8-1, 8-2 Exponential Functions Notes PDF

## 8-1: Exponential Functions An exponential function is a function of the form $f(x) = a \cdot b^x$. - What are the restrictions for the values of *a* and *b*? - $a \neq 0$ - $b > 0$ and $b \neq 1$ - What does the *a*-value represent in terms of the equation and graph? - initial val...

## 8-1: Exponential Functions An exponential function is a function of the form $f(x) = a \cdot b^x$. - What are the restrictions for the values of *a* and *b*? - $a \neq 0$ - $b > 0$ and $b \neq 1$ - What does the *a*-value represent in terms of the equation and graph? - initial value $\rightarrow$ y-intercept $(0, a)$ - What does the *b*-value represent in terms of the equation and graph? - multiplier/factor $\rightarrow$ constant ratio ## Properties of Exponents - $x^{a+b} = x^a \cdot x^b$ - $(x^a)^b = x^{a \cdot b}$ - $x^{-a} = \frac{1}{x^a}$ ## Complete the table for the exponential function $f(x) = 2^x$, then graph & list characteristics of the function. | x | f(x) | |---|---| | -2 | $2^{-2}$ = $\frac{1}{4}$ | | -1 | $2^{-1}$ = $\frac{1}{2}$ | | 0 | $2^0$ = 1 | | 1 | $2^1$ = 2 | | 2 | $2^2$ = 4| - x-intercept(s): none - y-intercept: (0, 1) - horizontal asymptote: y = 0 - domain: $IR (-∞, ∞)$ - range: $y > 0 (-∞, ∞)$ - increase or decrease: inc. ### Horizontal Asymptote - A horizontal line that a function gets closer and closer to, but never intersects/crosses. - Write as an equation of a horizontal line: $y = a$ ## 8-2: Transformations of Exponential Functions - Transforming $f(x) = b^x$: - The graph of $g(x) = b^x + k$ is the graph of $f(x) = b^x$ translated vertically. - The graph of $g(x) = b^{x + h}$ is the graph of $f(x) = b^x$ translated horizontally. - The graph $g(x) = ab^x$ is the graph of $f(x) = b^x$ stretched or compressed vertically by a factor of $|a|$. - The graph $g(x) = b^{ax}$ is the graph of $f(x) = b^x$ stretched or compressed horizontally by a factor of $|a|$. - When an exponential function $f(x)$ is multiplied by $-1$, the result is a reflection across the x- or y-axis. ### Examples #### Write the function $g(x)$ that demonstrates the given transformation of $f(x) = 3^x$. | Translated 7 units: | Stretched by a factor of 2: | Reflected: | |---|---|---| | Up $g(x) = 3^x + 7$ | Vertically $g(x) = 2 \cdot 3^x$ | Over the x-axis $(y = 0)$ $g(x) = -3^x$ | | Down $g(x) = 3^x - 7$ | Horizontally $g(x) = 3^{2x}$ | Over the y-axis $(x = 0)$ $g(x) = 3^{-x}$ | | Right $g(x) = 3^{x - 7}$ | | | | Left $g(x) = 3^{x + 7}$ | | | #### Write the function $g(x)$ that demonstrates the given transformation of $f(x) = 5^x + 2$. | Translated 9 units: | Compressed by a factor of 4: | Reflected: | |---|---|---| | Up $g(x) = 5^x + 11$ | Vertically $g(x) = \frac{1}{4}(5^x + 2)$ | Over the x-axis $(y = 0)$ $g(x) = -(5^x + 2)$ | | Down $g(x) = 5^x - 7$ | Horizontally $g(x) = 5^{\frac{1}{4}x + 2}$ | Over the y-axis $(x = 0)$ $g(x) = 5^{-(x + 2)}$ | | Right $g(x) = 5^{x - 9} + 2$ | | | | Left $g(x) = 5^{x + 9} + 2$ | | | ## Complete the table for the exponential function $f(x) = 2^x - 3$, then graph & list characteristics of the function. | x | f(x) | |---|---| | -2 | $2^{-2} - 3$ = $\frac{1}{4} - 3$ | | -1 | $2^{-1} - 3$ = $\frac{1}{2} - 3$ | | 0 | $2^0 - 3$ = $1 - 3$ | | 1 | $2^1 - 3$ = $2 - 3$ | | 2 | $2^2 - 3$ = $4 - 3$ | - x-intercept(s): between -1 and -2, approximately (-1.58, 0) - y-intercept: (0, -2) - horizontal asymptote: y = -3 - domain: $IR$ - range: $y > -3$ - increase or decrease: decreases