Question

synthesis
how can you calculate the average rate of change
for an interval of a function?
use this graph if it helps to show your thinking

Explanation:

Step1: Recall the formula for average rate of change

The formula for 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: Identify two points on the interval

Let's say we want to find the average rate of change from \(x = 1.5\) to \(x=3.25\). Here, \(a = 1.5\), \(f(a)=75\), \(b = 3.25\), \(f(b)=130\).

Step3: Substitute into the formula

Substitute the values into \(\frac{f(b)-f(a)}{b - a}\), we get \(\frac{130 - 75}{3.25-1.5}=\frac{55}{1.75}=\frac{55\times4}{1.75\times4}=\frac{220}{7}\approx31.43\) (miles per hour).

Answer:

The average rate of change of a function \(y = f(x)\) over the interval \([a,b]\) is calculated using the formula \(\frac{f(b)-f(a)}{b - a}\). For example, for the points \((1.5,75)\) and \((3.25,130)\) on the given distance - time graph (where \(y\) represents distance traveled and \(x\) represents time), the average rate of change (which is the average speed in this context) is \(\frac{130 - 75}{3.25-1.5}=\frac{55}{1.75}\approx31.43\) units (miles per hour).