a ______________ is a ratio that compares the amount of change in a dependent variable to the amount of…

a ______________ is a ratio that compares the amount of change in a dependent variable to the amount of change in an independent variable. rate of change = \\frac{change in dependent variable}{change in independent variable} example 1: what is the average rate of change of f(x) = x² + 1 between -1,3? example 2: look at the graph of f(x) = x² + 1. what is the average rate of change between -1,3?

a ______________ is a ratio that compares the amount of change in a dependent variable to the amount of change in an independent variable. rate of change = \\frac{change in dependent variable}{change in independent variable} example 1: what is the average rate of change of f(x) = x² + 1 between -1,3? example 2: look at the graph of f(x) = x² + 1. what is the average rate of change between -1,3?

Answer

Explanation:

Step1: Recall the average - rate - of - change formula

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

Step2: Identify $a$, $b$, and the function

Here, $a=-1$, $b = 3$, and $f(x)=x^{2}+1$.

Step3: Calculate $f(a)$ and $f(b)$

First, find $f(-1)$: $f(-1)=(-1)^{2}+1=1 + 1=2$. Then, find $f(3)$: $f(3)=3^{2}+1=9 + 1=10$.

Step4: Calculate the average rate of change

Substitute into the formula $\frac{f(b)-f(a)}{b - a}$: $\frac{f(3)-f(-1)}{3-(-1)}=\frac{10 - 2}{3+1}=\frac{8}{4}=2$.

Answer:

The average rate of change of $f(x)=x^{2}+1$ between $[-1,3]$ is $2$.