Linear & Exponential Growth - PSAT Math

1. Find the slope of the line $4y=-2x+8$.

   $m=-\frac{1}{2}$. Rewrite as $y=-\frac{1}{2}x+2$ to identify slope.

2. What is the constant difference for the sequence $4,9,14,19,\dots$?

   $5$. Each term increases by $5$ from the previous term.

3. If $y=5\cdot 2^x$, what is $y$ when $x=3$?

   $40$. Calculate $5\cdot 2^3=5\cdot 8=40$.

4. Which model is appropriate when the amount changes by a constant ratio each step: linear or exponential?

   Exponential. Constant ratio between consecutive terms indicates exponential growth.

5. What does the slope $m$ represent in $y=mx+b$?

   Rate of change: $\frac{\Delta y}{\Delta x}$. Slope measures vertical change per unit horizontal change.

6. If a quantity increases by $15\%$ each step, what is the factor $b$?

   $b=1.15$. $b=1+0.15=1.15$ for 15% increase.

7. Find the $y$-intercept of $y=3x-7$.

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

8. A value starts at $80$ and is multiplied by $1.25$ each step. What is the value after $3$ steps?

   $156.25$. $80\times 1.25^3 = 80\times 1.953125 = 156.25$

9. A quantity is multiplied by $1.08$ each year. What is the yearly percent change?

   $8\%$ increase. Factor $1.08=1+0.08$ means $8\%$ growth.

10. Identify the initial value in $y=a\cdot b^x$ (the value when $x=0$).

    $a$. When $x=0$, $b^0=1$, leaving only $a$.

11. What is the initial value of $y=a\cdot b^x$ at $x=0$?

    $y(0)=a$. Since $b^0=1$, the function equals $a$ when $x=0$.

12. Which statement defines exponential growth in a table: constant difference or constant ratio?

    Constant ratio. Exponential sequences have equal ratios between consecutive terms.

13. What is $f(3)$ for the linear model $f(n)=10+4n$?

    $f(3)=22$. Substitute $n=3$: $f(3)=10+4(3)=10+12=22$.

14. Find the linear equation in $y=mx+b$ form with slope $3$ and $y$-intercept $-4$.

    $y=3x-4$. Direct substitution into slope-intercept form.

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

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

16. State the formula for a linear function with initial value $a$ and constant change $d$ per step.

    $f(n)=a+dn$. Linear functions add a constant $d$ to initial value $a$ for each step $n$.

17. What is the recursive form for linear growth with initial value $a$ and constant change $d$?

    $f(0)=a,\ f(n)=f(n-1)+d$. Each term equals the previous term plus constant $d$.

18. Find the next value in the exponential sequence $3,6,12,24,\ldots$.

    $48$. Common ratio is $2$, so $24\times 2=48$.

19. What condition on $b$ in $y=a\cdot b^x$ gives exponential decay?

    $0<b<1$. Values multiply by less than 1 each step.

20. What is the average rate of change of $f$ from $x=a$ to $x=b$?

    $\frac{f(b)-f(a)}{b-a}$. Measures how much $f$ changes per unit of $x$ on average.

21. What does the slope $m$ represent in $y=mx+b$?

    $m=\frac{\Delta y}{\Delta x}$ (constant rate of change). Change in $y$ over change in $x$ stays constant.

22. Which condition on $b$ in $y=a\cdot b^x$ gives exponential growth?

    $b>1$. Values increase when the base exceeds 1.

23. Identify the constant ratio property that indicates exponential growth in a table.

    Equal $y$-multipliers for equal $x$-steps. Exponential growth has constant ratios between consecutive $y$-values.

24. In $y=a\cdot b^x$, what condition on $b$ indicates growth?

    Growth when $b>1$. Values multiply by more than 1 each step, causing increase.

25. What condition on $b$ in $y=a\cdot b^x$ gives exponential growth?

    $b>1$. Values multiply by more than 1 each step.

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

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

27. What value of $r$ indicates exponential growth in $y=a\cdot r^t$ (with $a>0$)?

    $r>1$. Growth factor greater than 1 means the quantity increases.

28. If $f(n)=8\cdot(1.5)^n$, what is the growth factor and percent increase per step?

    $r=1.5$, $50\%$ increase. Growth factor 1.5 means $(1.5-1)\times 100\%=50\%$ increase.

29. What is the slope of $3x+2y=10$?

    $-\frac{3}{2}$. Rewrite as $y=-\frac{3}{2}x+5$; slope is coefficient of $x$.

30. If a quantity grows by $20\%$ each step, what is the growth factor $b$?

    $b=1.2$. $b=1+0.20=1.2$ for 20% growth.

31. What is the explicit form for linear growth with initial value $a$ and constant change $d$?

    $f(n)=a+dn$. Direct formula: start at $a$, add $d$ for each step.

