Interpretting Functions: Average Rate of Change (CCSS.F-IF.6)
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Common Core: High School - Functions › Interpretting Functions: Average Rate of Change (CCSS.F-IF.6)
Let $f(x) = x^2 - 3x + 2$. What is the average rate of change of $f$ between $x=1$ and $x=4$?
2
$\frac{1}{2}$
0
$\frac{3}{2}$
Explanation
Compute $f(4)=16-12+2=6$ and $f(1)=1-3+2=0$. Average rate of change $=\dfrac{f(4)-f(1)}{4-1}=\dfrac{6-0}{3}=2$. The option $\tfrac{1}{2}$ flips $\dfrac{\Delta y}{\Delta x}$ to $\dfrac{\Delta x}{\Delta y}$. The option 0 uses a single $y$-value divided by $\Delta x$ (e.g., $\tfrac{f(1)}{4-1}$). The option $\tfrac{3}{2}$ uses the wrong denominator (dividing by 4 instead of $\Delta x=3$).
Let $g(x)=\dfrac{x^2+1}{x}$ for $x\ne 0$. What is the average rate of change of $g$ between $x=1$ and $x=4$?
$\frac{4}{3}$
$\frac{17}{12}$
$\frac{3}{4}$
$\frac{5}{4}$
Explanation
Compute $g(4)=\dfrac{16+1}{4}=\dfrac{17}{4}$ and $g(1)=\dfrac{1+1}{1}=2$. Then $\dfrac{\Delta y}{\Delta x}=\dfrac{\tfrac{17}{4}-2}{4-1}=\dfrac{\tfrac{17}{4}-\tfrac{8}{4}}{3}=\dfrac{\tfrac{9}{4}}{3}=\dfrac{9}{12}=\tfrac{3}{4}$. The option $\tfrac{4}{3}$ flips $\dfrac{\Delta y}{\Delta x}$. The option $\tfrac{17}{12}$ uses a single endpoint value $\tfrac{g(4)}{\Delta x}$. The option $\tfrac{5}{4}$ is an arithmetic slip in simplifying $\dfrac{\tfrac{9}{4}}{3}$.
Let $h(x)=2^x$. What is the average rate of change of $h$ between $x=1$ and $x=4$?
$\frac{3}{14}$
$\frac{16}{3}$
4
$\frac{14}{3}$
Explanation
$h(4)=16$ and $h(1)=2$, so $\dfrac{\Delta y}{\Delta x}=\dfrac{16-2}{4-1}=\dfrac{14}{3}$. The option $\tfrac{3}{14}$ flips the ratio. The option $\tfrac{16}{3}$ uses a single endpoint value $\tfrac{h(4)}{\Delta x}$. The option 4 is an arithmetic slip computing $\dfrac{16-2}{?}$ with the wrong denominator.
The values of $g(x)$ are given in the table.
x: 1, 2, 4, 7 g(x): 3, 8, 20, 35
What is the average rate of change of $g$ between $x=2$ and $x=7$?
$\frac{27}{5}$
$\frac{5}{27}$
7
$\frac{28}{5}$
Explanation
$\Delta y=g(7)-g(2)=35-8=27$ and $\Delta x=7-2=5$, so $\dfrac{\Delta y}{\Delta x}=\tfrac{27}{5}$. The option $\tfrac{5}{27}$ flips the ratio. The option 7 uses a single endpoint value $\tfrac{g(7)}{\Delta x}$. The option $\tfrac{28}{5}$ is an arithmetic slip using $35-7$ instead of $35-8$.
For the function $f(x)=2x^2-3x+1$, what is the average rate of change between $x=1$ and $x=4$?
$\tfrac{1}{7}$
$\tfrac{21}{4}$
$\tfrac{20}{3}$
$7$
Explanation
Compute $f(4)=2(4)^2-3(4)+1=21$ and $f(1)=2(1)^2-3(1)+1=0$. Average rate of change $=\dfrac{f(4)-f(1)}{4-1}=\dfrac{21-0}{3}=7$. Distractors: $\tfrac{1}{7}$ flips $\Delta y$ and $\Delta x$; $\tfrac{21}{4}$ uses a single-point ratio $\tfrac{f(4)}{4}$; $\tfrac{20}{3}$ comes from an arithmetic slip (treating $f(1)$ as $1$ instead of $0$).
A function is given by the table of values $(x,f(x))$: $(-2,7)$, $(0,1)$, $(3,-8)$, $(5,-14)$. What is the average rate of change between $x=0$ and $x=5$?
$-3$
$-\tfrac{1}{3}$
$-\tfrac{14}{5}$
$-\tfrac{13}{5}$
Explanation
From the table, $f(5)=-14$ and $f(0)=1$. Average rate of change $=\dfrac{-14-1}{5-0}=\dfrac{-15}{5}=-3$. Distractors: $-\tfrac{1}{3}$ flips $\Delta y$ and $\Delta x$; $-\tfrac{14}{5}$ uses the single-point ratio $\tfrac{f(5)}{5}$; $-\tfrac{13}{5}$ arises from the subtraction error $-14-1=-13$.
For $h(x)=-3x+2$, what is the average rate of change between $x=-1$ and $x=5$?
$-\tfrac{1}{3}$
$-3$
$-\tfrac{13}{5}$
$3$
Explanation
Compute $h(5)=-15+2=-13$ and $h(-1)=3+2=5$. Average rate of change $=\dfrac{-13-5}{5-(-1)}=\dfrac{-18}{6}=-3$ (equal to the slope for a linear function). Distractors: $-\tfrac{1}{3}$ flips $\Delta y$ and $\Delta x$; $-\tfrac{13}{5}$ uses the single-point ratio $\tfrac{h(5)}{5}$; $3$ is a sign error.
A function is given by the table of values $(x,f(x))$: $(1,5)$, $(2,9)$, $(4,17)$, $(7,26)$. What is the average rate of change between $x=2$ and $x=7$?
$\tfrac{5}{17}$
$\tfrac{26}{7}$
$\tfrac{17}{5}$
$\tfrac{17}{7}$
Explanation
From the table, $f(7)=26$ and $f(2)=9$. Average rate of change $=\dfrac{26-9}{7-2}=\dfrac{17}{5}$. Distractors: $\tfrac{5}{17}$ flips $\Delta y$ and $\Delta x$; $\tfrac{26}{7}$ uses the single-point ratio $\tfrac{f(7)}{7}$; $\tfrac{17}{7}$ uses the wrong denominator.