https://file.aiquickdraw.com/mathbot/file/1726147825243nozgeogp.jpg

What is the average rate of change of y between x = 0 and x = 1?

The equation shown in the graph is \( y = -x^2 + 6x \). To find the **average rate of change** of \( y \) between \( x = 0 \) and \( x = 1 \), we can use the formula:

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

where \( x_1 = 0 \) and \( x_2 = 1 \).

First, let's find the values of \( y \) at \( x = 0 \) and \( x = 1 \):

1. For \( x = 0 \):
\[
y(0) = -(0)^2 + 6(0) = 0
\]

2. For \( x = 1 \):
\[
y(1) = -(1)^2 + 6(1) = -1 + 6 = 5
\]

Now, using the average rate of change formula:

\[
\text{Average rate of change} = \frac{y(1) - y(0)}{1 - 0} = \frac{5 - 0}{1 - 0} = 5
\]

Thus, the average rate of change of \( y \) between \( x = 0 \) and \( x = 1 \) is \( 5 \).

Would you like more details or have any questions? Here are some related questions you might find interesting:

1. What is the instantaneous rate of change of \( y \) at \( x = 1 \)?
2. How would you find the vertex of the parabola \( y = -x^2 + 6x \)?
3. What is the significance of the sign of the rate of change?
4. How do you determine the concavity of the graph from the equation?
5. Can you find the x-intercepts of the parabola?

**Tip**: The average rate of change of a function over an interval is the slope of the secant line that connects the points on the graph corresponding to the endpoints of the interval.

Learn how to calculate the average rate of change for the quadratic function y = -x^2 + 6x between x = 0 and x = 1. This problem covers key algebraic concepts, including quadratic functions and the secant line slope theorem. Suitable for students in Grades 8-10.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation