Compare Linear and Exponential Growth - Algebra 2

1. What is the common ratio $b$ using two exponential values $f(x)$ and $f(x+1)$?

   $b=\frac{f(x+1)}{f(x)}$. Consecutive exponential values have ratio equal to the base.

2. What is the difference between consecutive outputs for $f(x)=mx+b$ when $x$ increases by $h$?

   $mh$. Linear functions change by slope times interval length.

3. What is the slope $m$ in terms of two points $(x_1,y_1)$ and $(x_2,y_2)$ on a line?

   $m=\frac{y_2-y_1}{x_2-x_1}$. Standard slope formula using rise over run.

4. Identify the constant factor for $f(x)=5\cdot 2^x$ over a $1$-unit interval.

   $2$. The base $2$ is the constant factor per unit interval.

5. What is $\frac{f(x+1)}{f(x)}$ for $f(x)=ab^x$ after simplification?

   $b$. Simplifying $\frac{ab^{x+1}}{ab^x}$ gives $b$.

6. Find $m$ if $f(3)-f(-1)=16$ for a linear function.

   $4$. Using $\frac{\text{difference}}{\text{interval}} = \frac{16}{4} = 4$.

7. What does "equal factors over equal intervals" mean for a function table?

   For fixed $h$, $\frac{f(x+h)}{f(x)}$ is constant. The ratio between outputs stays the same for equal input gaps.

8. Identify the constant difference for $f(x)=\frac{1}{2}x+6$ over a $1$-unit interval.

   $\frac{1}{2}$. The coefficient $\frac{1}{2}$ is the slope.

9. Identify the constant difference for $f(x)=-4x+10$ over a $1$-unit interval.

   $-4$. The coefficient $-4$ is the slope, giving constant difference $-4$.

10. Find $f(x+5)-f(x)$ for $f(x)=2x-1$.

    $10$. Difference over 5 units is $5 \times 2 = 10$.

11. Find $f(x+1)-f(x)$ for $f(x)=\left(\frac{1}{3}\right)x-2$.

    $\frac{1}{3}$. The slope is the coefficient of $x$, which is $\frac{1}{3}$.

12. Identify the function type if outputs are multiplied by $1.25$ each time $x$ increases by $1$.

    Exponential. Multiplying by constant factors indicates exponential growth pattern.

13. Identify the constant difference for $f(x)=7x-3$ over a $1$-unit interval.

    $7$. The coefficient $7$ is the slope, giving constant difference $7$.

14. Find $m$ if $f(10)-f(6)=20$ for a linear function.

    $5$. Using $\frac{\text{difference}}{\text{interval}} = \frac{20}{4} = 5$.

15. Find $f(x+1)-f(x)$ for $f(x)=3x+8$.

    $3$. Linear functions have constant difference equal to the slope.

16. Which value is constant for exponential functions: $\frac{\Delta y}{\Delta x}$ or $\frac{y_2}{y_1}$ over equal $\Delta x$?

    $\frac{y_2}{y_1}$ over equal $\Delta x$. Exponential functions have constant ratios over equal input intervals.

17. Find $\frac{f(x+3)}{f(x)}$ for $f(x)=2\cdot \left(\frac{1}{2}\right)^x$.

    $\frac{1}{8}$. Factor over 3 units is $(\frac{1}{2})^3 = \frac{1}{8}$.

18. Find the base $b$ if $\frac{f(x+1)}{f(x)}=0.8$ for all $x$ and $f(x)\neq 0$.

    $0.8$. If constant ratio is 0.8, then base $b = 0.8$.

19. What is the constant ratio over $h$ units for $f(x)=ab^x$?

    $\frac{f(x+h)}{f(x)}=b^h$. Generalizes to any interval $h$ with factor $b^h$.

20. Identify the function type if $\frac{f(x+3)}{f(x)}$ is constant for all $x$ and $f(x)\neq 0$.

    Exponential. Constant ratio over any interval indicates exponential growth.

21. Identify the function type if outputs increase by $+6$ each time $x$ increases by $1$.

    Linear. Adding constant amounts indicates linear growth pattern.

22. Identify the base $b$ if an exponential function doubles each time $x$ increases by $1$.

    $2$. Doubling means multiplying by 2, so base $b = 2$.

23. Find $b$ if $\frac{f(5)}{f(2)}=27$ for an exponential function.

    $3$. Since $b^3 = 27$ and $3^3 = 27$, base is $3$.

24. Find $\frac{f(x+1)}{f(x)}$ for $f(x)=9\cdot 1.1^x$.

    $1.1$. Consecutive exponential values have ratio equal to base $1.1$.

25. Which value is constant for linear functions: $\frac{\Delta y}{\Delta x}$ or $\frac{y_2}{y_1}$?

    $\frac{\Delta y}{\Delta x}$. Linear functions have constant slope (rate of change).

