https://file.aiquickdraw.com/mathbot/file/1726026489887jc7hi6mj.jpg

Let's break down the problem based on the image.

The problem asks you to:
1. **Find the equation of the secant line** through the points where \( x \) has the given values.
2. **Find the equation of the tangent line** when \( x \) has the first value.

### The function provided is:
\[ f(x) = x^2 + 2x \]

### Given values for \( x \):
- \( x_1 = 3 \)
- \( x_2 = 5 \)

### Step 1: Equation of the Secant Line
The slope of the secant line between two points \( x_1 \) and \( x_2 \) is calculated by:
\[
m_{\text{secant}} = \frac{f(x_2) - f(x_1)}{x_2 - x_1}
\]

First, let's calculate the function values at these points:
- \( f(3) = 3^2 + 2(3) = 9 + 6 = 15 \)
- \( f(5) = 5^2 + 2(5) = 25 + 10 = 35 \)

Now, the slope of the secant line:
\[
m_{\text{secant}} = \frac{35 - 15}{5 - 3} = \frac{20}{2} = 10
\]

The equation of the secant line is:
\[
y - f(x_1) = m_{\text{secant}}(x - x_1)
\]
Substitute \( m_{\text{secant}} = 10 \), \( f(x_1) = 15 \), and \( x_1 = 3 \):
\[
y - 15 = 10(x - 3)
\]
Simplifying:
\[
y = 10x - 30 + 15 = 10x - 15
\]

### Step 2: Equation of the Tangent Line
The slope of the tangent line is the derivative of \( f(x) \) evaluated at \( x_1 = 3 \).

The derivative of \( f(x) = x^2 + 2x \) is:
\[
f'(x) = 2x + 2
\]

Evaluating at \( x = 3 \):
\[
f'(3) = 2(3) + 2 = 6 + 2 = 8
\]

The equation of the tangent line is:
\[
y - f(x_1) = f'(x_1)(x - x_1)
\]
Substitute \( f'(x_1) = 8 \), \( f(x_1) = 15 \), and \( x_1 = 3 \):
\[
y - 15 = 8(x - 3)
\]
Simplifying:
\[
y = 8x - 24 + 15 = 8x - 9
\]

### Final Results:
1. **Secant line equation**: \( y = 10x - 15 \)
2. **Tangent line equation**: \( y = 8x - 9 \)

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

Here are 5 related questions to explore:
1. How do we find the derivative of a function?
2. Why is the slope of the tangent line different from the secant line?
3. What are the geometric interpretations of secant and tangent lines?
4. How would these equations change if the function were \( f(x) = x^3 + x \)?
5. Can we find the secant line using another pair of points?

**Tip**: The tangent line always represents the instantaneous rate of change of the function at a specific point.

For each function, find (a) the equation of the secant line through the points where x has the given values, and (b) the equation of the tangent line when x has the first value. Function: f(x) = x^2 + 2x; x1 = 3, x2 = 5.

Learn how to find the equation of the secant line through given points and the tangent line at a specific point for the quadratic function f(x) = x^2 + 2x. This guide provides a step-by-step solution, including the use of derivatives and line equations, suitable for Grades 11-12 or early college students.

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