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Chapter 4: Problem 18

a. Generate a small table of values and plot the function \(y=|x|\) for \(-5 \leq x \leq 5\) b. On the same graph, plot the function \(y=|x-2|\).

Short Answer

Expert verified
Create tables for y=|x| and y=|x-2|, then plot both functions on the same graph, showing both V-shapes with vertices at (0,0) and (2,0) respectively.

Step by step solution

01

- Create the Table of Values for y=|x|

Create a table with two columns, one for the variable x and one for the function y=|x|. Fill in the values of x from -5 to 5. Calculate the corresponding y-values using the function y=|x|. For example, when x=-5, y=|-5|=5, and when x=0, y=|0|=0. Continue this for all x-values.
02

- Generate the Table of Values for y=|x-2|

Create another table with two columns, one for the variable x and one for the function y=|x-2|. Fill in the values of x from -5 to 5. Calculate the corresponding y-values using the function y=|x-2|. For example, when x=-5, y=|-5-2|=7, and when x=0, y=|0-2|=2. Continue this for all x-values.
03

- Plot the Function y=|x|

Using graph paper or graphing software, plot the points from the table of values for y=|x|. Draw the graph by joining the points smoothly. Make sure the graph creates a V-shape with the vertex at the origin (0,0).
04

- Plot the Function y=|x-2|

On the same graph, plot the points from the table of values for y=|x-2|. Draw the graph by joining the points smoothly. The graph will also form a V-shape, but the vertex will be shifted to (2,0).
05

- Label the Graphs

Ensure that both graphs are clearly labeled. Label the graph of y=|x| and the graph of y=|x-2|. Additionally, mark the important points like the vertices clearly.

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Key Concepts

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

function table
Creating a function table is essential for understanding absolute value functions. To construct a table of values for a function like y = |x|, start by choosing a range of x-values. In our exercise, we used values from -5 to 5. For each x-value, compute the corresponding y-value by applying the absolute value function.
For example:
  • When x = -5, y = |-5| = 5
  • When x = 0, y = |0| = 0
  • When x = 3, y = |3| = 3
Doing this for all x-values ensures you have a complete set of points to plot on a graph.
graphing functions
Graphing functions, like y = |x|, helps visualize how the function behaves. Once you have your table of values, use graphing paper or software to plot each point. For the function y = |x|, you'll notice that it creates a V-shaped graph.
Key characteristics of the graph include:
  • Vertex at the origin (0,0)
  • Symmetrical about the y-axis
By plotting each point accurately and connecting them smoothly, you make the graph's shape clear and understandable.
vertex shift
Understanding vertex shifts is crucial in graphing transformed functions. For the equation y = |x-2|, compare it to the parent function y = |x|. The vertex of y = |x-2| is shifted to (2,0). This happens because the subtraction inside the absolute value indicates a horizontal shift.
Key points to recognize about vertex shifts:
  • If the function is y = |x-a|, the vertex shifts to (a,0)
  • A positive 'a' shifts the vertex to the right
  • A negative 'a' shifts the vertex to the left
plotting data
Plotting data accurately is vital for correct graph representation. When you have your function tables ready for y = |x| and y = |x-2|, carefully plot each corresponding x and y value on the same graph.
Steps to plot data successfully include:
  • Mark the x- and y-axes clearly.
  • Choose a consistent scale for both axes.
  • Plot each point from your function tables consistently.
  • Join the points smoothly to reflect the function's shape.
By following these steps, you ensure your graph reflects the mathematical relationships accurately.

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Most popular questions from this chapter

Simplify each expression by removing all possible factors from the radical, then combining any like terms. a. \(2 \sqrt{50}+12 \sqrt{8}\) b. \(3 \sqrt{27}-2 \sqrt{75}\) c. \(10 \sqrt{32}-6 \sqrt{18}\) d. \(2 \sqrt[3]{16}+4 \sqrt[3]{54}\)

Radio waves, sent from a broadcast station and picked up by the antenna of your radio, are a form of electromagnetic (EM) radiation, as are microwaves, X-rays, and visible, infrared, and ultraviolet light. They all travel at the speed of light. Electromagnetic radiation can be thought of as oscillations like the vibrating strings of a violin or guitar or like ocean swells that have crests and troughs. The distance between the crest or peak of one wave and the next is called the wavelength. The number of times a wave crests per minute, or per second for fast-oscillating waves, is called its frequency. Wavelength and frequency are inversely proportional: the longer the wavelength, the lower the frequency, and vice versa-the faster the oscillation, the shorter the wavelength. For radio waves and other \(\mathrm{EM}\), the number of oscillations per second of a wave is measured in hertz, after the German scientist who first demonstrated that electrical waves could transmit information across space. One cycle or oscillation per second equals 1 hertz \((\mathrm{Hz})\). For the following exercise you may want to find an old radio or look on a stereo tuner at the AM and FM radio bands. You may see the notation \(\mathrm{kHz}\) beside the AM band and MHz beside the FM band. AM radio waves oscillate at frequencies measured in the kilohertz range, and FM radio waves oscillate at frequencies measured in the megahertz range. a. The Boston FM rock station WBCN transmits at \(104.1 \mathrm{MHz}\). Write its frequency in hertz using scientific notation. b. The Boston AM radio news station WBZ broadcasts at 1030 \(\mathrm{kHz}\). Write its frequency in hertz using scientific notation. The wavelength \(\lambda\) (Greek lambda) in meters and frequency \(\mu\) (Greek mu) in oscillations per second are related by the formula \(\lambda=\frac{c}{\mu}\) where \(c\) is the speed of light in meters per second. c. Estimate the wavelength of the WBCN FM radio transmission. d. Estimate the wavelength of the WBZ AM radio transmission. e. Compare your answers in parts (c) and (d), using orders of magnitude, with the length of a football field (approximately 100 meters).

Earth travels in an approximately circular orbit around the sun. The average radius of Earth's orbit around the sun is \(9.3 \cdot 10^{7}\) miles. Earth takes one year, or 365 days, to complete one orbit. a. Use the formula for the circumference of the circle to determine the distance the Earth travels in one year. b. How many hours are in one year? c. Speed is distance divided by time. Find the orbital speed of Earth in miles per hour.

Each of the following simplifications contains an error made by students on a test. Find the error and correct the simplification so that the expression becomes true. a. \(\left[\left(x^{2}\right)^{3}\right]^{5}=\left[x^{5}\right]^{5}=x^{25}\) b. \(\frac{7 x^{2} y^{6}}{(x y)^{2}}=\frac{7 x^{2} y^{6}}{x^{2} y^{2}}=7 x^{4} y^{8}\) c. \(\left(\frac{4 x^{3} y^{5}}{6 x y^{4}}\right)^{3}=\left(\frac{2 x^{2} y}{3}\right)^{3}=\frac{2}{3} x^{6} y^{3}\) d. \(\left(1.1 \cdot 10^{6}\right) \cdot\left(1.1 \cdot 10^{4}\right)=1.1 \cdot 10^{6}\) e. \(\frac{4 \cdot 10^{6}}{8 \cdot 10^{3}}=0.5 \cdot 10^{3}=5.0 \cdot 10^{4}\) f. \(6 \cdot 10^{3}+7 \cdot 10^{5}=13 \cdot 10^{8}\)

Using rules of exponents, show that \(\frac{9^{5}}{27^{-7}}=3^{31}\).
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