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Chapter 1: Problem 7

For each function, sketch (on the same set of coordinate axes) a graph of each function for \(c = -1\), \(1\), and \(3\). (a) \(f(x) = |x| + c\) (b) \(f(x) = |x - c|\) (c) \(f(x) = |x + 4| + c\)

Short Answer

Expert verified
The graphs of the functions for different values of c show that c dictates shifts up or down (Function a), left or right (Function b) on the Cartesian plane, or combination of a constant 4 units leftward shift and vertical shift according to the value of c (Function c).

Step by step solution

01

Sketch Function (a)

Given \(f(x) = |x| + c\), the graph of the function will be a v-shaped curve that opens upward starting from point (0,c). The slope changes at x = 0. To draw this, sketch a v-shaped graph of \(|x|\), then vertically shift it c units: upwards if c > 0, downwards if c < 0, and no shift if c = 0. Do this for c = -1, 1, and 3.
02

Sketch Function (b)

Given \(f(x) = |x - c|\), the graph of the function will also be a v-shaped curve that opens upwards, but this time shifted to the right if c > 0 and to the left if c < 0. The slope changes at x = c. To draw this, sketch a v-shaped graph of \(|x|\), then horizontally shift it c units to the right if c > 0 and to the left if c < 0. Do this for c = -1, 1, and 3.
03

Sketch Function (c)

Given \(f(x) = |x + 4| + c\), the function first shifts 4 units to the left because of the \(|x + 4|\) part, and then shifts c units upwards or downwards according to the value of c. Vertical shift depends on c in the same way as in function (a) and horizontal shift is always 4 units to the left. The slope changes at x = -4. Sketch a v-shaped graph of \(|x| + c\), then shift it 4 units to the left and shift it vertically in accordance with the value of c. Do this for c = -1, 1, and 3.

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Key Concepts

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

Absolute Value Functions
The absolute value function is one of the most fundamental functions in mathematics. It is defined as \( f(x) = |x| \), which means that it takes any real number \( x \) and converts it to a non-negative value. This function transforms both positive and negative inputs into positive outputs.
  • If \( x \) is positive or zero, \( |x| = x \).
  • If \( x \) is negative, \( |x| = -x \).
This function creates a distinct V-shaped graph and is important for graph transformations, which include shifts and reflections. The vertex of the V-shaped graph represents the point where the function value is zero, and it plays a crucial role in understanding piecewise functions.
Graph Transformations
Graph transformations help us visualize how different changes affect the shape and position of a graph. For an absolute value function, transformations include shifts (both vertical and horizontal), reflections, and stretches. These adjustments allow us to move or change the graph while preserving its shape.
  • Vertical transformations change the graph's height or move it up and down.
  • Horizontal transformations focus on moving the graph left or right.
  • Reflections might flip the graph over a line, such as the x-axis or y-axis.
Understanding these transformations is essential for sketching graphs of absolute value functions with different formulas.
V-Shaped Graph
The graph of an absolute value function is notably V-shaped. This shape occurs because the function involves taking the absolute value of \( x \), producing two linear pieces that form a sharp corner.
The V-shape opens upwards and has a vertex where the function switches from decreasing to increasing. For example, in \( f(x) = |x| \), the vertex occurs at the origin (0,0). By changing the equation slightly, you can move the vertex to different locations without altering the overall V pattern.
This property of absolute value functions makes it easier to predict how changes in equations will affect the graph's appearance, especially when applying transformations.
Vertical Shifts
Vertical shifts involve moving the graph of a function up or down without changing its shape or orientation. For an absolute value function like \( f(x) = |x| + c \), the term \( c \) determines the vertical shift.
  • If \( c > 0 \), the graph shifts upward by \( c \) units.
  • If \( c < 0 \), it shifts downward by \( |c| \) units.
  • If \( c = 0 \), there is no vertical shift.
The slope and the V-shape remain consistent, while only the starting point changes. This shift is intuitive鈥攊magine lifting or lowering the entire graph while keeping the V intact.
Horizontal Shifts
Horizontal shifts move the graph left or right, adjusting the function's input to change the graph's starting position. This transformation occurs in functions such as \( f(x) = |x - c| \).
  • If \( c > 0 \), the graph shifts to the right by \( c \) units.
  • If \( c < 0 \), it shifts to the left by \( |c| \) units.
For example, with \( f(x) = |x - 2| \), the V-shape moves to the right, centering at \( x = 2 \). These shifts change where the V-shape's vertex is located along the x-axis, showing how horizontal transformations influence graph positions.

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Most popular questions from this chapter

MEASUREMENT When buying gasoline, you notice that 14 gallons of gasoline is approximately the same amount of gasoline as 53 liters. Use this information to find a linear model that relates liters \(y\) to gallons \(x\). Then use the model to find the numbers of liters in 5 gallons and 25 gallons.

The coiled spring of a toy supports the weight of a child. The spring is compressed a distance of 1.9 inches by the weight of a 25-pound child. The toy will not work properly if its spring is compressed more than 3 inches. What is the weight of the heaviest child who should be allowed to use the toy?

SPORTS The lengths (in feet) of the winning men's discus throws in the Olympics from 1920 through 2008 are listed below. (Source: International Olympic Committee) 1920 146.6 1924 151.3 1928 155.3 1932 162.3 1936 165.6 1948 173.2 1952 180.5 1956 184.9 1960 194.2 1964 200.1 1968 212.5 1972 211.3 1976 221.5 1980 218.7 1984 218.5 1988 225.8 1992 213.7 1996 227.7 2000 227.3 2004 229.3 2008 225.8 (a) Sketch a scatter plot of the data. Let \(y\) represent the length of the winning discus throw (in feet) and let \(t=20\) represent 1920. (b) Use a straightedge to sketch the best-fitting line through the points and find an equation of the line. (c) Use the \(regression\) feature of a graphing utility to find the least squares regression line that fits the data. (d) Compare the linear model you found in part (b) with the linear model given by the graphing utility in part (c). (e) Use the models from parts (b) and (c) to estimate the winning men's discus throw in the year 2012.

The mathematical model \(y = \frac{k}{x}\) is an example of ________ variation.

THINK ABOUT IT In Exercises 77-86, restrict the domain of the function \(f\) so that the function is one-to-one and has an inverse function. Then find the inverse function \(f^{-1}\). State the domains and ranges of \(f\) and \(f^{-1}\). Explain your results. (There are many correct answers.) \(f(x) = |x-4| + 1\)
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