Linear & Exponential Growth - SAT Math

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

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

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

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

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

1. The constant term is the y-intercept.

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

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

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

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

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

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

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

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

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

1. The constant term gives the y-intercept.

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

1.5. The base of the exponential expression.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

1. No vertical change means zero slope.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

1.5. The base of the exponential expression.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

1. The constant term gives the y-intercept.

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

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

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

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

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

1. The constant term is the y-intercept.

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

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

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

1. No vertical change means zero slope.

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

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

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

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

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

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

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

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

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

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