y=2x+3, what is the average rate of change on the interval [1,3]

To find the average rate of change of the function \( y = 2x + 3 \) on the interval \([1, 3]\), we use the formula:

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

where:
- \( f(x) = 2x + 3 \)
- \( a = 1 \) and \( b = 3 \)

Now, calculate \( f(3) \) and \( f(1) \):

\[
f(3) = 2(3) + 3 = 6 + 3 = 9
\]
\[
f(1) = 2(1) + 3 = 2 + 3 = 5
\]

Substitute these values into the formula:

\[
\text{Average rate of change} = \frac{9 - 5}{3 - 1} = \frac{4}{2} = 2
\]

So, the average rate of change is \( 2 \).

Would you like further details or have any questions?

Here are some related questions to explore:
1. What is the interpretation of the average rate of change in this context?
2. How does the average rate of change relate to the slope of a line?
3. How would the average rate of change change if the interval was \([2, 5]\)?
4. What would the average rate of change be for a non-linear function like \( y = x^2 \)?
5. How is the concept of average rate of change used in calculus?

**Tip:** The average rate of change of a linear function is constant and equal to the slope of the line.

y = 2x + 3, what is the average rate of change on the interval [1,3]

Explore how to find the average rate of change for the linear function y = 2x + 3 on the interval [1, 3]. This example demonstrates key algebraic concepts such as slope, rates of change, and the average rate of change formula. Ideal for students in Grades 8-10.

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