Question

what is the average rate of change between:
x = 1 and x = 2?
x = 2 and x = 3?
x = 3 and x = 4?
done

Explanation:

Step1: Recall the formula for average rate of change

The average rate of change of a function \(y = f(x)\) over the interval \([a,b]\) is \(\frac{f(b)-f(a)}{b - a}\)

Step2: Find \(y\) - values from the graph

From the graph:

• When \(x = 1\), \(y=2\); when \(x = 2\), \(y = 4\)
• When \(x = 2\), \(y = 4\); when \(x = 3\), \(y=8\)
• When \(x = 3\), \(y = 8\); when \(x = 4\), \(y=16\)

Step3: Calculate average rate of change for \(x = 1\) and \(x = 2\)

\(a = 1\), \(b=2\), \(f(1)=2\), \(f(2)=4\)
\(\frac{f(2)-f(1)}{2 - 1}=\frac{4 - 2}{1}=2\)

Step4: Calculate average rate of change for \(x = 2\) and \(x = 3\)

\(a = 2\), \(b = 3\), \(f(2)=4\), \(f(3)=8\)
\(\frac{f(3)-f(2)}{3 - 2}=\frac{8 - 4}{1}=4\)

Step5: Calculate average rate of change for \(x = 3\) and \(x = 4\)

\(a=3\), \(b = 4\), \(f(3)=8\), \(f(4)=16\)
\(\frac{f(4)-f(3)}{4 - 3}=\frac{16 - 8}{1}=8\)

Answer:

For \(x = 1\) and \(x = 2\): \(2\)
For \(x = 2\) and \(x = 3\): \(4\)
For \(x = 3\) and \(x = 4\): \(8\)