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Chapter 1: Problem 3

A nonrigid transformation of \(y=f(x)\) represented by \(g(x)=c f(x)\) is a ______ ______when \(c>1\) and a ______ ______ when \(0

Short Answer

Expert verified
The nonrigid transformation \(g(x) = c f(x)\) results in a vertical stretch when \(c > 1\) and a vertical shrink when \(0 < c < 1\).

Step by step solution

01

Definition of transformations

In mathematics, a transformation refers to the operation of constructing a new mathematical object from an existing one. When we multiply the function \(f(x)\) by a constant \(c\), we are performing a nonrigid transformation on \(f(x)\), also known as a dilation.
02

Identify the transformation when \(c>1\)

When \(c>1\), the graph of \(f(x)\) is stretched vertically. This type of transformation where the graph of a function is stretched or compressed vertically is called a vertical stretch.
03

Identify the transformation when \(0

On the other hand, when \(0<c<1\), the graph of \(f(x)\) is compressed vertically. This type of transformation is referred to as a vertical shrink.

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Key Concepts

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

Dilation
When we talk about dilation in the context of functions, we mean changing the shape of the graph of a function by multiplying it by a constant value. This operation scales the graph either up or down in a proportional manner. It鈥檚 like zooming in or out on the graph without altering its overall shape.

In mathematical terms, if you have a function \(f(x)\) and you multiply it by a constant \(c\), creating a new function \(g(x) = c \cdot f(x)\), you are performing a dilation.

This transformation is sometimes referred to as a 'nonrigid transformation' because the graph can stretch or shrink vertically based on the value of \(c\). Dilation plays a crucial role in understanding how functions behave when scaled, which is essential in both theoretical and applied mathematics.
Vertical Stretch
A vertical stretch occurs when you multiply a function by a constant greater than 1 (\(c > 1\)). This transformation makes the graph of the function appear taller as it effectively stretches the graph away from the x-axis.

For example, if \(c = 2\) and your original function is \(f(x)\), the transformed function \(g(x) = 2f(x)\) will have its y-values doubled. This causes the points on the graph to move farther apart vertically.

Key points to understand a vertical stretch include:
  • The higher the value of \(c\), the greater the stretch.
  • If the graph of the original function has a peak at a certain y-value, a vertical stretch will make this peak taller.
Understanding vertical stretches is useful in creating models that emphasize or exaggerate certain features of data.
Vertical Shrink
Vertical shrinkage is the opposite of vertical stretching. It occurs when the function is multiplied by a constant between 0 and 1 (\(0 < c < 1\)). This makes the graph look flatter, as it compresses the graph towards the x-axis.

Consider a function \(f(x)\) and a constant \(c = 0.5\). The new function \(g(x) = 0.5f(x)\) has the y-values halved, bringing them closer to the x-axis.

To identify a vertical shrink, consider:
  • The smaller the value of \(c\) (but still greater than zero), the more the graph compresses.
  • If \(f(x)\) had a high peak, \(g(x)\) after a vertical shrink will show a less prominent peak.
This transformation is especially important in scenarios where it's beneficial to lessen the impact of certain data features.

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Most popular questions from this chapter

A rectangle is bounded by the \(x\) -axis and the semicircle \(y=\sqrt{36-x^{2}}\) (see figure). Write the area \(A\) of the rectangle as a function of \(x,\) and graphically determine the domain of the function.

The table shows the numbers of tax returns (in millions) made through e-file from 2003 through \(2010 .\) Let \(f(t)\) represent the number of tax returns made through e-file in the year \(t .\) (Source: Internal Revenue Service) $$\begin{array}{|c|c|}\hline \text { Year } & \text { Number of Tax Returns Made Through E-File } \\\\\hline 2003 & 52.9 \\\2004 & 61.5 \\\2005 & 68.5 \\\2006 & 73.3 \\\2007 & 80.0 \\\2008 & 89.9 \\\2009 & 95.0 \\\2010 & 98.7 \\\\\hline\end{array}$$ (a) Find \(\frac{f(2010)-f(2003)}{2010-2003}\) and interpret the result in the context of the problem. (b) Make a scatter plot of the data. (c) Find a linear model for the data algebraically. Let \(N\) represent the number of tax returns made through e-file and let \(t=3\) correspond to 2003 (d) Use the model found in part (c) to complete the table. $$\begin{array}{|l|l|l|l|l|l|l|l|l|l|l|}\hline t & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 \\\\\hline N & & & & & & & & \\ \hline\end{array}$$ (e) Compare your results from part (d) with the actual data. (f) Use a graphing utility to find a linear model for the data. Let \(x=3\) correspond to \(2003 .\) How does the model you found in part (c) compare with the model given by the graphing utility?

The function \(F(y)=149.76 \sqrt{10} y^{5 / 2}\) estimates the force \(F\) (in tons) of water against the face of a dam, where \(y\) is the depth of the water (in feet). (a) Complete the table. What can you conclude from the table? $$\begin{array}{|l|l|l|l|l|l|}\hline y & 5 & 10 & 20 & 30 & 40 \\\\\hline F(y) & & & & & \\\\\hline\end{array}$$ (b) Use the table to approximate the depth at which the force against the dam is \(1,000,000\) tons. (c) Find the depth at which the force against the dam is \(1,000,000\) tons algebraically.

Use a graphing utility to graph the function. Be sure to choose an appropriate viewing window. $$f(x)=3 x^{2}-1.75$$

Write a sentence using the variation terminology of this section to describe the formula. Surface area of a sphere: \(S=4 \pi r^{2}\)
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