Unit 4

v Exponential functions

$$\begin{array}{rl}& f(x)=a{b}^{k(x-d)}+c\end{array}$$ $$\begin{array}{rl}& \text{1. If }a>1\text{ or }a<1,\text{ it stretches or compresses the graph vertically. If }a<0,\text{ it reflects over the x-axis.}\\ & \text{2. If }b>1,\text{ the function grows exponentially. If }0<b<1,\text{ the function decays exponentially.}\\ & \text{3. If }k>1,\text{ it compresses the graph horizontally. If }0<k<1,\text{ it stretches horizontally. If }k<0,\text{ it reflects over the y-axis.}\\ & \text{4. If }d>0,\text{ the graph is translated }d\text{ units to the right. If }d<0,\text{ the graph is translated }d\text{ units to the left.}\\ & \text{5. If }c>0,\text{ the graph is translated }c\text{ units upward. If }c<0,\text{ the graph is translated }c\text{ units downward.}\end{array}$$ $$\begin{array}{rl}& 2^{3x+5}=64^{x-7}\\ & 2^{3x+5}=2^{6^{x-7}}\\ & 2^{3x+5}=2^{6x-42}\\ & 3x+5=6x-42\\ & 5+42=6x-3x\\ & 47=3x\\ & x=\frac{47}{3}\end{array}$$ $$\begin{array}{rl}& 5^{x-12}={\left(\frac{1}{125}\right)}^{\frac{12}{5}}\\ & \dots \text{we know that }1/125=5^{-3}\end{array}$$