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 a decreasing curve that approaches zero as x increases, which is characteristic of an exponential decay function.
2. Step 2: Recall the given function options: \(f(x) = 3^x\), \(g(x) = 3^{x-1}\), \(h(x) = 3^x - 1\), \(f(x) = -3^x\), \(G(x) = 3^{-x}\), and \(H(x) = -3^{-x}\). Notice that functions with \$3^x\( grow exponentially, while those with \)3^{-x}$ decay exponentially.
3. Step 3: Since the graph is decreasing and approaches zero as x increases, focus on functions with negative exponents, such as \(G(x) = 3^{-x}\) and \(H(x) = -3^{-x}\).
4. Step 4: Check the sign of the function values. The graph is above the x-axis (positive values), so the function is positive. This rules out \(H(x) = -3^{-x}\), which would be negative.
5. Step 5: Verify the y-intercept by substituting \(x=0\) into \(G(x) = 3^{-x}\). This gives \(G(0) = 3^0 = 1\). Compare this with the graph's y-intercept to confirm the match.

Key Concepts

Exponential Functions and Their Graphs

Exponential functions have the form f(x) = a^x, where the base a is positive and not equal to 1. Their graphs show rapid growth or decay, depending on the base and the exponent's sign. Understanding the shape and behavior of these graphs helps identify the function from its graph.

Transformations of Exponential Functions

Transformations include shifts and reflections. Horizontal shifts occur when the exponent is modified (e.g., 3^(x-1) shifts the graph right by 1), vertical shifts occur when a constant is added or subtracted outside the function (e.g., 3^x - 1 shifts down by 1), and reflections occur when the function is multiplied by -1 or the exponent is negated.

Identifying Decreasing vs. Increasing Exponential Functions

An exponential function with a base greater than 1 and a positive exponent is increasing, while if the exponent is negative or the function is multiplied by -1, the graph decreases. Recognizing whether the graph is increasing or decreasing is key to matching it with the correct function.