32. What is the meaning of slope $m$ in $y=mx+b$?

    $m=\frac{\Delta y}{\Delta x}$ (rate of change per $1$ in $x$). Slope measures vertical change per unit horizontal change.

33. In $y=a\cdot b^x$, what condition on $b$ indicates decay?

    Decay when $0<b<1$. Values multiply by less than 1 each step, causing decrease.

34. A population is $200$ and grows by $5\%$ per year. What is the expression after $t$ years?

    $200(1.05)^t$. $5\%$ growth means multiply by $1.05$ each year.

35. State the exponential model for initial value $a$ with growth rate $r$ per step.

    $y=a(1+r)^x$. Growth rate $r$ means multiply by $(1+r)$ each step.

36. What is the relationship between percent rate $p$ and growth factor $r$ for growth?

    $r=1+\frac{p}{100}$. Convert percent increase to decimal and add 1.

37. What condition on $r$ gives exponential decay in $f(n)=a\cdot r^n$?

    $0<r<1$. Values decrease when multiplying by factor between 0 and 1.

38. Convert a $35\%$ decrease per step into an exponential multiplier $b$.

    $b=0.65$. Subtract the decay rate from 1: $b=1-0.35=0.65$.

39. What condition on $r$ gives exponential growth in $f(n)=a\cdot r^n$?

    $r>1$. Values increase when multiplying by factor greater than 1.

40. Find the value of $y$ for $y=10\cdot (2)^x$ when $x=3$.

    $80$. $y=10\cdot 2^3=10\cdot 8=80$.

41. Find $b$ in $y=mx+b$ for the line with $m=2$ passing through $(3,7)$.

    $b=1$. Substitute $(3,7)$: $7 = 2(3) + b$, so $b = 1$.

42. What factor $b$ corresponds to a decrease of $p\%$ per step?

    $b=1-\frac{p}{100}$. Subtract the percent as a decimal from 1.

43. What is the percent rate $r$ if an exponential model uses factor $b$ each step?

    $r=(b-1)\times 100\%$. Convert growth factor to percent: subtract 1, multiply by 100.

44. What is the meaning of $b$ in $y=mx+b$ when $x=0$?

    The $y$-intercept (initial value). When $x=0$, $y=m(0)+b=b$, so $b$ is where the line crosses the $y$-axis.

45. State the percent-change form of an exponential model with initial value $a$ and rate $r$.

    $f(t)=a(1+r)^t$. Growth factor $(1+r)$ represents $100\%$ plus the growth rate $r$.

46. Identify whether $f(n)=12+3n$ is linear or exponential.

    Linear. Has form $a+dn$ with constant change per step.

47. What is the growth factor $b$ if a quantity decreases by $p\%$ each step?

    $b=1-\frac{p}{100}$. A $p\%$ decrease means multiplying by $(100-p)/100$.

48. What is the meaning of slope $m$ in $y=mx+b$?

    $m=\frac{\Delta y}{\Delta x}$ (constant rate of change). Slope measures vertical change per horizontal unit.

49. What is the slope of the line $3x-2y=8$?

    $\frac{3}{2}$. Rewrite as $y=\frac{3}{2}x-4$ to find slope.

50. What is the growth factor $r$ if the percent rate is $p\%$?

    $r=1+\frac{p}{100}$. Add percent rate (as decimal) to 1 for growth factor.

51. What is the meaning of slope $m$ in $y=mx+b$?

    $m=\frac{\Delta y}{\Delta x}$ (constant rate of change). Slope measures rise over run between any two points.

52. What is the formula for slope between $(x_1,y_1)$ and $(x_2,y_2)$?

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

53. If $y=5(1.2)^x$, what is the percent increase for each increase of $1$ in $x$?

    $20\%$ increase. Factor $1.2$ means $20\%$ increase since $(1.2-1)\times 100\%=20\%$.

54. What is the meaning of $b$ in $y=mx+b$?

    $b$ is the $y$-intercept (value when $x=0$). The y-intercept is where the line crosses the y-axis.

55. Identify the constant difference property that indicates linear growth in a table.

    Equal $y$-differences for equal $x$-steps. Linear growth adds the same amount for equal $x$-intervals.

56. Find the percent change per step for $y=8\cdot 0.93^x$.

    $-7\%$. $0.93 = 1 - 0.07$, so 7% decrease.

57. What is the recursive form for a geometric sequence with first term $a_1$ and ratio $r$?

    $a_n=ra_{n-1}$ with $a_1$ given. Each term equals previous term times constant ratio.

58. What is the growth factor $b$ if a quantity increases by $p\%$ each step?

    $b=1+\frac{p}{100}$. A $p\%$ increase means multiplying by $(100+p)/100$.

59. What condition on $b$ gives exponential growth in $y=a(b)^x$ (with $a>0$)?

    $b>1$. Values multiply by more than 1, causing growth.

