Average Rate of Change - Algebra 1

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

   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.

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

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

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

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

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

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

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

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

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

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

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

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

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

   Negative. Decreasing function means negative change in output.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

25. What is the average rate of change between points $(5,9)$ and $(1,1)$?

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

37. Estimate the rate of change from a graph by using what geometric measurement?

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

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

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

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

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

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

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

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

    Positive. Increasing function means positive change in output.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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