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Magazine Chapter 10: Problem 36
Graph each of the exponential functions. $$ f(x)=2^{x-1} $$
Short Answer
Expert verified
The graph shifts right 1 unit and rises rapidly from left to right, never touching \( y = 0 \).
Step by step solution
01
Identify Base and Transformations
The function given is \( f(x) = 2^{x-1} \). In this function, the base is 2, which indicates that this is an exponential growth function. The exponent is \( x-1 \), which suggests a horizontal shift. Specifically, "\(-1\)" in \( x-1 \) suggests a shift to the right by 1 unit.
02
Create a Table of Values
To graph \( f(x) = 2^{x-1} \), first make a table of values by choosing a few \( x \) values, computing \( f(x) \) for each, and plotting these points. Choose \( x \) values such as -1, 0, 1, 2, and 3:- When \( x = -1 \), \( f(x) = 2^{-2} = \frac{1}{4} \).- When \( x = 0 \), \( f(x) = 2^{-1} = \frac{1}{2} \).- When \( x = 1 \), \( f(x) = 2^{0} = 1 \).- When \( x = 2 \), \( f(x) = 2^{1} = 2 \).- When \( x = 3 \), \( f(x) = 2^{2} = 4 \).
03
Plot the Points on a Coordinate Plane
Plot each pair from your table of values on the coordinate plane. For example, plot \((-1, \frac{1}{4})\), \((0, \frac{1}{2})\), \((1, 1)\), \((2, 2)\), and \((3, 4)\). These points will help you visualize the shape of the graph.
04
Draw the Graph
After plotting the points, draw a smooth curve through them. Since this is an exponential function, the graph should increase rapidly after the horizontal shift. The curve should get closer to the y-axis but never touch or cross it, reflecting the asymptote at \( x = 1 \).
05
Check the Asymptote and Direction
There's a horizontal asymptote as \( y \) approaches 0, which means the graph will never touch or cross the line \( y = 0 \). The function is increasing, reflecting exponential growth from left to right after the horizontal shift by 1 to the right.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
graphing exponential functions
Graphing exponential functions involves visually representing these functions' behavior on a coordinate plane.
This is crucial as it helps in understanding their growth patterns and characteristics over the x and y axes. To graph an exponential function, like \( f(x) = 2^{x-1} \), follow these steps:
This is crucial as it helps in understanding their growth patterns and characteristics over the x and y axes. To graph an exponential function, like \( f(x) = 2^{x-1} \), follow these steps:
- Start by identifying the base, in this case, 2. This helps determine the growth rate.
- Recognize any transformations, such as shifts or reflections.
- Create a table of values, by choosing several x-values and calculating the corresponding y-values.
- Plot these points on the coordinate plane to form the graph.
- Draw a smooth curve through the points, noting that exponential functions will typically curve upwards or downwards, avoiding the y-axis entirely unless shifted vertically.
exponential growth
Exponential growth describes how values increase rapidly over time.
In the function \( f(x) = 2^{x-1} \), this growth is evident as the values begin small and expand swiftly as x increases.
In the function \( f(x) = 2^{x-1} \), this growth is evident as the values begin small and expand swiftly as x increases.
- With a base greater than 1 (e.g., 2), exponential growth shows how the function values double for every increase in x.
- Such growth is dynamic and continuous, unlike linear growth, which increases at a constant rate.
- Exponential functions like this often model real-world phenomena, such as population growth or compound interest, where the rate of change accelerates rather than remaining fixed.
horizontal shift in graphs
A horizontal shift in graphs occurs when a function moves sideways, left or right, across the coordinate plane.
In \( f(x) = 2^{x-1} \), the expression "\(x-1\)" in the exponent indicates a shift.
In \( f(x) = 2^{x-1} \), the expression "\(x-1\)" in the exponent indicates a shift.
- The "-1" signifies shifting the graph one unit to the right.
- Understanding shifts is key to transforming the function accurately on a graph.
- Instead of merely moving the function vertically, a horizontal shift alters each point's x-coordinate.
coordinate plane
The coordinate plane is the fundamental grid that allows us to graph functions like \( f(x) = 2^{x-1} \).
It consists of two perpendicular axes: the horizontal x-axis, and the vertical y-axis.
It consists of two perpendicular axes: the horizontal x-axis, and the vertical y-axis.
- Each point on this plane is defined by an (x, y) coordinate, where you plot respective values from the function.
- It's essential for plotting exponential functions, ensuring accuracy in visualizing changes, such as growth or shifts.
- The grid structure helps students understand how mathematical functions behave in two dimensions, serving as a guide to explore patterns.