60. A population decreases by $15\%$ each year. What is the exponential factor $b$?

    $0.85$. Decreasing by $15\%$ means multiplying by $1-0.15=0.85$.

61. A quantity decays $15\%$ per step. What is the decay factor $r$?

    $0.85$. Decay factor $=1-\frac{15}{100}=0.85$.

62. What is the general form of an exponential growth/decay function?

    $y=a\cdot b^x$ with $b>0$ and $b\ne 1$. $a$ is initial value; $b$ is the growth/decay factor.

63. State the formula for an exponential function with initial value $a$ and growth factor $r$ per step.

    $f(n)=a\cdot r^n$. Exponential functions multiply initial value $a$ by factor $r$ for each step $n$.

64. Which model fits: value increases by $5\%$ each year starting at $200$?

    $V(t)=200\cdot 1.05^t$. Exponential model multiplies by $1.05$ (5% increase) each year.

65. Which model is linear and which is exponential: $y=3x+1$ and $y=3\cdot 1.1^x$?

    $y=3x+1$ is linear; $y=3\cdot 1.1^x$ is exponential. Linear has $x$; exponential has $x$ in exponent.

66. What does the base $b$ represent in $y=a(b)^x$?

    Multiplicative factor per $1$ increase in $x$. Each unit increase in $x$ multiplies $y$ by factor $b$.

67. If a quantity decays by $20\%$ each step, what is the decay factor $r$?

    $r=0.8$. Decay by 20% means multiply by $1-0.2=0.8$.

68. What is the recursive form for exponential growth with initial value $a$ and growth factor $r$?

    $f(0)=a,\ f(n)=r\,f(n-1)$. Each term equals the previous term times factor $r$.

69. State the formula for an exponential function with initial value $a$ and growth factor $b$ per step $t$.

    $f(t)=ab^t$. Exponential functions multiply by factor $b$ for each unit increase in $t$.

70. What is the meaning of slope $m$ in $y=mx+b$?

    $m=\frac{\Delta y}{\Delta x}$ (rate of change per $1$ in $x$). Slope measures vertical change per unit horizontal change.

71. Identify the model type: a quantity increases by $+7$ each step. Is it linear or exponential?

    Linear. Constant addition indicates linear growth.

72. Find $f(5)$ for the linear function $f(x)=12-3x$.

    $-3$. Substitute: $f(5)=12-3(5)=12-15=-3$.

73. Identify whether $y=2\cdot 5^x$ is linear or exponential.

    Exponential. Has form $y = a \cdot b^x$ with variable raised to power.

74. What is the common ratio for the sequence $3,6,12,24,\dots$?

    $2$. Each term is $2$ times the previous term.

75. What is the meaning of slope $m$ in $y=mx+b$?

    $m=\frac{\Delta y}{\Delta x}$ (constant rate of change). Slope measures vertical change per unit horizontal change.

76. Which model fits the table $x:0,1,2$ and $y:3,6,12$?

    Exponential. Values double each step ($\frac{6}{3}=\frac{12}{6}=2$), indicating exponential.

77. Identify the model type: $y=7x+3$ versus $y=7\cdot 3^x$.

    $y=7x+3$ is linear; $y=7\cdot 3^x$ is exponential. First has constant difference; second has constant ratio.

78. Which condition on $b$ gives exponential growth in $y=a\,b^x$?

    $b>1$. Values multiply by more than 1 each step.

79. What does the base $b$ indicate in $y=a\cdot b^x$?

    $b>1$ growth; $0<b<1$ decay. Base greater than 1 multiplies; less than 1 divides.

80. Find $g(3)$ for the exponential function $g(n)=5\cdot 2^n$.

    $40$. Substitute: $g(3)=5\cdot 2^3=5\cdot 8=40$.

81. What is the growth factor $r$ if the percent change each step is $p\%$?

    $r=1+\frac{p}{100}$. Convert percent increase to decimal and add 1.

82. What is the general exponential growth/decay form with initial value $a$ and factor $b$?

    $y=a\,b^x$. Standard exponential form with base $b$.

83. What is the meaning of slope $m$ in $y=mx+b$ for linear growth?

    Constant rate of change per $1$ unit of $x$. Slope tells how much $y$ changes when $x$ increases by $1$.

84. What is the initial value of $y=7\cdot (0.8)^x$?

    $7$. The coefficient $a=7$ is the initial value.

85. What is the meaning of $b$ in $y=mx+b$?

    $b$ is the $y$-intercept (value when $x=0$). The y-intercept is where the line crosses the y-axis.

86. Which model is linear or exponential if values go $3,6,12,24$ for $t=0,1,2,3$?

    Exponential. Each value doubles the previous, showing constant ratio of $2$.

