Example: Graphing $f(x) = 3^x$ and $g(x) = 3^x - 2$

Example: Graphing $f(x) = 3^x$ and $g(x) = 3^x + 2$

An operations manager uses the function $f(x) = 1.08^x$ to model the exponential growth of a company's energy usage over $x$ months. To represent a new energy-saving initiative that reduces the monthly usage by a fixed amount of 50 units, the manager creates the updated function $g(x) = 1.08^x - 50$. Based on the rules of vertical translation, how is the graph of $f(x)$ shifted to create the graph of $g(x)$?

An operations analyst is adjusting growth models to account for different fixed baseline costs and subsidies. Match each adjusted function with the correct description of its vertical translation from the base growth function $f(x) = 1.05^x$.

A data analyst at a utilities company is adjusting an energy usage model $f(x) = 1.04^x$ to account for a new solar subsidy that subtracts 15 units from the total monthly usage, resulting in the updated model $g(x) = 1.04^x - 15$.

True or False: According to the rules of vertical translation, this adjustment shifts the horizontal asymptote of the graph from $y = 0$ to $y = -15$.

Inventory Cost Model Transformation

An inventory manager models the exponential growth of warehouse stock using the function $f(x) = 3^x$. To account for a fixed daily shipment of 50 units leaving the warehouse, the model is adjusted to $g(x) = 3^x - 50$. Recognizing the pattern of vertical translation, this subtraction causes the graph of the baseline function, including its horizontal asymptote, to shift vertically ____ by 50 units.

A financial analyst is preparing a visual report on projected server maintenance costs. The original exponential growth model is given by the function $C(t) = 3^t$ (in thousands of dollars). To account for a newly negotiated flat-rate reduction of 5 thousand dollars per month, the analyst updates the function to $N(t) = 3^t - 5$. Based on the rules of vertical translation, arrange the steps the analyst must recall and follow to correctly graph the new cost model from the original.

Visualizing Operational Cost and Volume Reductions

Graphing the Downward Vertical Translation of $f(x) = 4^x$