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Ch. 4 - Exponential and Logarithmic Functions
Blitzer - College Algebra 8th Edition
Blitzer8th EditionCollege AlgebraISBN: 9780136970514Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoBlitzer 8th EditionCh. 4 - Exponential and Logarithmic FunctionsProblema 23
Capitolo 5, Problema 23

The graph of an exponential function is given. Select the function for each graph from the following options:
f(x)=3x,g(x)=3x−1,h(x)=3x−1,f(x)=−3x,G(x)=3−x,H(x)=−3−x.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}.

Guida verificata passo dopo passo
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.
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.
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 \(H(x) = -3^{-x}\).
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.
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.

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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.
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Graphs of Exponential Functions

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.
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Transformations of Exponential Graphs

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.
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Graphs of Exponential Functions
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