Question

name:
section:
precalculus
1.4 exit ticket: (complete the questions independently and quietly)

1. in your own words, what does the average rate of change tell you about a function?
2. find the average rate of change of f(x)=-2x² + 3x - 1 on 0,2.
3. explain why the average rate of change of a linear function is constant.

Explanation:

1.

The average rate of change of a function over an interval tells you the overall steepness or slope of the function on that interval. It represents the average amount by which the function's output (y - values) changes per unit change in the input (x - values) within that interval.

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 $\frac{f(b)-f(a)}{b - a}$. Here, $a = 0$, $b = 2$, and $f(x)=-2x^{2}+3x - 1$.

Step2: Calculate $f(0)$

Substitute $x = 0$ into $f(x)$: $f(0)=-2(0)^{2}+3(0)-1=-1$.

Step3: Calculate $f(2)$

Substitute $x = 2$ into $f(x)$: $f(2)=-2(2)^{2}+3(2)-1=-2\times4 + 6-1=-8 + 6-1=-3$.

Step4: Calculate the average rate of change

Using the formula $\frac{f(2)-f(0)}{2 - 0}$, we substitute $f(0)=-1$ and $f(2)=-3$: $\frac{-3-(-1)}{2}=\frac{-3 + 1}{2}=\frac{-2}{2}=-1$.

A linear function has the form $y=mx + b$, where $m$ is the slope and $b$ is the y - intercept. The average rate of change of a function over an interval $[a,b]$ is $\frac{f(b)-f(a)}{b - a}$. For $y=mx + b$, $f(a)=ma + b$ and $f(b)=mb + b$. Then $\frac{f(b)-f(a)}{b - a}=\frac{(mb + b)-(ma + b)}{b - a}=\frac{mb - ma}{b - a}=\frac{m(b - a)}{b - a}=m$. Since $m$ is a constant, the average rate of change of a linear function is constant.

Answer:

The average rate of change of a function over an interval gives the average slope of the function on that interval, showing how much the function's value changes on average for each unit change in the input variable.

2.