Constructing Linear and Exponential Functions - Algebra 1

1. What is the linear function with slope $4$ that passes through $(3,-2)$?

   $y=4x-14$. Use point-slope form: $y-(-2)=4(x-3)$, then simplify.

2. What is the initial value $a$ of $y=9\left(\frac{1}{3}\right)^x$?

   $9$. The coefficient $a$ is the initial value when $x=0$.

3. What is the exponential function with $a=3$ and factor $b=2$?

   $y=3\cdot 2^x$. Direct substitution into the exponential form $y=a\cdot b^x$.

4. What is $a_5$ for the geometric sequence with $a_1=3$ and $r=2$?

   $a_5=48$. Use formula: $a_5=3\cdot 2^{5-1}=3\cdot 16=48$.

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

   $a_n=a_1\cdot r^{n-1}$. General term formula using first term and common ratio.

6. What is an explicit formula for the arithmetic sequence $7,12,17,22,\dots$?

   $a_n=7+5(n-1)$. First term $a_1=7$ with common difference $d=5$.

7. What is the exponential function if it starts at $20$ and triples each step?

   $y=20\cdot 3^x$. Initial value 20 with multiplicative factor 3 per step.

8. What is the slope of the line through $(1,4)$ and $(5,12)$?

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

9. What is the factor $b$ for $12\%$ decay per step?

   $b=0.88$. Calculate $b=1-0.12=0.88$ for 12% decay.

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

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

11. What is the linear function through $(2,5)$ with slope $3$?

    $y=3x-1$. Use point-slope form: $y-5=3(x-2)$, then simplify.

12. Identify whether $y=7-3x$ has slope $-3$ or $3$.

    $-3$. Rewrite as $y=-3x+7$ to identify the slope coefficient.

13. What is the exponential function through $(0,12)$ and $(2,3)$ in the form $y=a\cdot b^x$?

    $y=12\cdot \left(\frac{1}{2}\right)^x$. Use $a=12$ and solve $12b^2=3$ to get $b=\frac{1}{2}$.

14. What is the exponential model for percent decay rate $r$ per step?

    $y=a(1-r)^x$. Convert decay rate $r$ to multiplicative factor $1-r$.

15. Identify the slope of the linear function $y=mx+b$.

    $m$. The coefficient of $x$ represents the rate of change.

16. What is the common ratio definition for a geometric sequence?

    Constant ratio: $\frac{a_{n+1}}{a_n}=r$. Each term is multiplied by the same constant ratio $r$.

17. What is the linear function rule if a table shows $(0,2)$ and $(3,11)$?

    $y=3x+2$. Calculate slope $m=\frac{11-2}{3-0}=3$ and use y-intercept $b=2$.

18. What is the slope formula for two points $(x_1,y_1)$ and $(x_2,y_2)$?

    $m=\frac{y_2-y_1}{x_2-x_1}$. Rise over run formula for calculating slope between two points.

19. What is an explicit formula for the geometric sequence $10,5,2.5,1.25,\dots$?

    $a_n=10\left(\frac{1}{2}\right)^{n-1}$. First term $a_1=10$ with common ratio $r=\frac{1}{2}$.

20. What is the exponential function rule if a table shows $(0,6)$ and $(2,24)$?

    $y=6\cdot 2^x$. Use $a=6$ and solve $6b^2=24$ to get $b=2$.

21. What is the linear function passing through $(5,1)$ and $(5,9)$?

    $x=5$. Vertical line since both points have the same x-coordinate.

22. What is the factor $b$ for $25\%$ growth per step?

    $b=1.25$. Calculate $b=1+0.25=1.25$ for 25% growth.

23. Identify whether the sequence $81,27,9,3,\dots$ is arithmetic or geometric.

    Geometric. Constant ratio of $\frac{1}{3}$ between consecutive terms.

24. Identify whether the sequence $5,9,13,17,\dots$ is arithmetic or geometric.

    Arithmetic. Constant difference of 4 between consecutive terms.

