https://file.aiquickdraw.com/mathbot/file/1726147465788bt645hdl.jpg

What is the average rate of change of y between x = 0 and x = 6?

The equation provided on the graph is \( y = -x^2 + 6x \).

To find the **average rate of change** of \( y \) between \( x = 0 \) and \( x = 6 \), we will use the formula for the average rate of change:

\[
\text{Average Rate of Change} = \frac{y_2 - y_1}{x_2 - x_1}
\]

Where:
- \( x_1 = 0 \), \( x_2 = 6 \)
- \( y_1 = f(x_1) = f(0) \)
- \( y_2 = f(x_2) = f(6) \)

Let's calculate this step by step.

### Step 1: Calculate \( y_1 = f(0) \)

\[
f(0) = -(0)^2 + 6(0) = 0
\]

### Step 2: Calculate \( y_2 = f(6) \)

\[
f(6) = -(6)^2 + 6(6) = -36 + 36 = 0
\]

### Step 3: Calculate the Average Rate of Change

\[
\text{Average Rate of Change} = \frac{0 - 0}{6 - 0} = \frac{0}{6} = 0
\]

Thus, the **average rate of change** of \( y \) between \( x = 0 \) and \( x = 6 \) is **0**.

Would you like further details or clarification?

### Related questions:
1. How is the average rate of change related to the slope of the secant line?
2. What is the significance of having an average rate of change of zero?
3. Can we calculate the instantaneous rate of change at a specific point using derivatives?
4. What does the graph of \( y = -x^2 + 6x \) tell us about its vertex and symmetry?
5. How would the average rate of change change if the interval was between \( x = 0 \) and \( x = 3 \)?

### Tip:
The average rate of change gives you the slope of the secant line between two points on a curve. A zero slope indicates the function has the same value at both points, often related to symmetry or a maximum/minimum in certain cases.

Learn how to calculate the average rate of change for the quadratic equation y = -x^2 + 6x between x = 0 and x = 6. This algebra problem involves understanding rate of change concepts and quadratic equations. Suitable for students in Grades 9-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation