Average Rate of Change - Algebra
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Find the average rate of change if $f(2)=7$ and $f(8)=19$ on $[2,8]$.
Find the average rate of change if $f(2)=7$ and $f(8)=19$ on $[2,8]$.
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$2$. Using $\frac{19-7}{8-2} = \frac{12}{6} = 2$.
$2$. Using $\frac{19-7}{8-2} = \frac{12}{6} = 2$.
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Find the average rate of change if $f(0)=10$ and $f(4)=2$ on $[0,4]$.
Find the average rate of change if $f(0)=10$ and $f(4)=2$ on $[0,4]$.
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$-2$. Using $\frac{2-10}{4-0} = \frac{-8}{4} = -2$.
$-2$. Using $\frac{2-10}{4-0} = \frac{-8}{4} = -2$.
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Find the average rate of change if $f(3)=-1$ and $f(9)=-1$ on $[3,9]$.
Find the average rate of change if $f(3)=-1$ and $f(9)=-1$ on $[3,9]$.
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$0$. Using $\frac{-1-(-1)}{9-3} = \frac{0}{6} = 0$.
$0$. Using $\frac{-1-(-1)}{9-3} = \frac{0}{6} = 0$.
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What is the average rate of change from a table if $f(1)=4$ and $f(5)=12$?
What is the average rate of change from a table if $f(1)=4$ and $f(5)=12$?
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$2$. Using $\frac{12-4}{5-1} = \frac{8}{4} = 2$.
$2$. Using $\frac{12-4}{5-1} = \frac{8}{4} = 2$.
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What is the average rate of change from a table if $f(-2)=6$ and $f(2)=-2$?
What is the average rate of change from a table if $f(-2)=6$ and $f(2)=-2$?
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$-2$. Using $\frac{-2-6}{2-(-2)} = \frac{-8}{4} = -2$.
$-2$. Using $\frac{-2-6}{2-(-2)} = \frac{-8}{4} = -2$.
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What is the average rate of change from a table if $f(0)=3$ and $f(6)=0$?
What is the average rate of change from a table if $f(0)=3$ and $f(6)=0$?
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$-\frac{1}{2}$. Using $\frac{0-3}{6-0} = \frac{-3}{6} = -\frac{1}{2}$.
$-\frac{1}{2}$. Using $\frac{0-3}{6-0} = \frac{-3}{6} = -\frac{1}{2}$.
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What is the average rate of change from a table if $f(2)=1$ and $f(10)=5$?
What is the average rate of change from a table if $f(2)=1$ and $f(10)=5$?
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$\frac{1}{2}$. Using $\frac{5-1}{10-2} = \frac{4}{8} = \frac{1}{2}$.
$\frac{1}{2}$. Using $\frac{5-1}{10-2} = \frac{4}{8} = \frac{1}{2}$.
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Identify the average rate of change if $f$ rises $15$ units while $x$ increases by $5$ units.
Identify the average rate of change if $f$ rises $15$ units while $x$ increases by $5$ units.
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$3$. Rate equals change in output divided by change in input: $\frac{15}{5}$.
$3$. Rate equals change in output divided by change in input: $\frac{15}{5}$.
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Identify the average rate of change if $f$ drops $12$ units while $x$ increases by $3$ units.
Identify the average rate of change if $f$ drops $12$ units while $x$ increases by $3$ units.
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$-4$. Rate equals change in output divided by change in input: $\frac{-12}{3}$.
$-4$. Rate equals change in output divided by change in input: $\frac{-12}{3}$.
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What is the average rate of change if $f(b)-f(a)=9$ and $b-a=3$?
What is the average rate of change if $f(b)-f(a)=9$ and $b-a=3$?
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$3$. Direct application of the average rate formula: $\frac{9}{3}$.
$3$. Direct application of the average rate formula: $\frac{9}{3}$.
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What is the average rate of change if $f(b)-f(a)=-8$ and $b-a=2$?
What is the average rate of change if $f(b)-f(a)=-8$ and $b-a=2$?