26. Identify the base $b$ if an exponential function is halved each time $x$ increases by $1$.

    $\frac{1}{2}$. Halving means multiplying by $\frac{1}{2}$, so base $b = \frac{1}{2}$.

27. What is the general form of a linear function used to show equal differences?

    $f(x)=mx+b$. Standard linear form where $m$ is the slope (constant difference).

28. What is the factor between outputs for $f(x)=ab^x$ when $x$ increases by $h$?

    $b^h$. Exponential functions multiply by base raised to interval length.

29. What does "equal differences over equal intervals" mean for a function table?

    For fixed $h$, $f(x+h)-f(x)$ is constant. The difference between outputs stays the same for equal input gaps.

30. What expression gives the constant change over $1$ unit for $f(x)=mx+b$?

    $f(x+1)-f(x)=m$. Shows the slope $m$ as the constant difference per unit.

31. Find $\frac{f(x+1)}{f(x)}$ for $f(x)=12\cdot \left(\frac{5}{2}\right)^x$.

    $\frac{5}{2}$. The base is $\frac{5}{2}$ from the exponential form.

32. What is $f(x+1)-f(x)$ for $f(x)=mx+b$ after simplification?

    $m$. Expanding $(mx+m+b)-(mx+b)$ gives $m$.

33. Identify the constant difference for $f(x)=-4x+10$ over a 1-unit interval.

    $-4$. The coefficient $-4$ is the slope, giving constant difference $-4$.

34. Which test identifies exponential growth in a table: constant $f(x+1)-f(x)$ or constant $\frac{f(x+1)}{f(x)}$?

    Constant $\frac{f(x+1)}{f(x)}$. Exponential functions have constant ratios between consecutive outputs.

35. What expression gives the constant factor over $1$ unit for $f(x)=ab^x$?

    $\frac{f(x+1)}{f(x)}=b$. Shows the base $b$ as the constant multiplicative factor per unit.

36. Identify the function type if $f(x+2)-f(x)$ is constant for all $x$.

    Linear. Constant difference over any interval indicates linear growth.

37. What is the key condition needed to use $\frac{f(x+h)}{f(x)}$ as a test?

    $f(x)\neq 0$. Division by zero is undefined, so $f(x) \neq 0$ is required.

38. Find the constant difference over $4$ units if $m=-2$ for a linear function.

    $-8$. Difference over 4 units is $4 \times (-2) = -8$.

39. Find $f(x+3)-f(x)$ for $f(x)=-6x+4$.

    $-18$. Difference over 3 units is $3 \times (-6) = -18$.

40. What is the slope $m$ in terms of two points $ ($x_1$, $y_1$) $ and $ ($x_2$, $y_2$) $ on a line?

    $m=\frac{y_2-y_1}{x_2-x_1}$. Standard slope formula using rise over run.

41. Which test identifies linear growth in a table: constant $f(x+1)-f(x)$ or constant $\frac{f(x+1)}{f(x)}$?

    Constant $f(x+1)-f(x)$. Linear functions have constant differences between consecutive outputs.

42. Find $b$ if $\frac{f(4)}{f(1)}=\frac{1}{8}$ for an exponential function.

    $\frac{1}{2}$. Since $b^3 = \frac{1}{8}$ and $(\frac{1}{2})^3 = \frac{1}{8}$, base is $\frac{1}{2}$.

43. Find the constant factor over $2$ units if $b=3$ for an exponential function.

    $9$. Factor over 2 units is $b^2 = 3^2 = 9$.

44. Identify the interval length $h$ used in $f(x+h)-f(x)$ when comparing $x=2$ to $x=7$.

    $5$. Distance from 2 to 7 is $7-2 = 5$ units.

45. Choose the correct statement for linear functions: constant difference or constant ratio?

    Constant difference. Linear functions add the same amount over equal intervals.

46. Find the slope $m$ if $f(x+1)-f(x)=12$ for all $x$.

    $12$. If constant difference is 12, then slope $m = 12$.

47. What is the constant difference over $h$ units for $f(x)=mx+b$?

    $f(x+h)-f(x)=mh$. Generalizes to any interval $h$ with difference $mh$.

48. Choose the correct statement for exponential functions: constant difference or constant ratio?

    Constant ratio. Exponential functions multiply by the same factor over equal intervals.

49. What is the general form of an exponential function used to show equal factors?

    $f(x)=ab^x$. Standard exponential form where $b$ is the base (constant factor).

50. Identify the interval length $h$ used in $\frac{f(x+h)}{f(x)}$ when comparing $x=-1$ to $x=3$.

    $4$. Distance from $-1$ to $3$ is $3 - (-1) = 4$ units.

51. Identify the constant factor for $f(x)=3\cdot \left(\frac{1}{4}\right)^x$ over $1$ unit.

    $\frac{1}{4}$. The base $\frac{1}{4}$ is the constant factor per unit.

52. Find $\frac{f(x+2)}{f(x)}$ for $f(x)=7\cdot 3^x$.

    $9$. Factor over 2 units is $3^2 = 9$.

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