Sketch the graphs of f(x) = 4^x, f(x) = 4 ullet 4^x and f(x) = \frac{1}{4} ullet 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 the given exponential functions and describe their similarities and differences. Specifically, we need to analyze how the different coefficients and bases affect the shape and position of each graph on a coordinate axis.
Answer
The graphs of $f(x) = 4^x$, $f(x) = 4 \cdot 4^x$, and $f(x) = \frac{1}{4} \cdot 4^x$ illustrate exponential growth with different y-intercepts: (0, 1), (0, 4), and (0, 0.25), respectively.
Answer for screen readers
The sketches exhibit exponential growth for all functions with respective y-intercepts of:
- $f(x) = 4^x$: (0, 1)
- $f(x) = 4 \cdot 4^x$: (0, 4)
- $f(x) = \frac{1}{4} \cdot 4^x$: (0, 0.25)
Steps to Solve
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Understand the functions We have three functions to analyze:
- $f(x) = 4^x$
- $f(x) = 4 \cdot 4^x$
- $f(x) = \frac{1}{4} \cdot 4^x$
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Sketch the graph of $f(x) = 4^x$ The function $f(x) = 4^x$ is an exponential function with a base greater than 1. It passes through the point (0, 1) and increases rapidly as $x$ increases.
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Sketch the graph of $f(x) = 4 \cdot 4^x$ This function can be written as $f(x) = 4^{x+1}$, which is just a vertical stretch of the first function. It shifts the graph of $f(x) = 4^x$ upwards by a factor of 4, passing through (0, 4).
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Sketch the graph of $f(x) = \frac{1}{4} \cdot 4^x$ This function simplifies to $f(x) = 4^{x-1}$. It's a vertical compression of the first function, translating downwards so that it passes through (0, 0.25).
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Analyze similarities and differences
- Similarities: All three graphs exhibit exponential growth since they all have a base greater than 1.
- Differences: Their vertical shifts affect the y-intercept. The first graph crosses at (0, 1), the second at (0, 4), and the third at (0, 0.25).
The sketches exhibit exponential growth for all functions with respective y-intercepts of:
- $f(x) = 4^x$: (0, 1)
- $f(x) = 4 \cdot 4^x$: (0, 4)
- $f(x) = \frac{1}{4} \cdot 4^x$: (0, 0.25)
More Information
The graphs show how modifications of the base or coefficient in exponential functions influence the y-intercept and overall shape. Exponential functions are characteristically defined by their rapid growth or decay depending on their coefficients.
Tips
- Misinterpreting the coefficient as affecting the base rather than the overall vertical stretch or compression.
- Confusing the translations caused by the addition or subtraction in the exponent.
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