Exponential Functions Algebra

Cheatsheet Content

### Exponential Functions Overview Exponential functions are of the form $Y = ab^x$, where: - $a$ is the initial amount or starting value (the y-intercept when $x=0$). - $b$ is the growth/decay factor (constant multiplier). - If $b > 1$, it's exponential growth. - If $0 ### Identifying Components of $Y = ab^x$ Given an equation or scenario: - **Initial amount ($a$):** The value of Y when $x=0$. - **Growth/Decay Factor ($b$):** The number by which Y is multiplied for each unit increase in $x$. - **Percentage Change:** - Growth: $b = 1 + r$ (where $r$ is the growth rate as a decimal). - Decay: $b = 1 - r$ (where $r$ is the decay rate as a decimal). ### Graphing Exponential Functions - **Y-intercept:** Always $(0, a)$. - **X-intercept:** Usually none, unless specified (e.g., in a limited domain). - **Horizontal Asymptote:** A line that the graph approaches but never touches. For $Y = ab^x$, the asymptote is typically $y=0$. - **Domain:** All real numbers (unless restricted by context), $(-\infty, \infty)$. - **Range:** - If $a > 0$: $(0, \infty)$ (all positive real numbers). - If $a ### Solving Exponential Equations - **Strategy 1: Common Base** - If $b^x = b^y$, then $x=y$. - Example: $2^x = 8 \Rightarrow 2^x = 2^3 \Rightarrow x=3$. - **Strategy 2: Graphing Calculator** - Graph $Y_1 = \text{LHS}$ and $Y_2 = \text{RHS}$. Find the intersection point. ### Simple vs. Compound Interest - **Simple Interest:** Interest calculated only on the principal amount. $A = P(1 + rt)$. - $A$: total amount, $P$: principal, $r$: annual interest rate, $t$: time in years. - **Compound Interest:** Interest calculated on the principal *and* accumulated interest. $A = P(1 + \frac{r}{n})^{nt}$. - $n$: number of times interest is compounded per year. - If compounded continuously: $A = Pe^{rt}$. ### Writing Exponential Equations #### From Tables 1. Identify $a$ (the Y-value when $x=0$). 2. Find the constant multiplier ($b$) by dividing consecutive Y-values. 3. Substitute $a$ and $b$ into $Y = ab^x$. #### From Graphs 1. Identify $a$ (the Y-intercept). 2. Choose another clear point $(x, Y)$. 3. Substitute $a$, $x$, and $Y$ into $Y = ab^x$ and solve for $b$. #### From Word Problems/Scenarios 1. Identify the initial amount ($a$). 2. Determine if it's growth or decay and find the percentage rate ($r$). 3. Calculate the growth/decay factor $b = 1+r$ or $b = 1-r$. 4. Substitute $a$ and $b$ into $Y = ab^x$. ### Simplifying Exponential Expressions **Using only positive exponents:** - **Product Rule:** $x^m \cdot x^n = x^{m+n}$ - **Quotient Rule:** $\frac{x^m}{x^n} = x^{m-n}$ - **Power Rule:** $(x^m)^n = x^{mn}$ - **Negative Exponent Rule:** $x^{-n} = \frac{1}{x^n}$ and $\frac{1}{x^{-n}} = x^n$ - **Zero Exponent Rule:** $x^0 = 1$ (for $x \neq 0$) - **Power of a Product:** $(xy)^n = x^n y^n$ - **Power of a Quotient:** $(\frac{x}{y})^n = \frac{x^n}{y^n}$ **Example:** Simplify $\frac{(2x^3y^{-2})^2}{4x^2y^5}$ 1. Apply Power Rule: $\frac{2^2(x^3)^2(y^{-2})^2}{4x^2y^5} = \frac{4x^6y^{-4}}{4x^2y^5}$ 2. Simplify constants and apply Quotient Rule: $x^{6-2}y^{-4-5} = x^4y^{-9}$ 3. Apply Negative Exponent Rule: $\frac{x^4}{y^9}$