Constructing Linear and Exponential Functions - Algebra 2

1. What is the linear function for "starts at $30$ and decreases by $2$ each hour"?

   $y=-2x+30$. Linear with slope $-2$ and $y$-intercept $30$.

2. What is the slope of the line $5x-10y=20$?

   $m=\frac{1}{2}$. Rewrite as $y = \frac{1}{2}x - 2$, so slope is $\frac{1}{2}$.

3. What is the decay factor $b$ if an exponential model keeps $80\%$ each time $x$ increases by $1$?

   $b=0.8$. Keeping $80\%$ means multiplying by $0.8$ each step.

4. What is the common difference $d$ for the arithmetic sequence $5,9,13,17,\dots$?

   $d=4$. Each term increases by $9-5 = 4$.

5. What is $a$ in $y=a\cdot b^x$ if the function passes through $(0,-8)$?

   $a=-8$. The initial value when $x = 0$ is $a$.

6. What is the standard form of a linear equation (with integer coefficients)?

   $Ax+By=C$. Linear equation with integer coefficients $A$, $B$, $C$.

7. What is $a_5$ for the geometric sequence with $a_1=7$ and $r=2$?

   $a_5=112$. $a_5 = 7 \cdot 2^{5-1} = 7 \cdot 16 = 112$.

8. What is the exponential function through $(1,12)$ and $(3,48)$ in the form $y=a\cdot b^x$?

   $y=6\cdot 2^x$. From ratio $\frac{48}{12} = 4 = 2^2$, so $b = 2$, $a = 6$.

9. What is the linear function through points $(0,-3)$ and $(4,5)$ in slope-intercept form?

   $y=2x-3$. Initial value $-3$, slope $\frac{5-(-3)}{4-0} = 2$.

10. What is the initial value in $y=mx+b$?

    $b$ (the $y$-intercept, value when $x=0$). Where the line crosses the $y$-axis.

11. What is $b$ if $y=a\cdot b^x$ passes through $(0,2)$ and $(3,16)$?

    $b=2$. $a = 2$, $16 = 2 \cdot b^3$, so $b^3 = 8$ and $b = 2$.

12. What is the explicit formula for a geometric sequence with first term $a_1$ and ratio $r$?

    $a_n=a_1\cdot r^{n-1}$. Start with $a_1$, multiply by $r$ raised to $(n-1)$ power.

13. What is the slope of the line through $(2,5)$ and $(6,13)$?

    $m=2$. $m = \frac{13-5}{6-2} = \frac{8}{4} = 2$.

14. What is the recursive formula for an arithmetic sequence with difference $d$?

    $a_n=a_{n-1}+d$ (with a given $a_1$). Each term equals the previous term plus constant $d$.

15. What is $b$ in $y=a\cdot b^x$ if the function passes through $(0,9)$ and $(1,3)$?

    $b=\frac{1}{3}$. At $(1,3)$: $3 = 9 \cdot b$, so $b = \frac{1}{3}$.

16. What is the slope of the line passing through $(0,4)$ and $(5,4)$?

    $m=0$. Horizontal line has zero slope (no rise).

17. What is the point-slope form of a line through $(x_1,y_1)$ with slope $m$?

    $y-y_1=m(x-x_1)$. Uses a known point and slope to build the equation.

18. What is the slope of a line that falls $10$ units when $x$ increases by $5$?

    $m=-2$. Slope equals rise over run: $\frac{-10}{5} = -2$.

19. What is the recursive formula for a geometric sequence with ratio $r$?

    $a_n=r\cdot a_{n-1}$ (with a given $a_1$). Each term equals the previous term times constant $r$.

20. What condition on $b$ gives exponential growth in $y=a\cdot b^x$ (assuming $a>0$)?

    $b>1$. Base greater than 1 means increasing exponential function.

21. What is the exponential function through $(0,6)$ and $(1,15)$ in the form $y=a\cdot b^x$?

    $y=6\cdot\left(\frac{5}{2}\right)^x$. $a = 6$ from $(0,6)$, $b = \frac{15}{6} = \frac{5}{2}$.

22. In $y=a\cdot b^x$, what does $b$ represent?

    $b$ is the growth/decay factor per $1$ unit of $x$. How much the output multiplies for each unit increase in $x$.

23. What is the slope of the line passing through $(2,-1)$ and $(2,6)$?

    Undefined slope (vertical line $x=2$). Vertical lines have undefined slope (zero run).

24. What is the $y$-intercept of the line $3x+2y=8$?

    $b=4$. Solve $3(0) + 2y = 8$ to get $y = 4$.

25. What is the exponential function for "starts at $200$ and decreases by $5\%$ per month"?

    $y=200\cdot(0.95)^x$. Initial $200$, decay factor $1 - 0.05 = 0.95$.

26. What is the exponential function through $(0,4)$ and $(2,36)$ in the form $y=a\cdot b^x$?

    $y=4\cdot 3^x$. $a = 4$, $36 = 4 \cdot b^2$, so $b^2 = 9$ and $b = 3$.

