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Chapter 5: Problem 3

Make a hand-drawn graph of each of the following. Then check your work using a graphing calculator. $$x=\left(\frac{1}{2}\right)^{y}$$

Short Answer

Expert verified
Plot points from a table of values (e.g., (4, -2), (2, -1)), draw the decreasing curve, and verify using a calculator.

Step by step solution

01

Understand the Equation

The equation given is in the form of a power function where the output, x, is an exponential function of the input, y. Specifically, the equation is: \[ x = \left(\frac{1}{2}\right)^{y} \]
02

Create a Table of Values

To plot the graph, create a table of values by choosing different values for y, and then calculating the corresponding values of x. For example:\[ \begin{array}{c|c} y & x \ \hline -2 & 4 \ -1 & 2 \ 0 & 1 \ 1 & 0.5 \ 2 & 0.25 \end{array} \]
03

Plot the Points

Using the table created, plot the points (x, y) on the Cartesian plane. These points are: (4, -2), (2, -1), (1, 0), (0.5, 1), and (0.25, 2).
04

Draw the Graph

Connect the points smoothly, recognizing that the graph will approach the x-axis but never actually touch it. The graph will be decreasing and will show the nature of exponential decay.
05

Verify Using a Graphing Calculator

Enter the equation into a graphing calculator to confirm that the hand-drawn graph is accurate. Plot the equation \(x = \left(\frac{1}{2}\right)^y\) and compare the graph on the calculator to your hand-drawn graph.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

hand-drawn graph
One critical step in understanding exponential functions is to draw the graph by hand. This helps you see the behavior of the function more intuitively. When you hand-draw a graph, make sure to:
  • First, identify key points using a table of values, picking both positive and negative inputs for the variable.
  • Next, plot these points on your graph paper.
  • Finally, connect the points smoothly, following the curvature indicated by the plotted points. Keep an eye on the graph's trend as it approaches but never touches the x-axis, demonstrating how the function behaves near zero.
Hand-drawing the graph provides a deeper understanding of how the function behaves under various inputs, highlighting key aspects like exponential decay clearly.
table of values
Creating a table of values is a methodological approach to understanding and graphing any function, especially exponential ones. To develop a table of values for \(\frac{1}{2}^y\), follow these steps:
  • Select a range of y-values, both positive and negative, because exponential functions can span much broader ranges.
  • Calculate the corresponding x-values using the function \(\frac{1}{2}^y\).
  • For example:
    • If y=-2, x = \(4\)
    • If y=-1, x = \(2\)
    • If y=0, x = \(1\)
    • If y=1, x = \(0.5\)
    • If y=2, x = \(0.25\)
Organizing these values into a table allows you to see how x changes as y changes and provides precise points to plot on your graph.
exponential decay
Exponential functions can exhibit exponential growth or decay. When dealing with \(\frac{1}{2}^y\), we are looking at exponential decay. This means as y increases, the value of x decreases rapidly. Characteristics of exponential decay include:
  • The function decreases rapidly at first and then levels off but never actually reaches zero.
  • For negative values of y, x increases exponentially.
  • This behavior is evident in the table of values and is clearly illustrated in the hand-drawn graph.
  • Recognize that the closer y gets to positive infinity, the closer x gets to zero, but it will never actually be zero.
Understanding these properties helps in identifying and graphing exponential decay functions accurately.
graphing calculator
Using a graphing calculator can assist in verifying the accuracy of your hand-drawn graph. Here鈥檚 how to use it effectively:
  • Enter the function \(x = \left(\frac{1}{2}\right)^y\) into the calculator.
  • Generate the graph and observe the points plotted.
  • Compare these points with your hand-drawn graph to ensure alignment.
  • The calculator will precisely show the curve of the function, including highlighting the exponential decay behavior and the asymptote approaching the x-axis.
Graphing calculators offer a visual confirmation and can provide additional insights into complex behaviors which might be trickier to accurately represent by hand, helping to solidify your understanding.

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