Algebra Flashcards: Average Rate Of Change

What this deck covers

This deck focuses on Average Rate Of Change, giving you a quick way to review the definitions, rules, and examples that matter most for Algebra.

How to use these flashcards

Work through these flashcards in short sessions. Try to answer each prompt before flipping the card, then revisit any cards you miss until the explanation feels automatic.

Flashcard 1: Identify what $\frac{f(x_2)-f(x_1)}{x_2-x_1}$ represents on the graph of $y=f(x)$.

Answer: 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.

Flashcard 2: When estimating from a graph, what two points should you choose to find average rate on $[a,b]$?

Answer: The points $(a,f(a))$ and $(b,f(b))$. Use the interval endpoints to find the secant slope.

Flashcard 3: Identify the average rate of change if $f$ drops $12$ units while $x$ increases by $3$ units.

Answer: $-4$. Rate equals change in output divided by change in input: $\frac{-12}{3}$.

Flashcard 4: What is another name for the average rate of change of $f$ on $[a,b]$?

Answer: Slope of the secant line through $(a,f(a))$ and $(b,f(b))$. The secant line connects two points on the function's graph.

Flashcard 5: What is the average rate of change between points $(2,5)$ and $(6,1)$?

Answer: $-1$. Using slope formula: $\frac{1-5}{6-2} = \frac{-4}{4} = -1$.

Flashcard 6: What is the average rate of change from a table if $f(-2)=6$ and $f(2)=-2$?

Answer: $-2$. Using $\frac{-2-6}{2-(-2)} = \frac{-8}{4} = -2$.

Flashcard 7: What is the average rate of change of $f(x)=x^2-4x$ on $[1,5]$?

Answer: $2$. Using $\frac{f(5)-f(1)}{5-1} = \frac{5-(-3)}{4} = 2$.

Flashcard 8: Identify the sign of average rate of change when $f$ decreases as $x$ increases on $[a,b]$.

Answer: Negative. Decreasing function means negative change in output.

Flashcard 9: What is the average rate of change of $f(x)=x^2-1$ on $[-1,2]$?

Answer: $1$. Using $\frac{f(2)-f(-1)}{2-(-1)} = \frac{3-0}{3} = 1$.

Flashcard 10: Identify the units of average rate of change if $x$ is seconds and $f(x)$ is meters.

Answer: Meters per second. Units are output units divided by input units.

Flashcard 11: Find the average rate of change if $f(3)=-1$ and $f(9)=-1$ on $[3,9]$.

Answer: $0$. Using $\frac{-1-(-1)}{9-3} = \frac{0}{6} = 0$.

Flashcard 12: State the formula for the average rate of change of $f$ on $[a,b]$.

Answer: $\frac{f(b)-f(a)}{b-a}$. Standard formula: change in output divided by change in input.

Flashcard 13: What is the average rate of change of $f(x)=-(x^2)$ on $[1,4]$?

Answer: $-5$. Using $\frac{f(4)-f(1)}{4-1} = \frac{-16-(-1)}{3} = -5$.

Flashcard 14: What is the average rate of change of $f(x)=|x|$ on $[0,3]$?

Answer: $1$. On positive side of vertex, $|x|$ has slope $1$.

Flashcard 15: What is the average rate of change between points $(-3,4)$ and $(1,12)$?

Answer: $2$. Using slope formula: $\frac{12-4}{1-(-3)} = \frac{8}{4} = 2$.

Flashcard 16: What is the average rate of change between points $(0,-2)$ and $(4,6)$?

Answer: $2$. Using slope formula: $\frac{6-(-2)}{4-0} = \frac{8}{4} = 2$.

Flashcard 17: What does the average rate of change of $f$ on $[a,b]$ measure in context?

Answer: Average change in output per $1$ unit change in input. Describes how much the output changes per unit of input change.

Flashcard 18: What is the average rate of change if $f(b)-f(a)=-8$ and $b-a=2$?

Answer: $-4$. Direct application of the average rate formula: $\frac{-8}{2}$.

Flashcard 19: Find the average rate of change if $f(0)=10$ and $f(4)=2$ on $[0,4]$.

Answer: $-2$. Using $\frac{2-10}{4-0} = \frac{-8}{4} = -2$.

Flashcard 20: Identify the average rate of change if $f$ rises $15$ units while $x$ increases by $5$ units.

Answer: $3$. Rate equals change in output divided by change in input: $\frac{15}{5}$.

Flashcard 21: What is the average rate of change from a table if $f(-2)=6$ and $f(2)=-2$?

Answer: $-2$. Using $\frac{-2-6}{2-(-2)} = \frac{-8}{4} = -2$.

Flashcard 22: What is the average rate of change of $f(x)=\frac{1}{2}x-4$ on $[2,10]$?

Answer: $\frac{1}{2}$. Linear function's average rate equals its slope $\frac{1}{2}$.

Flashcard 23: If a graph shows points approximately $(0,3)$ and $(8,1)$, what is the estimated rate of change?

Answer: $-\frac{1}{4}$. Using slope: $\frac{1-3}{8-0} = \frac{-2}{8} = -\frac{1}{4}$.

Flashcard 24: Interpret $\frac{f(10)-f(2)}{10-2}=-3$ in words about $f$ over $[2,10]$.

Answer: $f$ decreases by $3$ units per $1$ unit increase in $x$, on average. Negative rate means decreasing function over the interval.

Flashcard 25: If a graph shows points approximately $(-2,7)$ and $(2,-1)$, what is the estimated rate of change?

Answer: $-2$. Using slope: $\frac{-1-7}{2-(-2)} = \frac{-8}{4} = -2$.