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$-4$. Direct application of the average rate formula: $\frac{-8}{2}$.
$-4$. Direct application of the average rate formula: $\frac{-8}{2}$.
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Choose the expression that equals average rate of change on $[a,b]$: $\frac{f(a)-f(b)}{a-b}$ or $\frac{f(a)-f(b)}{b-a}$?
Choose the expression that equals average rate of change on $[a,b]$: $\frac{f(a)-f(b)}{a-b}$ or $\frac{f(a)-f(b)}{b-a}$?
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$\frac{f(a)-f(b)}{a-b}$. This expression also equals $\frac{f(b)-f(a)}{b-a}$ due to sign changes.
$\frac{f(a)-f(b)}{a-b}$. This expression also equals $\frac{f(b)-f(a)}{b-a}$ due to sign changes.
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Find and correct the error: average rate of change on $[a,b]$ written as $\frac{f(b)-f(a)}{a-b}$.
Find and correct the error: average rate of change on $[a,b]$ written as $\frac{f(b)-f(a)}{a-b}$.
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Correct: $\frac{f(b)-f(a)}{b-a}$. The denominator should be $b-a$, not $a-b$.
Correct: $\frac{f(b)-f(a)}{b-a}$. The denominator should be $b-a$, not $a-b$.
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Identify what $\frac{f(x_2)-f(x_1)}{x_2-x_1}$ represents on the graph of $y=f(x)$.
Identify what $\frac{f(x_2)-f(x_1)}{x_2-x_1}$ represents on the graph of $y=f(x)$.
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Slope of the secant line through $(x_1,f(x_1))$ and $(x_2,f(x_2))$. This formula gives the slope of the secant line.
Slope of the secant line through $(x_1,f(x_1))$ and $(x_2,f(x_2))$. This formula gives the slope of the secant line.
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Interpret $\frac{f(10)-f(2)}{10-2}=-3$ in words about $f$ over $[2,10]$.
Interpret $\frac{f(10)-f(2)}{10-2}=-3$ in words about $f$ over $[2,10]$.
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$f$ decreases by $3$ units per $1$ unit increase in $x$, on average. Negative rate means decreasing function over the interval.
$f$ decreases by $3$ units per $1$ unit increase in $x$, on average. Negative rate means decreasing function over the interval.
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If a graph shows points approximately $(0,3)$ and $(8,1)$, what is the estimated rate of change?
If a graph shows points approximately $(0,3)$ and $(8,1)$, what is the estimated rate of change?
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$-\frac{1}{4}$. Using slope: $\frac{1-3}{8-0} = \frac{-2}{8} = -\frac{1}{4}$.
$-\frac{1}{4}$. Using slope: $\frac{1-3}{8-0} = \frac{-2}{8} = -\frac{1}{4}$.
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If a graph shows points approximately $(-2,7)$ and $(2,-1)$, what is the estimated rate of change?
If a graph shows points approximately $(-2,7)$ and $(2,-1)$, what is the estimated rate of change?
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$-2$. Using slope: $\frac{-1-7}{2-(-2)} = \frac{-8}{4} = -2$.
$-2$. Using slope: $\frac{-1-7}{2-(-2)} = \frac{-8}{4} = -2$.
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If a graph shows points approximately $(1,2)$ and $(5,10)$, what is the estimated rate of change?
If a graph shows points approximately $(1,2)$ and $(5,10)$, what is the estimated rate of change?
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$2$. Using slope: $\frac{10-2}{5-1} = \frac{8}{4} = 2$.
$2$. Using slope: $\frac{10-2}{5-1} = \frac{8}{4} = 2$.
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When estimating from a graph, what two points should you choose to find average rate on $[a,b]$?
When estimating from a graph, what two points should you choose to find average rate on $[a,b]$?
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The points $(a,f(a))$ and $(b,f(b))$. Use the interval endpoints to find the secant slope.
The points $(a,f(a))$ and $(b,f(b))$. Use the interval endpoints to find the secant slope.
