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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

Graph each of the functions with a graphing utility. Determine whether the function is \(even\), \(odd\), or \(neither\). \(f(x) = x^2 - x^4\) \(g(x) = 2x^3 + 1\) \(h(x) = x^5 - 2x^3 + x\) \(j(x) = 2 - x^6 - x^8\) \(k(x) = x^5 - 2x^4 + x - 2\) \(p(x) = x^9 + 3x^5 - x^3 + x\) What do you notice about the equations of functions that are odd? What do you notice about the equations off unctions that are even? Can you describe a way to identify a function as odd or even by inspecting the equation? Can you describe a way to identify a function as neither odd nor even by inspecting the equation?

GRAPHICAL ANALYSIS In Exercises 103-106, identify any relationships that exist among the lines, and then use a graphing utility to graph the three equations in the same viewing window. Adjust the viewing window so that the slope appears visually correct\(-\)that is, so that parallel lines appear parallel and perpendicular lines appear to intersect at right angles. (a) \(y = 2x\) (b) \(y = -2x\) (c) \(y = \frac{1}{2}x\)

In Exercises 39-46, determine the intervals over which the function is increasing, decreasing, or constant. \(f(x) = x^3 - 3x^2 + 2\)

COLLEGE ENROLLMENT The University of Florida had enrollments of 46,107 students in 2000 and 51,413 students in 2008. (Source: University of Florida) (a) What was the average annual change in enrollment from 2000 to 2008? (b) Use the average annual change in enrollment to estimate the enrollments in 2002, 2004, and 2006. (c) Write the equation of a line that represents the given data in terms of the year \(t\), where \(t = 0\) corresponds to 2000. What is its slope? Interpret the slope in the context of the problem. (d) Using the results of parts (a)-(c) write a short paragraph discussing the concepts of \(slope\) and \(average rate of change\).

In Exercises 103-110, find the difference quotient and simplify your answer. \(f(x) = x^3+3x\), \(\frac{f(x+h)-f(x)}{h}\), \(h \neq 0\)
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