Linear & Exponential Growth - SAT Math

1. What is the initial value in the function $y = 7 \times 3^x$?

   7. The coefficient $a$ represents the starting value.

2. What is the value of $y$ when $x = 3$ in $y = 2x + 1$?

   7. Substitute $x = 3$: $y = 2(3) + 1 = 7$.

3. Which function represents linear growth?

   $y = mx + b$. Constant rate of change defines linear functions.

4. Identify the initial value in $y = 8 \times 2^x$.

   8. The coefficient $a$ is the starting value.

5. Which function represents exponential decay?

   $y = a \times b^x$ where $0 < b < 1$. When $0 < b < 1$, the function decreases as $x$ increases.

6. Determine if $y = -3 \times (0.5)^x$ is growth or decay.

   Decay. Base $0.5 < 1$ indicates exponential decay.

7. What does the graph of $y = -2^x$ represent?

   Reflect exponential decay. Negative coefficient reflects the exponential across x-axis.

8. What is the slope of a horizontal line?

   0. No vertical change means zero slope.

9. Find the slope of the line through points $(1, 2)$ and $(3, 8)$.

   3. Using slope formula: $\frac{8-2}{3-1} = \frac{6}{2} = 3$.

10. Identify the growth rate if $y = a \times (1 + r)^x$ and $r = 0.08$.

    8%. Growth rate $r = 0.08 = 8\%$.

11. What is the formula for exponential growth?

    $y = a(1 + r)^t$. Where $a$ is initial value, $r$ is growth rate, $t$ is time.

12. Identify the slope in the equation $y = 3x + 7$.

    3. The coefficient of $x$ represents the slope.

13. Which function represents exponential growth?

    $y = a \times b^x$ where $b > 1$. When $b > 1$, the function increases as $x$ increases.

14. What does $b$ represent in the linear equation $y = mx + b$?

    Y-intercept. The value where the line crosses the y-axis.

15. What does $r$ represent in the exponential growth formula $y = a(1 + r)^t$?

    Growth rate. The decimal or percentage increase per time unit.

16. Determine the y-intercept of $y = -3x + 6$.

    6. The constant term is the y-intercept.

17. What is the general formula for an exponential function?

    $y = a \times b^x$. Where $a$ is initial value and $b$ is the growth/decay factor.

18. Identify the y-intercept in $y = -2x + 4$.

    4. The constant term gives the y-intercept.

19. Identify the growth factor in the function $y = 7(1.5)^t$.

    1.5. The base of the exponential expression.

20. Calculate $y$ when $x = 2$ for $y = 3 \times 2^x$.

    12. Substitute: $y = 3 \times 2^2 = 3 \times 4 = 12$.

21. What characteristic of a line does $b$ in $y = mx + b$ represent?

    Y-intercept. Where the line crosses the y-axis at $(0, b)$.

22. Determine the y-intercept of the line $y = -2x + 4$.

    4. The constant term gives the y-intercept value.

23. Identify the growth factor in $y = 7 \times 4^x$.

    4. The base $b$ determines the multiplication factor.

24. Find the slope of the line $y = \frac{1}{2}x + 2$.

    $\frac{1}{2}$. The coefficient of $x$ gives the slope.

25. Identify the growth type: $y = 5(1.08)^t$.

    Exponential growth. Base 1.08 > 1 indicates growth with 8% rate.

26. Find the slope in the equation $y = -5x + 3$.

    -5. The coefficient of $x$ is the slope.

27. Which function shows exponential decay? $y = 2(0.8)^x$ or $y = 2(1.2)^x$?

    $y = 2(0.8)^x$. Base $0.8 < 1$ indicates decay, while $1.2 > 1$ indicates growth.

28. What does $r$ represent in $y = a \times (1 + r)^x$?

    Growth rate. The percentage rate of growth per time period.

29. What is the slope of a vertical line?

    Undefined. Division by zero in slope formula makes it undefined.

30. What is the y-intercept in $y = 0.5 \times 3^x$?

    0.5. The initial value $a$ when $x = 0$.

31. Find the y-intercept of the line passing through $(0, -3)$.

    -3. The y-coordinate when $x = 0$.

32. Determine the slope of the line through $ (1, 2) $ and $ (3, 6) $.

    2. Using slope formula: $\frac{6-2}{3-1} = \frac{4}{2} = 2$.

33. What is the formula to find the slope between 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.

34. What is the effect of $b = 1$ in $y = a \times b^x$?

    Constant function. When $b = 1$, $b^x = 1$ for all $x$, so $y = a$.

35. State the initial value in the exponential function $y = 5(1.2)^x$.

    5. The coefficient before the parentheses is the initial value.

36. Which function represents exponential decay?

    $y = a \times b^x$ with $0 < b < 1$. Base between 0 and 1 causes exponential decay.

37. Calculate the slope of a line through $(2, 4)$ and $(6, 8)$.

    1. Using slope formula: $\frac{8-4}{6-2} = \frac{4}{4} = 1$.

38. What is the formula for linear growth?

    $y = mx + b$. Standard form where $m$ is slope and $b$ is y-intercept.

39. Identify the function type: $y = 5x^2 - 3x + 1$.

    Quadratic. Highest power of $x$ is 2, making it quadratic.

40. Determine the growth factor in $y = 2 \times 5^x$.

    $5$. The base $b$ determines how fast the function grows or decays.

41. What type of function does $y = 2(x - 3)^2 + 1$ represent?

    Quadratic. Contains $x^2$ term making it a second-degree polynomial.

42. Identify the slope in the equation $y = 3x + 5$.

    3. The coefficient of $x$ is the slope in slope-intercept form.

43. What is the formula to find the slope between 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.

44. What does the graph of $y = 2^x$ look like?

    Exponential growth. Exponential function with base greater than 1 shows growth.

45. Which function represents exponential decay?

    $y = a(1 - r)^t$. Uses $(1-r)$ where $0 < (1-r) < 1$ for decay.

46. State the y-intercept of $y = a \times b^x$ where $b > 0$.

    $a$. Initial value when $x = 0$, since $b^0 = 1$.

47. Identify the exponential function with $a = 5$ and $b = 0.5$.

    $y = 5 \times 0.5^x$. Exponential form with given initial value and base.

48. What does $m$ represent in the linear equation $y = mx + b$?

    Slope. Rate of change per unit increase in $x$.

49. What is the general formula for a linear function?

    $y = mx + b$. Standard slope-intercept form where $m$ is slope and $b$ is y-intercept.

50. Find $y$ when $x = 0$ in $y = 4 \times 10^x$.

    4. When $x = 0$: $y = 4 \times 10^0 = 4 \times 1 = 4$.

51. Identify the growth type: $y = 3x + 2$.

    Linear growth. Has constant slope of 3 and y-intercept of 2.

52. Find $y$ when $x = 1$ for $y = 3^x + 2$.

    5. Substitute $x = 1$: $y = 3^1 + 2 = 3 + 2 = 5$.

53. Identify the y-intercept in the equation $y = 4x - 7$.

    -7. The constant term is the y-intercept in slope-intercept form.

54. Determine the slope of the line through $(1, 2)$ and $(3, 6)$.

    2. Using slope formula: $\frac{6-2}{3-1} = \frac{4}{2} = 2$.

55. Determine the equation if a line passes through $(2, 3)$ with slope $4$.

    $y = 4x - 5$. Using point-slope form: $y - 3 = 4(x - 2)$.

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