Flashcard 26: What is the average rate of change between points $(5,9)$ and $(1,1)$?

Answer: $2$. Using slope formula: $\frac{1-9}{1-5} = \frac{-8}{-4} = 2$.

Flashcard 27: What is the average rate of change of $f(x)=3x+2$ on $[1,5]$?

Answer: $3$. For linear functions, average rate equals the slope $m$.

Flashcard 28: State the average rate of change of a constant function $f(x)=c$ on $[a,b]$.

Answer: $0$. Constant functions have no change in output.

Flashcard 29: What is the average rate of change of $f(x)=x^2+2x$ on $[1,2]$?

Answer: $5$. Using $\frac{f(2)-f(1)}{2-1} = \frac{8-3}{1} = 5$.

Flashcard 30: What is the average rate of change of $f(x)=x^2$ on $[0,h]$ in terms of $h$?

Answer: $h$. Using $\frac{h^2-0^2}{h-0} = \frac{h^2}{h} = h$.

Flashcard 31: Find the average rate of change if $f(2)=7$ and $f(8)=19$ on $[2,8]$.

Answer: $2$. Using $\frac{19-7}{8-2} = \frac{12}{6} = 2$.

Flashcard 32: If a graph shows points approximately $(1,2)$ and $(5,10)$, what is the estimated rate of change?

Answer: $2$. Using slope: $\frac{10-2}{5-1} = \frac{8}{4} = 2$.

Flashcard 33: What is the average rate of change of $f(x)=|x|$ on $[-3,0]$?

Answer: $-1$. On negative side of vertex, $|x|$ has slope $-1$.

Flashcard 34: What is the average rate of change of $f(x)=x^2-4x$ on $[0,4]$?

Answer: $0$. Using $\frac{f(4)-f(0)}{4-0} = \frac{0-0}{4} = 0$.

Flashcard 35: What is the average rate of change of $f(x)=|x|$ on $[-2,2]$?

Answer: $0$. Symmetric interval around vertex gives zero net change.

Flashcard 36: What is the average rate of change of $f(x)=x^2+2x$ on $[-1,1]$?

Answer: $2$. Using $\frac{f(1)-f(-1)}{1-(-1)} = \frac{3-(-1)}{2} = 2$.

Flashcard 37: What is the average rate of change of $f(x)=x^2$ on $[3,1]$?

Answer: $4$. Order doesn't matter: $\frac{f(1)-f(3)}{1-3} = \frac{1-9}{-2} = 4$.

Flashcard 38: Estimate the rate of change from a graph by using what geometric measurement?

Answer: Slope (rise over run) of a secant or tangent line. Visual estimation uses slope between two graph points.

Flashcard 39: Find and correct the error: average rate of change on $[a,b]$ written as $\frac{f(b)-f(a)}{a-b}$.

Answer: Correct: $\frac{f(b)-f(a)}{b-a}$. The denominator should be $b-a$, not $a-b$.

Flashcard 40: What is the average rate of change from a table if $f(1)=4$ and $f(5)=12$?

Answer: $2$. Using $\frac{12-4}{5-1} = \frac{8}{4} = 2$.

Flashcard 41: What is the average rate of change of $f(x)=x^2$ on $[1,3]$?

Answer: $4$. Using $\frac{f(3)-f(1)}{3-1} = \frac{9-1}{2} = 4$.

Flashcard 42: Identify the sign of average rate of change when $f$ increases as $x$ increases on $[a,b]$.

Answer: Positive. Increasing function means positive change in output.

Flashcard 43: What is the average rate of change of $f(x)=x^2$ on $[2,2+h]$ in terms of $h$?

Answer: $4+h$. Using $\frac{(2+h)^2-2^2}{h} = \frac{4+4h+h^2-4}{h} = 4+h$.

Flashcard 44: Find the average rate of change if $f(-1)=3$ and $f(5)=-9$ on $[-1,5]$.

Answer: $-2$. Using $\frac{-9-3}{5-(-1)} = \frac{-12}{6} = -2$.

Flashcard 45: State the average rate of change of a linear function $f(x)=mx+b$ on any interval.

Answer: $m$. Linear functions have constant rate of change equal to slope.

Flashcard 46: What is the average rate of change from a table if $f(2)=1$ and $f(10)=5$?

Answer: $\frac{1}{2}$. Using $\frac{5-1}{10-2} = \frac{4}{8} = \frac{1}{2}$.

Flashcard 47: What is the average rate of change if $f(b)-f(a)=9$ and $b-a=3$?

Answer: $3$. Direct application of the average rate formula: $\frac{9}{3}$.

Flashcard 48: What is the average rate of change from a table if $f(0)=3$ and $f(6)=0$?

Answer: $-\frac{1}{2}$. Using $\frac{0-3}{6-0} = \frac{-3}{6} = -\frac{1}{2}$.

Flashcard 49: Choose the expression that equals average rate of change on $[a,b]$: $\frac{f(a)-f(b)}{a-b}$ or $\frac{f(a)-f(b)}{b-a}$?

Answer: $\frac{f(a)-f(b)}{a-b}$. This expression also equals $\frac{f(b)-f(a)}{b-a}$ due to sign changes.

Flashcard 50: What is the average rate of change of $f(x)=-2x+7$ on $[-1,4]$?

Answer: $-2$. Linear function's average rate equals its slope $-2$.

Flashcard 51: What is the average rate of change of $f(x)=2x^2$ on $[0,2]$?

Answer: $4$. Using $\frac{f(2)-f(0)}{2-0} = \frac{8-0}{2} = 4$.