What are the characteristics of the function f(x) = (1/4)^x and how does its graph look?
Understand the Problem
The question involves identifying the characteristics of an exponential function represented by the equation f(x) = (1/4)^x, and possibly interpreting its graph.
Answer
The graph of \( f(x) = \left(\frac{1}{4}\right)^x \) is a decreasing curve with a y-intercept at (0, 1) and a horizontal asymptote at \( y = 0 \).
Answer for screen readers
The graph of the function ( f(x) = \left(\frac{1}{4}\right)^x ) is a decreasing exponential function with a y-intercept at (0, 1) and a horizontal asymptote at ( y = 0 ).
Steps to Solve
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Determine the Base and Its Effect The function ( f(x) = \left(\frac{1}{4}\right)^x ) has a base of ( \frac{1}{4} ), which is less than 1. This indicates that the function is decreasing.
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Identify the Y-Intercept To find the y-intercept, substitute ( x = 0 ) into the function: $$ f(0) = \left(\frac{1}{4}\right)^0 = 1 $$ This shows the graph will intersect the y-axis at (0, 1).
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Analyze the Asymptote Exponential functions of this form have a horizontal asymptote at ( y = 0 ). As ( x ) approaches infinity, ( f(x) ) approaches 0 but never touches it.
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Evaluate Function Values Calculate a couple of additional values to plot:
- For ( x = 1 ): $$ f(1) = \left(\frac{1}{4}\right)^1 = \frac{1}{4} $$
- For ( x = 2 ): $$ f(2) = \left(\frac{1}{4}\right)^2 = \frac{1}{16} $$
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Sketch the Graph Using the points (0, 1), (1, 0.25), and (2, 0.0625), along with the asymptote, sketch the curve that decreases from the y-intercept toward the horizontal asymptote of ( y = 0 ).
The graph of the function ( f(x) = \left(\frac{1}{4}\right)^x ) is a decreasing exponential function with a y-intercept at (0, 1) and a horizontal asymptote at ( y = 0 ).
More Information
Exponential functions of the form ( a^x ) (where ( 0 < a < 1 )) always decrease as ( x ) increases. The base ( \frac{1}{4} ) plays a crucial role in the shape of the graph.
Tips
- Incorrectly assuming the function increases instead of decreases due to misinterpreting the base.
- Forgetting to consider the horizontal asymptote or the y-intercept when sketching the graph.
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