87. What is the $y$-intercept $b$ of the line with slope $m=4$ passing through $(2,11)$?

    $b=3$. Use $y=mx+b$: $11=4(2)+b$, so $b=11-8=3$.

88. Which function has the greater value at $x=5$: $y=3x+2$ or $y=2^x$?

    $y=2^x$. At $x=5$: linear gives $17$, exponential gives $32$.

89. Which model is exponential: $y=7-2x$ or $y=7\cdot (0.8)^x$?

    $y=7\cdot (0.8)^x$. Has form $y=a\cdot b^x$ with multiplicative change.

90. Find the slope of the line through $(2,5)$ and $(6,13)$.

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

91. Find $f(2)$ for the exponential function $f(t)=6\cdot 2^t$.

    $f(2)=24$. Substitute $t=2$: $f(2)=6\cdot 2^2=6\cdot 4=24$.

92. What is the percent change per step if the exponential factor is $b$?

    $(b-1)\times 100\%$. Convert factor to percent: subtract 1 and multiply by 100.

93. What is the general form of an exponential function showing growth or decay?

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

94. Which expression represents exponential growth with initial value $a$ and factor $r$?

    $f(n)=a\cdot r^n$. Multiplies by factor $r$ for each unit increase in $n$.

95. Find $f(3)$ for the linear function $f(t)=12-4t$.

    $f(3)=0$. Substitute $t=3$: $f(3)=12-4(3)=12-12=0$.

96. In $y=80\left(0.9\right)^t$, what is the initial value at $t=0$?

    $80$. When $t=0$, any number to the zero power equals 1.

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

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

98. What is the constant difference for the linear sequence $5,9,13,17,\dots$?

    $d=4$. Each term increases by 4: $9-5=4$, $13-9=4$, etc.

99. If a quantity decreases by $15\%$ per step, what is the exponential factor $b$?

    $0.85$. Decreasing by $15\%$ means keeping $85\%$, so $b=0.85$.

100. An amount is $200$ and increases by $5$ each day. What is the linear model after $d$ days?

     $y=200+5d$. Start at 200, add 5 for each day (constant change).

101. What is the meaning of $b$ in $y=mx+b$?

     $b$ is the $y$-intercept (value when $x=0$). The y-intercept is where the line crosses the y-axis.

102. What is the meaning of $b$ in $y=mx+b$?

     $b$ is the $y$-intercept (value when $x=0$). The y-intercept is where the line crosses the y-axis.

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

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

104. Find the output for $y=4\cdot 3^x$ when $x=2$.

     $36$. $y = 4 \cdot 3^2 = 4 \cdot 9 = 36$.

105. A line has slope $-4$ and passes through $(1,2)$. What is its equation in slope-intercept form?

     $y=-4x+6$. Use point-slope: $y-2=-4(x-1)$, then simplify to $y=-4x+6$.

106. State the formula for slope through 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 finding slope between two points.

107. Identify the model type: $y=3\cdot (1.2)^x$ (linear or exponential)?

     Exponential. Has form $y=a\cdot b^x$ with $a=3$ and $b=1.2$.

108. Which model fits: value increases by $8$ each year starting at $120$?

     $V(t)=120+8t$. Linear model adds constant $8$ per year to initial $120$.

109. What is the linear model for initial value $y_0$ at $x=0$ and constant change $d$ per step?

     $y=y_0+dx$. Linear form starting at $y_0$ with constant change $d$ per unit.

110. What is the slope of the line through points $(x_1,y_1)$ and $(x_2,y_2)$?

     $m=\frac{y_2-y_1}{x_2-x_1}$. Slope is the change in $y$ divided by the change in $x$.

111. Evaluate the exponential function $f(n)=5\cdot 2^n$ at $n=3$.

     $40$. Substitute: $f(3)=5\cdot 2^3=5\cdot 8=40$.

112. What is $g(2)$ for the exponential model $g(n)=5\cdot 2^n$?

     $g(2)=20$. Substitute $n=2$: $g(2)=5\cdot 2^2=5\cdot 4=20$.

113. State the formula for the slope through points $(x_1,y_1)$ and $(x_2,y_2)$.

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

114. What is the linear growth model for initial value $a$ and change $d$ per step?

     $f(n)=a+dn$. Adds constant $d$ for each unit increase in $n$.

115. What is the exponential factor $b$ if a quantity increases by $p\%$ each step?

     $b=1+\frac{p}{100}$. Add percent as decimal to 1 to get growth factor.

116. Identify the $y$-intercept in the equation $y=mx+b$.

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

117. What is the explicit form for exponential growth with initial value $a$ and growth factor $r$?

     $f(n)=a\,r^n$. Direct formula: start at $a$, multiply by $r^n$.

118. Identify the growth factor per step for $y=5\cdot 1.2^x$.

     $1.2$. The base being raised to power $x$.

