Problem 36 - Algebra for College Students

Question

Graph each of the exponential functions. $$ f(x)=2^{x-1} $$

Step-by-Step Solution

Answer

The graph shifts right 1 unit and rises rapidly from left to right, never touching \( y = 0 \).

Step 1: 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.

Step 2: 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 \( x = 3 \), \( f(x) = 2^{2} = 4 \).

Step 3: 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.

Step 4: 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 \( x = 1 \).

Step 5: 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.

Key Concepts

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:

• 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.

Remember, the "smooth curve" is essential, as it reflects how the function’s values continuously increase or decrease, based on the function parameters.

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.

• 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.

It's crucial to recognize this growth in graphing since the curve will steepen rapidly, reflecting the ever-increasing rate at which the function's values soar.

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.

• 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.

This ensures that the base's impact (the rate of growth or decay) isn't misunderstood. Through identifying the exact nature of the shift, graphing becomes more precise and predictive for complex functions.

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.

• 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.

Understanding the coordinate plane's role ensures you can accurately translate abstract mathematical equations into tangible graphs, aiding comprehension and analysis of exponential behaviors.