The figure shows the graphs of y = 2^x , y = e^x , y = 10^x...

The figure shows the graphs of $ y = 2^x $, $ y = e^x $, $ y = 10^x $, $ y = 2^{-x} $, $ y = e^{-x} $, and $ y = 10^{-x} $. Match each function with its graph. [The graphs are labeled (a) through (f).] Explain your reasoning.

Key Concepts

Graphical Analysis Techniques

Graphical analysis involves examining key features such as the y-intercept, asymptotic behavior, and the steepness of the graph. Understanding these features allows one to match functions to their corresponding graphs by comparing expected behaviors, such as the rate of growth or decay and horizontal reflections.

Exponential Functions

Exponential functions are mathematical functions defined by the form y = a^x, where a is a positive constant. These functions exhibit distinctive behavior such as rapid growth when a > 1 and rapid decay when 0 < a < 1. They play a crucial role in modeling real-world phenomena such as population growth, radioactive decay, and compound interest.

Growth Rate Comparison

The growth rate of an exponential function is determined by its base. With bases such as 2, e, and 10, higher bases yield faster increasing functions for x > 0 and more rapid decreasing functions for x < 0. Thus, comparing how steeply a graph rises or falls helps in identifying which function corresponds to which base.

Reflection Over the y-axis

When the exponent in an exponential function is negated, as in y = a^(-x), the graph is reflected about the y-axis. This reflection interchanges the behavior of growth and decay; functions that normally grow for positive x now decay, and vice versa. Recognizing this transformation is key in distinguishing between functions like y = a^x and y = a^(-x).

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