119. Identify whether $y=5x-2$ is linear or exponential.

     Linear. Has form $y=mx+b$ with constant slope.

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

     $a$ is the initial value (value when $x=0$). When $x=0$, $b^0=1$, so $y=a\cdot 1=a$.

121. State the formula for a linear function with initial value $a$ and constant change $d$ per step $t$.

     $f(t)=a+dt$. Linear functions add a constant $d$ for each unit increase in $t$.

122. Identify the function type: constant difference in outputs for equal $x$-steps.

     Linear. Equal steps in $x$ produce equal changes in $y$.

123. Convert a $12\%$ increase per step into an exponential multiplier $b$.

     $b=1.12$. Add the growth rate to 1: $b=1+0.12=1.12$.

124. Find the explicit formula for the linear sequence with $a_1=10$ and $d=-3$.

     $a_n=10-3(n-1)$. Use $a_n=a_1+d(n-1)$ with $a_1=10$ and $d=-3$.

125. What factor $b$ corresponds to an increase of $p\%$ per step?

     $b=1+\frac{p}{100}$. Add the percent as a decimal to 1.

126. Find the linear function $f(t)$ with $f(0)=7$ and constant change $-2$ per step.

     $f(t)=7-2t$. Start at $7$, decrease by $2$ each step: $f(t)=7+(-2)t$.

127. What is the slope between the points $(2,7)$ and $(6,19)$?

     $3$. Slope $=\frac{\Delta y}{\Delta x}=\frac{19-7}{6-2}=\frac{12}{4}=3$.

128. Identify whether $g(n)=12\cdot 3^n$ is linear or exponential.

     Exponential. Has form $a\cdot r^n$ with constant ratio between terms.

129. What does the slope $m$ represent in $y=mx+b$?

     $m=\frac{\Delta y}{\Delta x}$ (constant rate of change). Slope measures vertical change per unit horizontal change.

130. Which model is linear: $y=3x+1$ or $y=3\cdot 2^x$?

     $y=3x+1$. Has form $y=mx+b$ with constant rate of change.

131. Identify the model type: constant difference between outputs for equal $x$-steps.

     Linear. Equal steps produce equal differences in linear models.

132. A quantity grows $8\%$ per step. What is the growth factor $r$?

     $1.08$. Growth factor $=1+\frac{8}{100}=1.08$.

133. What is the slope of the line $3x-2y=10$?

     $m=\frac{3}{2}$. Rewrite as $y=\frac{3}{2}x-5$; coefficient of $x$ is slope.

134. What is the percent rate $r$ if the exponential factor is $b=1+r$?

     $r=b-1$. Percent rate is factor minus 1.

135. What condition on $b$ gives exponential decay in $y=a(b)^x$ (with $a>0$)?

     $0<b<1$. Values multiply by less than 1, causing decay.

136. Find the equation of the line with slope $3$ and $y$-intercept $-4$.

     $y=3x-4$. Direct substitution into slope-intercept form.

137. Which model is appropriate when the amount changes by a constant difference each step: linear or exponential?

     Linear. Constant difference between consecutive terms indicates linear growth.

138. What is the model for exponential change by a constant percent $r$ over $t$ steps?

     $y=a\left(1+r\right)^t$. Compound growth formula with rate $r$ as decimal.

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

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

140. Identify the y-intercept of the linear function $y=5x-12$.

     $-12$. The y-intercept is the constant term when $x=0$.

141. In $y=6\cdot 3^x$, what is the value of $y$ when $x=2$?

     $54$. $y=6\cdot 3^2=6\cdot 9=54$

142. Identify the model type: a quantity is multiplied by $1.08$ each step. Is it linear or exponential?

     Exponential. Constant multiplication indicates exponential growth.

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

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

144. What is the percent growth rate $r$ if the growth factor is $b$?

     $r=(b-1)\cdot 100\%$. Convert growth factor to percent by subtracting 1 and multiplying by 100.

145. What is the slope between $(x_1,y_1)$ and $(x_2,y_2)$?

     $m=\frac{y_2-y_1}{x_2-x_1}$. Calculate rise over run between two points.

146. What is the decay factor $r$ for a decrease of $p\%$ each step?

     $r=1-\frac{p}{100}$. Subtract percent rate (as decimal) from 1 for decay factor.

147. If a quantity decreases by $30\%$ each step, what is the factor $b$?

     $b=0.70$. $b=1-0.30=0.70$ for 30% decrease.

148. What is the value of $y$ after $t$ steps if it starts at $A$ and grows $r$ per step?

     $y=A(1+r)^t$. Growth formula where $r$ is the decimal growth rate.

149. Identify the model type: $f(n)=12+5n$ is linear or exponential?

     Linear. Has form $a+bn$ with constant term and linear term.

