In Exercises 19–24, the graph of an exponential function is given. Select the function for each graph from the following options:
$f(x) = 3^x, \(\quad\) g(x) = 3^{x-1}, \(\quad\) h(x) = 3^x - 1, \(\f\)(x) = -3^x, \(\quad\) G(x) = 3^{-x}, \(\quad\) H(x) = -3^{-x}.$

Verified step by step guidance

1. Step 1: Identify the general shape of the graph. The graph shows exponential growth, which means the base of the exponential function is greater than 1, and the function increases as x increases.
2. Step 2: Check the y-intercept of the graph. The y-intercept occurs when x = 0. From the graph, observe the value of the function at x = 0.
3. Step 3: Compare the y-intercept to the given functions. For example, for f(x) = 3^x, the y-intercept is 3^0 = 1. For g(x) = 3^(x-1), the y-intercept is 3^{-1} = 1/3. For h(x) = 3^x - 1, the y-intercept is 1 - 1 = 0.
4. Step 4: Look at the behavior of the graph for negative x-values. If the graph approaches zero but never touches the x-axis, it matches the behavior of 3^x or its transformations. If the graph is reflected or shifted, it will affect this behavior.
5. Step 5: Based on the y-intercept and the shape, match the graph to the correct function from the options: f(x) = 3^x, g(x) = 3^(x-1), h(x) = 3^x - 1, f(x) = -3^x, G(x) = 3^{-x}, or H(x) = -3^{-x}.

Key Concepts

Exponential Functions

An exponential function has the form f(x) = a^x, where the base a is a positive constant not equal to 1. These functions exhibit rapid growth or decay depending on the base and the exponent's sign. Understanding the general shape and behavior of exponential graphs is essential for identifying specific functions.

Horizontal and Vertical Shifts

Shifts in exponential functions occur when constants are added or subtracted inside the exponent or outside the function. For example, f(x) = 3^(x-1) shifts the graph right by 1 unit, while f(x) = 3^x - 1 shifts it down by 1 unit. Recognizing these shifts helps match graphs to their corresponding equations.

Reflections and Negative Exponents

A negative sign in front of the function reflects the graph across the x-axis, while a negative exponent reflects it across the y-axis. For instance, f(x) = -3^x flips the graph vertically, and g(x) = 3^(-x) flips it horizontally. Identifying these reflections is key to distinguishing between similar exponential functions.