The graphs labeled (a)-(d) in the figure represent y=3^x y=5^x, y=((1)/(3))^x, and y=((1)/(5))^x

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The graphs labeled (a)-(d) in the figure represent y=3^x...

The graphs labeled (a)-(d) in the figure represent $y=3^{x}$ $y=5^{x}, y=\left(\frac{1}{3}\right)^{x},$ and $y=\left(\frac{1}{5}\right)^{x},$ but not necessarily in that order. Which is which? Describe the process that enables you to make this decision.
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Key Concepts

Exponential Functions

Exponential functions are mathematical functions in which the independent variable appears in the exponent, normally written as y = a^x. They model situations where quantities grow or decay at rates proportional to their current size, making them key to understanding phenomena in many fields of science and finance.

Growth versus Decay

Exponential functions can either be increasing or decreasing. When the base a is greater than 1, the function represents exponential growth, meaning it increases as x increases. Conversely, when the base is between 0 and 1, the function represents exponential decay, decreasing as x increases. Recognizing whether a graph is rising or falling helps in classifying the function as growth or decay.

Rate of Change/Steepness

The rate of change in exponential functions depends on the base: a larger base leads to a faster, steeper increase in the case of growth, and a smaller base (closer to 0) results in a faster, steeper decrease in the case of decay. This property is vital when comparing multiple exponential graphs because it helps determine which graph corresponds to which base.

Graphical Analysis Techniques

Graphical analysis of exponential functions involves examining features such as the y-intercept, asymptotic behavior, and the steepness of the curve. All exponential functions without shifts will pass through (0,1) and approach a horizontal asymptote (typically y = 0), but differences in steepness and direction (increasing or decreasing) are used to distinguish between functions with different bases.

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