150. What is the percent change per $1$ unit of $x$ for $y=a\cdot b^x$?

     $(b-1)\times 100\%$. Convert growth factor to percent: subtract 1, multiply by 100.

151. What is the explicit form for linear growth with initial value $a$ and constant change $d$?

     $f(n)=a+dn$. Direct formula: start at $a$, add $d$ for each step.

152. What is the exponential growth/decay form with initial value $a$ and factor $b$?

     $y=a\cdot b^x$. Base $b$ is repeatedly multiplied as $x$ increases.

153. What is the decay factor $b$ for a decrease of $r\%$ per step?

     $b=1-\frac{r}{100}$. Subtract percent decrease (as decimal) from 1.

154. What is the growth/decay factor $b$ if the percent rate is $r\%$?

     $b=1+\frac{r}{100}$. Add percent rate (as decimal) to 1 to get growth factor.

155. Identify the slope of the line $4x-2y=10$.

     $m=2$. Rewrite as $y=2x-5$ to identify slope coefficient.

156. Find the growth factor $b$ for a $12\%$ increase each step.

     $b=1.12$. 12% increase means multiply by $1+0.12=1.12$.

157. Find the explicit formula for the exponential sequence with $a_1=6$ and $r=\frac{1}{2}$.

     $a_n=6\left(\frac{1}{2}\right)^{n-1}$. Use $a_n=a_1\cdot r^{n-1}$ with $a_1=6$ and $r=\frac{1}{2}$.

158. Which condition indicates exponential decay in $y=a\cdot b^x$?

     $0<b<1$. Base between 0 and 1 means the function decreases.

159. Identify the slope of the linear function $y=-\frac{3}{4}x+2$.

     $-\frac{3}{4}$. The slope is the coefficient of $x$ in slope-intercept form.

160. Identify the model type: constant ratio between outputs for equal $x$-steps.

     Exponential. Equal steps produce equal ratios in exponential models.

161. What is the percent change per step if an exponential model has factor $b$ each step?

     $\text{percent change}=(b-1)\times 100\%$. Convert growth factor to percent by subtracting 1 and multiplying by 100.

162. What is the slope of the line $4x-2y=10$?

     $2$. Rewrite as $y=2x-5$; coefficient of $x$ is the slope.

163. Identify the model type: $g(n)=3\cdot(1.2)^n$ is linear or exponential?

     Exponential. Has form $a\cdot r^n$ with base raised to power $n$.

164. Which statement defines linear growth in a table: constant difference or constant ratio?

     Constant difference. Linear sequences have equal differences between consecutive terms.

165. Find $f(3)$ for the exponential function $f(x)=2\cdot 5^x$.

     $250$. Calculate $f(3)=2\cdot 5^3=2\cdot 125=250$.

166. What is the percent growth rate when the growth factor is $r$?

     $(r-1)\times 100\%$. Convert growth factor to percent by subtracting 1 and multiplying by 100.

167. What is the recursive form for linear growth with initial value $a$ and constant change $d$?

     $f(0)=a,\ f(n)=f(n-1)+d$. Each term equals the previous term plus constant $d$.

168. What is the common ratio for the sequence $3,6,12,24,\dots$?

     $2$. Each term is $2$ times the previous term.

169. What is the standard form of exponential growth or decay with rate $k$ and initial value $A$?

     $y=A(1+k)^t$. When $k>0$, it's growth; when $k<0$, it's decay.

170. Identify the model type: $y=5x-12$ (linear or exponential)?

     Linear. Has form $y=mx+b$ with $m=5$ and $b=-12$.

171. Which model doubles each step: $f(n)=7+2n$ or $g(n)=7\cdot 2^n$?

     $g(n)=7\cdot 2^n$. Exponential $2^n$ doubles; linear $2n$ adds 2.

172. What is the common ratio for the exponential sequence $2,6,18,54,\dots$?

     $r=3$. Each term is 3 times the previous: $\frac{6}{2}=\frac{18}{6}=3$.

173. What is the $y$-intercept of the line $y=-3x+7$?

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

174. What is the exponential factor $b$ for a $12\%$ increase per step?

     $b=1.12$. 12% increase means multiply by 1.12.

175. What is the slope-intercept form of a linear function with slope $m$ and intercept $b$?

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

176. What is the equation of the line with slope $3$ and $y$-intercept $-2$?

     $y=3x-2$. Substitute directly into $y=mx+b$.

177. If $y=81\cdot\left(\frac{1}{3}\right)^x$, what is $y$ when $x=2$?

     $9$. Calculate $81\cdot\left(\frac{1}{3}\right)^2=81\cdot\frac{1}{9}=9$.

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

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

179. State the point-slope form of a line through $(x_1,y_1)$ with slope $m$.

     $y-y_1=m(x-x_1)$. Used when you know one point and the slope.

