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Chapter 0: Problem 40

Use a computer or a graphing calculator in Problems \(37-40 .\) Let \(f(x)=1 /\left(x^{2}+1\right)\). Using the same axes, draw the graphs of \(y=f(x), y=f(2 x),\) and \(y=f(x-2)+0.6,\) all on the domain [-4,4]

Short Answer

Expert verified
Graph \( f(x) = \frac{1}{x^2+1} \), \( f(2x) = \frac{1}{4x^2+1} \), \( f(x-2)+0.6 = \frac{1}{(x-2)^2+1}+0.6 \) over \([-4, 4]\).

Step by step solution

01

Define the Functions

First, let's clearly define each function we need to graph. We have three functions:- The original function is \( f(x) = \frac{1}{x^2 + 1} \).- The transformed function \( y = f(2x) \) means we will evaluate the function at \( 2x \), giving us \( y = \frac{1}{(2x)^2 + 1} = \frac{1}{4x^2 + 1} \).- The function \( y = f(x-2) + 0.6 \) implies a horizontal shift to the right by 2 and a vertical shift upwards by 0.6, so \( y = \frac{1}{(x-2)^2 + 1} + 0.6 \).
02

Set Up the Graphing Range

We are instructed to graph all functions over the domain \([-4, 4]\). This means for each function, we'll evaluate and plot the function values starting at \( x = -4 \) and ending at \( x = 4 \).
03

Graph the First Function

Using your calculator or graphing software, graph \( y = \frac{1}{x^2 + 1} \). Note that this function is symmetric about the y-axis and peaks at \( x = 0 \) with a maximum value of 1. It decreases as \( |x| \) increases, approaching 0 as \( x \) approaches 卤4.
04

Graph the Second Function

Now graph \( y = \frac{1}{4x^2 + 1} \). This transformation compresses the graph horizontally by a factor of 2. This means the maximum value remains the same at 1, but the function decreases more rapidly than \( y = \frac{1}{x^2 + 1} \) as \( |x| \) increases.
05

Graph the Third Function

For the final function, graph \( y = \frac{1}{(x-2)^2 + 1} + 0.6 \). This graph is shifted 2 units to the right, so it now peaks at \( x = 2 \), and the whole graph is raised by 0.6. This means the maximum value is 1.6 at \( x = 2 \).
06

Analyze the Graphs

Observe the differences in each transformation:- The compression with \( y = f(2x) \) results in a narrower graph.- The horizontal and vertical shift in \( y = f(x-2) + 0.6 \) moves the peak to \( x = 2 \) and lifts the entire curve upwards.

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Key Concepts

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

Horizontal Shift
A horizontal shift affects the position of the graph along the x-axis. When we look at the function transformation from \(f(x) = \frac{1}{x^2 + 1}\) to \(y = f(x-2) + 0.6\), we see a horizontal shift to the right by 2 units. This is caused by replacing \(x\) with \(x-2\) inside the function. Visually, every point on the graph moves 2 units to the right.
  • The term \(x-2\) means that the graph is shifted rightward by 2 units.
  • This keeps the shape of the graph the same, since the overall relationship between variables is unchanged.
  • It鈥檚 crucial to remember that a negative sign in \(x-a\) results in movement to the right.
After the shift, the graph of \(y = \frac{1}{(x-2)^2 + 1}\) peaks at \(x = 2\) instead of at \(x = 0\). Understanding this type of transformation helps to predict and analyze how a graph will move on the x-axis.
Vertical Shift
Vertical shifts change the position of the graph along the y-axis. In the function transformation \(y = f(x-2) + 0.6\), there's a vertical shift because of the "+0.6" outside the function. This means every point on the graph moves upwards by 0.6 units.
  • The "+0.6" indicates an upward shift of the entire graph.
  • This doesn鈥檛 change the shape or the width of the graph, only its vertical position.
  • Vertical shifts are straightforward: a plus sign moves the graph up, and a minus sign would move it down.
Consequently, the peak of the graph shifts from a maximum value of 1 to 1.6. Vertical shifts are essential for adjusting the baseline or zero level of the graph on a coordinate plane.
Function Compression
Function compression refers to the squeezing or stretching of a graph along the x-axis. In the transformation from \(f(x) = \frac{1}{x^2 + 1}\) to \(y = f(2x)\), we observe horizontal compression. This happens when we perform a transformation by multiplying the \(x\) variable inside the function by a constant, in this case, 2.
  • The transformation \(f(2x)\) means that each x-value is halved, causing the graph to compress.
  • This results in a steeper graph that reaches its peak more quickly and declines faster than the original.
  • Horizontal compression or stretching affects how 鈥榳ide鈥� or 鈥榥arrow鈥� the graph appears.
This transformation changes how rapidly the function values change as \(x\) moves away from zero. Recognizing this can help you better understand the effects of different transformations on function graphs.

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