Let ​f(x)equals3 x minus 6.
​(a)  Find the average rate of change from 2 to 8.
​(b)  Find an equation of the secant line containing left parenthesis 2 comma f left parenthesis 2 right parenthesis right parenthesis and left parenthesis 8 comma f left parenthesis 8 right parenthesis right parenthesis .

We are given the function \( f(x) = 3x - 6 \), and we need to:

### (a) Find the average rate of change from \( x = 2 \) to \( x = 8 \).

The formula for the average rate of change of a function between two points \( x_1 \) and \( x_2 \) is given by:

\[
\text{Average rate of change} = \frac{f(x_2) - f(x_1)}{x_2 - x_1}
\]

Here, \( x_1 = 2 \) and \( x_2 = 8 \). So, we first compute \( f(2) \) and \( f(8) \).

- \( f(2) = 3(2) - 6 = 6 - 6 = 0 \)
- \( f(8) = 3(8) - 6 = 24 - 6 = 18 \)

Now we substitute into the average rate of change formula:

\[
\text{Average rate of change} = \frac{f(8) - f(2)}{8 - 2} = \frac{18 - 0}{8 - 2} = \frac{18}{6} = 3
\]

So, the average rate of change from \( x = 2 \) to \( x = 8 \) is **3**.

---

### (b) Find an equation of the secant line containing the points \( (2, f(2)) \) and \( (8, f(8)) \).

The slope \( m \) of the secant line is the same as the average rate of change, which we found to be 3. 

The equation of a line is given by the point-slope form:

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

We can use the point \( (2, f(2)) = (2, 0) \) and the slope \( m = 3 \) to write the equation of the secant line:

\[
y - 0 = 3(x - 2)
\]

Simplifying:

\[
y = 3(x - 2) = 3x - 6
\]

Thus, the equation of the secant line is:

\[
y = 3x - 6
\]

### Summary:

- (a) The average rate of change from \( x = 2 \) to \( x = 8 \) is **3**.
- (b) The equation of the secant line is \( y = 3x - 6 \).

Would you like more details on any of these steps or have any questions?

---

Here are 5 related questions to extend this concept:
1. How is the average rate of change related to the derivative of the function?
2. What would be the slope of the tangent line at \( x = 2 \) for the function \( f(x) = 3x - 6 \)?
3. How does the secant line differ from the tangent line geometrically?
4. What is the average rate of change of \( f(x) = 3x - 6 \) over the interval \( [4, 10] \)?
5. How do you find the equation of a line given two arbitrary points?

**Tip**: The average rate of change of a linear function is always the same, as it represents the constant slope of the line.

Let f(x) = 3x - 6. (a) Find the average rate of change from 2 to 8. (b) Find an equation of the secant line containing the points (2, f(2)) and (8, f(8)).

Explore how to calculate the average rate of change for the linear function f(x) = 3x - 6 from x = 2 to x = 8, and how to find the equation of the secant line through points (2, f(2)) and (8, f(8)). This problem covers core algebra concepts like linear functions and rate of change, suitable for students in Grades 9-11.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation