]

\[
15 = 24 + b
\]

\[
b = 15 - 24 = -9
\]

Thus, the equation of the tangent line is:

\[
y = 8x - 9
\]

### Final Answers:
- The slope of the tangent line is **8**.
- The equation of the tangent line is \( y = 8x - 9 \).

Would you like further clarification or more details? Here are five related questions to deepen your understanding:

1. What is the geometric interpretation of the tangent line in this context?
2. How would the equation change if the point of tangency was at \( x = 2 \) instead?
3. Can you verify the value of \( f(3) \) to ensure the point \( (3, 15) \) lies on the graph of \( f(x) \)?
4. How does the second derivative \( f''(x) \) help describe the curvature of the graph near \( x = 3 \)?
5. How would you use the same process to find the tangent line for a cubic function like \( f(x) = x^3 - 2x + 1 \)?

**Tip:** The derivative gives the slope of the tangent line at any point on the curve, which is key in analyzing how the function behaves locally.

Find the slope of the tangent line and the equation of the tangent line to the graph of f(x) = 2x^2 - 4x + 9 at the point (3, 15).

Learn how to find the slope of the tangent line and the equation of the tangent line to the graph of f(x) = 2x^2 - 4x + 9 at the point (3, 15). The problem involves calculus concepts such as derivatives and the equation of a line.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation