Sketch the graphs of f(x) = 4^x, f(x) = 4 · 4^x, and f(x) = 1/4 · 4^x on the same set of axes. Then describe the similarities and differences among the graphs.
Understand the Problem
The question is asking us to sketch the graphs of four exponential functions and to analyze their similarities and differences. We'll approach this by understanding the transformation properties of each function.
Answer
The graphs all exhibit exponential growth, with variations in height and growth rates due to vertical and horizontal transformations.
Answer for screen readers
The graphs of the functions exhibit similarities in that they all pass through the point (0,1) and display exponential growth. Differences lie in their vertical stretches and horizontal scaling, affecting their growth rates.
Steps to Solve
- Identify the functions to be graphed
The functions to graph are:
- $f(x) = 4^x$
- $f(x) = 4 \cdot 4^x$
- $f(x) = \frac{1}{4} \cdot 4^x$
- $f(x) = 4^{(1/4)x}$
- Determine key characteristics of each function
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For $f(x) = 4^x$: This is the basic exponential function, which increases rapidly as $x$ increases and approaches $0$ as $x$ decreases.
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For $f(x) = 4 \cdot 4^x$: This function is a vertical stretch of the basic function by a factor of $4$. This means it will have the same general shape but will be higher on the graph.
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For $f(x) = \frac{1}{4} \cdot 4^x$: This is a vertical compression of the basic function, making it lower on the graph by a factor of $\frac{1}{4}$.
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For $f(x) = 4^{(1/4)x}$: This function has a base of $4$ raised to the power of $\frac{1}{4}x$, meaning it will grow slower than $4^x$, depicting a horizontal stretch.
- Sketch the functions
Plotting the functions on the same set of axes:
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For $f(x) = 4^x$: Start from (0,1), and as $x$ increases, points like (1,4) and (2,16) should be plotted.
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For $f(x) = 4 \cdot 4^x$: It starts higher, at (0,4), and has points like (1,16) and (2,64).
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For $f(x) = \frac{1}{4} \cdot 4^x$: Begin at (0,1) but goes to (1,4) and (2,16), but starts lower for negative $x$ values.
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For $f(x) = 4^{(1/4)x}$: Start at (0,1), but the growth is more gradual; for example, at (4,4) instead of (1,4).
- Analyze similarities and differences
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Similarities: All functions pass through the point (0,1). They exhibit exponential growth behavior as $x$ increases.
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Differences: The vertical stretches and compressions affect the height. The horizontal stretch or compression (in $f(x) = 4^{(1/4)x}$) affects how quickly the function grows.
The graphs of the functions exhibit similarities in that they all pass through the point (0,1) and display exponential growth. Differences lie in their vertical stretches and horizontal scaling, affecting their growth rates.
More Information
The basic function $f(x) = 4^x$ provides a foundation, while the other three functions demonstrate variations through vertical scaling or horizontal stretching, which are key concepts in understanding transformations of exponential functions.
Tips
- Forgetting to include the vertical stretch or compression when sketching modified exponential functions.
- Misinterpreting the horizontal stretch, leading to incorrect growth rates for the function $f(x) = 4^{(1/4)x}$.
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