27. What is the linear function with slope $m=-4$ and passing through $(2,1)$?

    $y=-4x+9$. Using point-slope form with $(2,1)$ and $m=-4$.

28. What is the slope of a line that rises $6$ units when $x$ increases by $2$?

    $m=3$. Slope equals rise over run: $\frac{6}{2} = 3$.

29. What is the slope-intercept form of a linear function?

    $y=mx+b$. Standard linear form with slope $m$ and $y$-intercept $b$.

30. In $y=a\cdot b^x$, what does $a$ represent?

    $a$ is the initial value $y(0)$. The output value when $x = 0$.

31. Identify the function type if the table has a constant ratio $\frac{y_{k+1}}{y_k}$ for equal steps in $x$.

    Exponential. Constant ratios indicate exponential relationship.

32. What condition on $b$ gives exponential decay in $y=a\cdot b^x$ (assuming $a>0$)?

    $0<b<1$. Base between 0 and 1 means decreasing exponential function.

33. What is the linear equation of the vertical line passing through $(3,0)$?

    $x=3$. Vertical lines have form $x = \text{constant}$.

34. What is the linear function through points $(1,7)$ and $(3,11)$ in slope-intercept form?

    $y=2x+5$. Slope $\frac{11-7}{3-1} = 2$, $y$-intercept when $x=0$ is $5$.

35. What is $a_6$ for the arithmetic sequence with $a_1=2$ and $d=5$?

    $a_6=27$. $a_6 = 2 + (6-1) \cdot 5 = 2 + 25 = 27$.

36. What is the constant ratio property of a geometric sequence?

    $\frac{a_{n+1}}{a_n}=r$ (constant, $a_n\neq 0$). Consecutive terms have the same ratio $r$.

37. Identify the function type if the graph is a straight line with constant slope.

    Linear. Straight lines have constant slope (linear functions).

38. What is the linear function for "starts at $12$ and increases by $4$ each step"?

    $y=4x+12$. Linear with slope $4$ and $y$-intercept $12$.

39. Identify the function type if the table has a constant first difference in $y$ for equal steps in $x$.

    Linear. Constant differences indicate linear relationship.

40. What is the growth factor $b$ if an exponential model doubles each time $x$ increases by $1$?

    $b=2$. Doubling means multiplying by $2$ each step.

41. What is the exponential function through $(0,81)$ and $(4,9)$ in the form $y=a\cdot b^x$?

    $y=81\cdot\left(\frac{1}{\sqrt{3}}\right)^x$. $a = 81$, $b^4 = \frac{9}{81} = \frac{1}{9}$, so $b = \frac{1}{\sqrt{3}}$.

42. What is the constant difference property of an arithmetic sequence?

    $a_{n+1}-a_n=d$ (constant). Consecutive terms have the same difference $d$.

43. What is the exponential function for "starts at $50$ and increases by $3\%$ per year"?

    $y=50\cdot(1.03)^x$. Initial $50$, growth factor $1 + 0.03 = 1.03$.

44. What is the slope formula $m$ using two points $(x_1,y_1)$ and $(x_2,y_2)$?

    $m=\frac{y_2-y_1}{x_2-x_1}$. The rise over run formula using coordinate differences.

45. Identify the function type if the graph curves and multiplies by a constant factor each $1$ step in $x$.

    Exponential. Curved growth by constant factors indicates exponential.

46. What is the general form of an exponential function showing initial value and growth factor?

    $y=a\cdot b^x$. Initial value $a$ times growth factor $b$ raised to power $x$.

47. What is the linear equation of the horizontal line passing through $(0,-7)$?

    $y=-7$. Horizontal lines have form $y = \text{constant}$.

48. What is the exponential function through $(0,10)$ and $(2,40)$ in the form $y=a\cdot b^x$?

    $y=10\cdot 2^x$. $a = 10$, $b^2 = 4$ so $b = 2$.

49. What is the explicit formula for the geometric sequence with $a_1=5$ and $r=\frac{1}{3}$?

    $a_n=5\cdot\left(\frac{1}{3}\right)^{n-1}$. Using the explicit geometric sequence formula.

50. What is the linear function through $(2,0)$ and $(0,6)$ in slope-intercept form?

    $y=-3x+6$. Slope $\frac{0-6}{2-0} = -3$, $y$-intercept $6$.

51. What is the explicit formula for an arithmetic sequence with first term $a_1$ and difference $d$?

    $a_n=a_1+(n-1)d$. Start with $a_1$, add $d$ for each step to term $n$.

52. What is the explicit formula for the arithmetic sequence with $a_1=-3$ and $d=2$?

    $a_n=-3+2(n-1)$. Using the explicit arithmetic sequence formula.

53. What is the common ratio $r$ for the geometric sequence $3,12,48,192,\dots$?

    $r=4$. Each term multiplies by $\frac{12}{3} = 4$.

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