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

    $a_n=a_1+(n-1)d$. General term formula using first term and common difference.

26. What is the equation of the line with slope $-\frac{1}{2}$ and $y$-intercept $6$?

    $y=-\frac{1}{2}x+6$. Direct substitution into slope-intercept form $y=mx+b$.

27. What is the common difference definition for an arithmetic sequence?

    Constant difference: $a_{n+1}-a_n=d$. Each term increases by the same constant amount $d$.

28. What is the average rate of change from $x=2$ to $x=6$ for points $(2,1)$ and $(6,9)$?

    $2$. Average rate of change equals $\frac{9-1}{6-2}=\frac{8}{4}=2$.

29. What is the common difference $d$ of $-2,1,4,7,\dots$?

    $d=3$. Calculate $d=1-(-2)=3$ from consecutive terms.

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

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

31. What is the exponential function through $(0,4)$ and $(1,10)$ in the form $y=a\cdot b^x$?

    $y=4\cdot 2.5^x$. Use $a=4$ and calculate $b=\frac{10}{4}=2.5$.

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

    $b>1$. Base greater than 1 causes exponential increase.

33. What is the common ratio $r$ of $2,6,18,54,\dots$?

    $r=3$. Calculate $r=\frac{6}{2}=3$ from consecutive terms.

34. What is the linear function passing through $(2,7)$ and $(6,7)$?

    $y=7$. Horizontal line since both points have the same y-coordinate.

35. What is the exponential model for percent growth rate $r$ per step?

    $y=a(1+r)^x$. Convert growth rate $r$ to multiplicative factor $1+r$.

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

    $a_n=a_{n-1}\cdot r$. Each term equals the previous term times the common ratio.

37. What is the general form of an exponential function with initial value $a$ and factor $b$?

    $y=a\cdot b^x$. Standard exponential form with initial value $a$ and base $b$.

38. In $y=a\cdot b^x$, what does $a$ represent when $x=0$?

    $a=y(0)$. When $x=0$, $b^0=1$, so $y=a\cdot 1=a$.

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

    Growth/decay factor per $1$ unit of $x$. The multiplier that determines how $y$ changes per unit increase in $x$.

40. What is the standard form of a linear equation?

    $Ax+By=C$. General linear equation form where $A$, $B$, $C$ are constants.

41. What is the slope of a line that rises $9$ units while running $-3$ units?

    $m=-3$. Slope equals rise over run: $m=\frac{9}{-3}=-3$.

42. What is the factor $b$ of $y=5\cdot 0.2^x$?

    $0.2$. The base $b$ is the multiplicative factor per unit increase.

43. What is the exponential function if it starts at $50$ and decreases by $10\%$ each step?

    $y=50\cdot 0.9^x$. Initial value 50 with decay factor $1-0.10=0.9$.

44. What is the equation of the line through $(0,-3)$ and $(4,5)$?

    $y=2x-3$. First find slope $m=2$, then use point $(0,-3)$ for y-intercept.

45. What is $a_6$ for the arithmetic sequence with $a_1=4$ and $d=-3$?

    $a_6=-11$. Use formula: $a_6=4+(6-1)(-3)=4-15=-11$.

46. Identify the $y$-intercept of the linear function $y=mx+b$.

    $b$. The constant term represents where the line crosses the y-axis.

47. Identify whether the table pattern is linear or exponential: $y$ increases by $-4$ each time $x$ increases by $1$.

    Linear. Constant additive change indicates linear relationship.

48. Identify whether the table pattern is linear or exponential: $y$ multiplies by $3$ each time $x$ increases by $1$.

    Exponential. Constant multiplicative factor indicates exponential relationship.

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

    $a_n=a_{n-1}+d$. Each term equals the previous term plus the common difference.

50. What is the factor $b$ for an exponential function with $y(0)=6$ and $y(3)=48$?

    $b=2$. Solve $6b^3=48$ to get $b^3=8$, so $b=2$.

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

    $0<b<1$. Base between 0 and 1 causes exponential decrease.

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