Example: Graphing $f(x) = 2^x$ and $g(x) = 2^{x+1}$

Example: Graphing $f(x) = 2^x$ and $g(x) = 2^{x-1}$

Example: Graphing $f(x) = 3^x$ and $g(x) = 3^{x+1}$

A market researcher is modeling the exponential adoption rate of a new software tool using the function $A(t) = 2^t$. If the company decides to launch the marketing campaign 1 month early, the researcher needs to translate the graph of the function horizontally 1 unit to the left. According to the rules of horizontal translation for exponential functions, which of the following modified functions correctly represents this shift?

An urban planner uses the exponential function $P(t) = {}1.03^t$ to model the population growth of a city over time $t$. If the planner updates the model to $P(t + 8)$ to adjust for a change in the reference year, the graph of the function will be translated 8 units to the ____.

A business analyst is adjusting an exponential growth model used to forecast monthly sales. Match each modified function with the correct description of its horizontal translation compared to the original model $f(x) = b^x$.

To shift the graph of a growth model $f(t) = b^t$ horizontally to the right by 3 units, the variable in the exponent should be changed to $(t + 3)$.

Adjusting Sequestration Timelines

An adult education center uses the exponential model $P(t) = {}2^t$ to project student enrollment growth over time $t$ (in terms). Different branches of the center modify this base growth function to account for earlier or delayed launch dates. Recall the algebraic rules for horizontal translations of exponential functions. Arrange the following modified enrollment functions in order of their horizontal translation, starting with the graph that is shifted furthest to the left and ending with the graph that is shifted furthest to the right compared to the base graph $P(t)$.

Documenting Timeline Shifts in a Depreciation Model