180. Which option is the linear equation with slope $-4$ and $y$-intercept $9$?

     $y=-4x+9$. Direct substitution into $y=mx+b$ form.

181. State the general exponential growth/decay model in terms of initial value $a$.

     $y=a\cdot b^x$. Standard form where $a$ is initial value and $b$ is growth/decay factor.

182. Find the $y$-intercept of the line with slope $3$ passing through $(2,-1)$.

     $-7$. Use $y=mx+b$: $-1=3(2)+b$, so $b=-7$.

183. Find the linear equation in slope-intercept form for slope $m=3$ and $y$-intercept $b=-4$.

     $y=3x-4$. Slope-intercept form directly uses given $m$ and $b$ values.

184. A balance is $500$ and decreases by $12\%$ per month. What is the expression after $t$ months?

     $500(0.88)^t$. $12\%$ decrease means multiply by $1-0.12=0.88$ each month.

185. Find the slope of the line through $(2,5)$ and $(6,1)$.

     $-1$. Use $m=\frac{1-5}{6-2}=\frac{-4}{4}=-1$.

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

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

187. What is the value of $y$ after $t$ steps if it starts at $A$ and decays $r$ per step?

     $y=A(1-r)^t$. Decay formula where $r$ is the decimal decay rate.

188. An amount is $200$ and increases by $5\%$ each day. What is the exponential model after $d$ days?

     $y=200(1.05)^d$. Start at 200, multiply by 1.05 each day (5% growth).

189. What is the slope of the line through $(2,7)$ and $(6,19)$?

     $m=3$. Use slope formula: $m=\frac{19-7}{6-2}=\frac{12}{4}=3$.

190. Identify the model type: constant additive change each step (linear or exponential)?

     Linear. Adding the same amount each step creates linear growth.

191. Find the $y$-intercept of the line $4x+5y=20$.

     $b=4$. Set $x=0$: $5y=20$, so $y=4$.

192. What is the percent rate $p$ if the growth factor is $r$?

     $p=(r-1)\cdot 100\%$. Convert growth factor to percent by subtracting 1 and multiplying by 100.

193. If $f(x)=5\cdot 2^x$, what is the ratio $\frac{f(x+1)}{f(x)}$?

     $2$. $\frac{5\cdot 2^{x+1}}{5\cdot 2^x}=\frac{2^{x+1}}{2^x}=2$.

194. If $y=200\cdot (1.05)^t$, what is $y$ when $t=0$?

     $200$. When $t=0$, $y=200\cdot 1=200$.

195. State the exponential model for initial value $a$ with decay rate $r$ per step.

     $y=a(1-r)^x$. Decay rate $r$ means multiply by $(1-r)$ each step.

196. Which condition on $b$ gives exponential decay in $y=a\,b^x$?

     $0<b<1$. Values multiply by less than 1 each step.

197. A quantity starts at $200$ and decays by $12\%$ per step. Write $f(t)$.

     $f(t)=200(0.88)^t$. $12\%$ decay means multiply by $1-0.12=0.88$ each step.

198. A quantity starts at $80$ and grows by $5\%$ per step. Write $f(t)$.

     $f(t)=80(1.05)^t$. $5\%$ growth means multiply by $1.05$ each step.

199. Identify the $y$-intercept in $y=mx+b$.

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

200. What is the recursive form for a linear sequence with first term $a_1$ and common difference $d$?

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

201. What is the growth factor $r$ when the percent rate is $p\%$ growth per step?

     $r=1+\frac{p}{100}$. Convert percent to growth factor by dividing by 100 and adding 1.

202. What is the equation for exponential growth with initial value $a$ and growth factor $b$?

     $f(x)=ab^x$. Initial value $a$ multiplied by factor $b$ raised to power $x$.

203. Find the decay factor $b$ for a $30\%$ decrease each step.

     $b=0.70$. 30% decrease means multiply by $1-0.30=0.70$.

204. What is the recursive form for exponential growth with initial value $a$ and growth factor $r$?

     $f(0)=a,\ f(n)=r\,f(n-1)$. Each term equals the previous term times factor $r$.

205. Identify the linear function with slope $3$ and $y$-intercept $-5$.

     $y=3x-5$. Substitute $m=3$ and $b=-5$ into slope-intercept form.

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

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

207. Identify the slope and $y$-intercept of $y=-4x+7$.

     $m=-4,\ b=7$. Already in slope-intercept form.

208. In $y=a\,b^x$, what do $a$ and $b$ represent?

     $a=y$ when $x=0$; $b$ is the per-$1$-unit multiplier. Initial value and growth/decay factor.

209. If $y=10+4x$, what is the value of $y$ when $x=6$?

     $34$. Substitute: $y=10+4(6)=10+24=34$.

210. Which condition on $b$ in $y=a\cdot b^x$ gives exponential decay?

     $0<b<1$. Values decrease when base is between 0 and 1.