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Estimate the rate of change from a graph by using what geometric measurement?
Estimate the rate of change from a graph by using what geometric measurement?
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Slope (rise over run) of a secant or tangent line. Visual estimation uses slope between two graph points.
Slope (rise over run) of a secant or tangent line. Visual estimation uses slope between two graph points.
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What is the average rate of change of $f(x)=x^2$ on $[0,h]$ in terms of $h$?
What is the average rate of change of $f(x)=x^2$ on $[0,h]$ in terms of $h$?
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$h$. Using $\frac{h^2-0^2}{h-0} = \frac{h^2}{h} = h$.
$h$. Using $\frac{h^2-0^2}{h-0} = \frac{h^2}{h} = h$.
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What is the average rate of change of $f(x)=x^2$ on $[2,2+h]$ in terms of $h$?
What is the average rate of change of $f(x)=x^2$ on $[2,2+h]$ in terms of $h$?
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$4+h$. Using $\frac{(2+h)^2-2^2}{h} = \frac{4+4h+h^2-4}{h} = 4+h$.
$4+h$. Using $\frac{(2+h)^2-2^2}{h} = \frac{4+4h+h^2-4}{h} = 4+h$.
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State the formula for the average rate of change of $f$ on $[a,b]$.
State the formula for the average rate of change of $f$ on $[a,b]$.
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$\frac{f(b)-f(a)}{b-a}$. Standard formula: change in output divided by change in input.
$\frac{f(b)-f(a)}{b-a}$. Standard formula: change in output divided by change in input.
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What is another name for the average rate of change of $f$ on $[a,b]$?
What is another name for the average rate of change of $f$ on $[a,b]$?
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Slope of the secant line through $(a,f(a))$ and $(b,f(b))$. The secant line connects two points on the function's graph.
Slope of the secant line through $(a,f(a))$ and $(b,f(b))$. The secant line connects two points on the function's graph.
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What does the average rate of change of $f$ on $[a,b]$ measure in context?
What does the average rate of change of $f$ on $[a,b]$ measure in context?
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Average change in output per $1$ unit change in input. Describes how much the output changes per unit of input change.
Average change in output per $1$ unit change in input. Describes how much the output changes per unit of input change.
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Identify the units of average rate of change if $x$ is seconds and $f(x)$ is meters.
Identify the units of average rate of change if $x$ is seconds and $f(x)$ is meters.
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Meters per second. Units are output units divided by input units.
Meters per second. Units are output units divided by input units.
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What is the average rate of change of $f(x)=3x+2$ on $[1,5]$?
What is the average rate of change of $f(x)=3x+2$ on $[1,5]$?
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$3$. For linear functions, average rate equals the slope $m$.
$3$. For linear functions, average rate equals the slope $m$.
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What is the average rate of change of $f(x)=x^2$ on $[1,3]$?
What is the average rate of change of $f(x)=x^2$ on $[1,3]$?
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$4$. Using $\frac{f(3)-f(1)}{3-1} = \frac{9-1}{2} = 4$.
$4$. Using $\frac{f(3)-f(1)}{3-1} = \frac{9-1}{2} = 4$.
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What is the average rate of change of $f(x)=x^2$ on $[3,1]$?
What is the average rate of change of $f(x)=x^2$ on $[3,1]$?
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$4$. Order doesn't matter: $\frac{f(1)-f(3)}{1-3} = \frac{1-9}{-2} = 4$.
$4$. Order doesn't matter: $\frac{f(1)-f(3)}{1-3} = \frac{1-9}{-2} = 4$.
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What is the average rate of change of $f(x)=x^2-1$ on $[-1,2]$?
What is the average rate of change of $f(x)=x^2-1$ on $[-1,2]$?
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$1$. Using $\frac{f(2)-f(-1)}{2-(-1)} = \frac{3-0}{3} = 1$.
$1$. Using $\frac{f(2)-f(-1)}{2-(-1)} = \frac{3-0}{3} = 1$.
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