Question

the graph of y = f(x) is given. show that the average rates of change of f over equal intervals of the domain follow a linear pattern.

Explanation:

Step1: Recall 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: Select equal - length intervals

Let's choose intervals of length $1$. For example, consider the intervals $[-7,-6]$, $[-6,-5]$, $[-5,-4]$, etc.
For the interval $[-7,-6]$: Let $a=-7$ and $b = - 6$. From the graph, if $f(-7)=0$ and $f(-6)=3$, then the average rate of change is $\frac{f(-6)-f(-7)}{-6-(-7)}=\frac{3 - 0}{1}=3$.
For the interval $[-6,-5]$: Let $a=-6$ and $b=-5$. If $f(-6)=3$ and $f(-5)=5$, then the average rate of change is $\frac{f(-5)-f(-6)}{-5-(-6)}=\frac{5 - 3}{1}=2$.
For the interval $[-5,-4]$: Let $a=-5$ and $b=-4$. If $f(-5)=5$ and $f(-4)=6$, then the average rate of change is $\frac{f(-4)-f(-5)}{-4-(-5)}=\frac{6 - 5}{1}=1$.
For the interval $[-4,-3]$: Let $a=-4$ and $b=-3$. If $f(-4)=6$ and $f(-3)=5$, then the average rate of change is $\frac{f(-3)-f(-4)}{-3-(-4)}=\frac{5 - 6}{1}=-1$.
For the interval $[-3,-2]$: Let $a=-3$ and $b=-2$. If $f(-3)=5$ and $f(-2)=3$, then the average rate of change is $\frac{f(-2)-f(-3)}{-2-(-3)}=\frac{3 - 5}{1}=-2$.
For the interval $[-2,-1]$: Let $a=-2$ and $b=-1$. If $f(-2)=3$ and $f(-1)=0$, then the average rate of change is $\frac{f(-1)-f(-2)}{-1-(-2)}=\frac{0 - 3}{1}=-3$.

Step3: Analyze the pattern

The average rates of change we found are $3,2,1,-1,-2,-3$. If we consider these values as $y$ - values and the mid - points of the intervals as $x$ - values, we can see that they follow a linear pattern. For example, if we plot the points (mid - point of the interval, average rate of change), we can observe that they lie on a straight line.

We have shown that the average rates of change of $f$ over equal intervals of the domain follow a linear pattern.

Answer:

Step1: Recall 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: Select equal - length intervals

Let's choose intervals of length $1$. For example, consider the intervals $[-7,-6]$, $[-6,-5]$, $[-5,-4]$, etc.
For the interval $[-7,-6]$: Let $a=-7$ and $b = - 6$. From the graph, if $f(-7)=0$ and $f(-6)=3$, then the average rate of change is $\frac{f(-6)-f(-7)}{-6-(-7)}=\frac{3 - 0}{1}=3$.
For the interval $[-6,-5]$: Let $a=-6$ and $b=-5$. If $f(-6)=3$ and $f(-5)=5$, then the average rate of change is $\frac{f(-5)-f(-6)}{-5-(-6)}=\frac{5 - 3}{1}=2$.
For the interval $[-5,-4]$: Let $a=-5$ and $b=-4$. If $f(-5)=5$ and $f(-4)=6$, then the average rate of change is $\frac{f(-4)-f(-5)}{-4-(-5)}=\frac{6 - 5}{1}=1$.
For the interval $[-4,-3]$: Let $a=-4$ and $b=-3$. If $f(-4)=6$ and $f(-3)=5$, then the average rate of change is $\frac{f(-3)-f(-4)}{-3-(-4)}=\frac{5 - 6}{1}=-1$.
For the interval $[-3,-2]$: Let $a=-3$ and $b=-2$. If $f(-3)=5$ and $f(-2)=3$, then the average rate of change is $\frac{f(-2)-f(-3)}{-2-(-3)}=\frac{3 - 5}{1}=-2$.
For the interval $[-2,-1]$: Let $a=-2$ and $b=-1$. If $f(-2)=3$ and $f(-1)=0$, then the average rate of change is $\frac{f(-1)-f(-2)}{-1-(-2)}=\frac{0 - 3}{1}=-3$.

Step3: Analyze the pattern

The average rates of change we found are $3,2,1,-1,-2,-3$. If we consider these values as $y$ - values and the mid - points of the intervals as $x$ - values, we can see that they follow a linear pattern. For example, if we plot the points (mid - point of the interval, average rate of change), we can observe that they lie on a straight line.

We have shown that the average rates of change of $f$ over equal intervals of the domain follow a linear pattern.