211. What is the standard exponential growth/decay form for a function of $x$?

     $y=a\cdot b^x$. Standard form where $a$ is initial value and $b$ is growth factor.

212. What is the growth factor $b$ for an increase of $r\%$ per step?

     $b=1+\frac{r}{100}$. Add percent increase (as decimal) to 1.

213. What is the exponential factor $b$ for a $15\%$ decrease per step?

     $b=0.85$. 15% decrease means multiply by 0.85.

214. If $y=3(2)^x$, what is the value of $y$ when $x=4$?

     $48$. Substitute: $y=3(2)^4=3(16)=48$.

215. What is the relationship between percent rate $p$ and growth factor $r$ for decay?

     $r=1-\frac{p}{100}$. Convert percent decrease to decimal and subtract from 1.

216. What is the equation of the line with slope $-2$ through $(3,7)$?

     $y-7=-2(x-3)$. Substitute $m=-2$ and $(x_1,y_1)=(3,7)$ into point-slope form.

217. Identify whether $y=7x+2$ is linear or exponential.

     Linear. Has form $y = mx + b$ with constant rate of change.

218. Find the next value in the linear sequence $4,9,14,19,\ldots$.

     $24$. Common difference is $5$, so $19+5=24$.

219. What is the decay factor $r$ if the percent decrease each step is $p\%$?

     $r=1-\frac{p}{100}$. Convert percent decrease to decimal and subtract from 1.

220. Identify the $y$-intercept of the line $y=5x-7$.

     $-7$. The y-intercept is the constant term when $x=0$.

221. Identify the function type: constant ratio in outputs for equal $x$-steps.

     Exponential. Equal steps in $x$ multiply output by same factor.

222. Which model is linear or exponential if values go $4,7,10,13$ for $t=0,1,2,3$?

     Linear. Constant differences of $3$ indicate linear growth.

223. What is the explicit form for exponential growth with initial value $a$ and growth factor $r$?

     $f(n)=a\cdot r^n$. Direct formula: start at $a$, multiply by $r^n$.

224. Evaluate the linear function $f(n)=7+4n$ at $n=6$.

     $31$. Substitute: $f(6)=7+4(6)=7+24=31$.

225. What is the $y$-intercept of the line $y=4x-7$?

     $-7$. The constant term when $x=0$.

226. Identify whether $y=3\cdot 2^x$ is linear or exponential.

     Exponential. Has form $y=a\cdot b^x$ with variable in exponent.

227. Find $f(4)$ for the linear function $f(n)=7+2n$.

     $15$. Substitute: $f(4)=7+2(4)=7+8=15$.

228. If a quantity decays by $15\%$ each step, what is the decay factor $b$?

     $b=0.85$. $b=1-0.15=0.85$ for 15% decay.

229. What is the exponential growth/decay form using initial value $a$ and factor $b$?

     $y=a\big(b\big)^x$. Standard exponential form with initial value $a$ and growth/decay factor $b$.

230. What is the common difference for the sequence $4,9,14,19,\dots$?

     $5$. Each term increases by $5$ from the previous term.

231. Find the exponential function $f(t)$ with $f(0)=5$ and growth factor $b=3$.

     $f(t)=5\cdot 3^t$. Initial value $5$ multiplied by $3$ for each step $t$.

232. Which condition indicates exponential growth in $y=a\cdot b^x$?

     $b>1$. Base greater than 1 means the function increases.

233. What is the meaning of slope $m$ in $y=mx+b$ in a real-world context?

     Constant rate of change per $1$ unit of $x$. Slope tells how much $y$ changes when $x$ increases by 1.

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

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

235. Identify the model type: constant multiplicative change each step (linear or exponential)?

     Exponential. Multiplying by the same factor each step creates exponential growth.

236. What value range of $r$ indicates exponential decay in $y=a\cdot r^t$ (with $a>0$)?

     $0<r<1$. Growth factor between 0 and 1 means the quantity decreases.

237. Find the slope of the line through $(2,3)$ and $(6,11)$.

     $2$. Use $m=\frac{11-3}{6-2}=\frac{8}{4}=2$.

238. What is the equation for linear growth with initial value $a$ and rate $d$ per step?

     $f(x)=a+dx$. Linear growth adds constant amount $d$ per unit increase in $x$.

239. What is the exponential factor $b$ if the percent change per step is $r\%$?

     $b=1+\frac{r}{100}$. Convert percent to factor: divide by 100 and add 1.

240. What is the percent change per $1$ unit in $x$ for $y=a\cdot b^x$?

     $\text{percent change}=(b-1)\times 100\%$. Convert growth factor to percent by subtracting 1.

241. Find the $y$-intercept of the line $2x+3y=12$.

     $4$. Set $x=0$: $3y=12$, so $y=4$.

Study these flashcards