The following figure shows the graphs of y=2^x, y=3^x, y=2^-x, and y=3^-x . Match each curve wit

step 1 In this problem, we are given the graph, which includes the graph of four different functions an

The following figure shows the graphs of y=2^x, y=3^x,...

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Key Concepts

Graph Transformations and Reflections

Transformations, including reflections, shifts, and stretches, play an important role in function analysis. For exponential functions, replacing x with -x creates a reflection over the y-axis, which turns a growing function into one that decays and vice versa.

Effect of the Base on Function Behavior

The base of the exponential function influences the rate at which the function increases or decreases. A function with a larger base will grow or decay more steeply compared to one with a smaller base, making it easier to distinguish different curves based on their steepness.

Exponential Functions

Exponential functions are defined by the form f(x) = a^x, where a is a positive constant not equal to 1. These functions model situations with constant percentage change, exhibiting rapid increases or decreases depending on the value of the base.

Exponential Growth and Decay

Exponential growth occurs when the base a is greater than 1, leading to a rapid increase of the function as x increases. Conversely, exponential decay happens when the function decreases as x increases. Recognizing these patterns is crucial in interpreting graphs of exponential functions.

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