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Chapter 3: Problem 13

Exer. 13-26: Sketch, on the same coordinate plane, the graphs of \(f\) for the given values of \(c\). (Make use of symmetry, shifting, stretching, compressing, or reflecting.) $$ f(x)=|x|+c ; \quad c=-3,1,3 $$

Short Answer

Expert verified
Graph \( |x|-3 \), \( |x|+1 \), and \( |x|+3 \) by vertically shifting \( |x| \).

Step by step solution

01

Understand the Function

The function given is \( f(x) = |x| + c \). This is an absolute value function shifted vertically. The absolute value function \( |x| \) is V-shaped and symmetrical about the y-axis.
02

Identify Shifts Based on Values of c

For different values of \( c \), the graph will be vertically shifted. When \( c = -3 \), it shifts down by 3 units. When \( c = 1 \), it shifts up by 1 unit. When \( c = 3 \), it shifts up by 3 units.
03

Graph the Function for each c

First, graph \( f(x) = |x| \), which is a V-shape with the vertex at the origin (0,0). For \( c = -3 \), shift this V-shape 3 units down, placing the vertex at (0, -3). For \( c = 1 \), shift the original V-shape 1 unit up, placing the vertex at (0, 1). For \( c = 3 \), shift the V-shape 3 units up, placing the vertex at (0, 3).
04

Sketch All Three Graphs

On the same coordinate plane, draw the three V-shaped graphs representing each function. Ensure each is properly positioned based on the vertical shifts identified in the previous step.

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Key Concepts

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

Vertical Shifts
In mathematics, vertical shifts refer to the movement of a graph up or down on the coordinate plane. For the function \( f(x) = |x| + c \), the constant \( c \) determines this vertical shift.
If \( c \) is positive, the graph moves up by \( c \) units, and if \( c \) is negative, it shifts down by the same number of units.
Vertical shifts do not change the shape of the graph but simply move it along the y-axis.
In our exercise, we have three different values for \( c \): -3, 1, and 3:
  • When \( c = -3 \), the graph is shifted 3 units down.
  • When \( c = 1 \), it moves 1 unit upwards.
  • Finally, for \( c = 3 \), the graph is shifted 3 units upwards.
This understanding helps in accurately sketching the graphs on a shared coordinate plane.
Graphing
Graphing an absolute value function like \( f(x) = |x| + c \) allows us to visualize the effect of vertical shifts. In this case, the absolute value function is V-shaped, and it begins typically at the origin.
To graph these functions, a baseline graph, \( f(x) = |x| \), is initially plotted. This baseline has a vertex at the origin, (0, 0).
Depending on the value of \( c \), you apply the determined vertical shift:
  • For \( c = -3 \), lower the graph three units down.
  • For \( c = 1 \), lift the graph one unit upwards.
  • For \( c = 3 \), raise it three units above the baseline.
Graphing these accurately involves simply plotting the V-shape for each adjusted vertex position, ensuring each graph maintains the V formation.
Symmetry
Symmetry in absolute value functions plays a vital role, especially since \( |x| \) is naturally symmetric about the y-axis. Symmetry means that if you fold the graph along the y-axis, both halves will match perfectly.
This property is preserved even with vertical shifts, as they do not alter the x-values or the shape of the graph. Therefore, all graphs of the form \( f(x) = |x| + c \) share this y-axis symmetry, making them visually balanced.
This symmetry helps in understanding and predicting the graph's behavior, as any transformation involving vertical shifts retains the V-shape's original symmetry.
Coordinate Plane
The coordinate plane is the stage where these absolute value functions are graphed, providing a structured eye view of how each function behaves.
It consists of two axes: the horizontal x-axis and the vertical y-axis, intersecting at the origin (0, 0).
Each point on the plane is defined by a pair (x, y), representing its location.
For our function \( f(x) = |x| + c \), the V-shaped graph's vertex changes position along the y-axis in the coordinate plane, depending upon the value of \( c \).
  • All movements remain parallel to the y-axis, only shifting up or down.
  • Visualizing these shifts on the coordinate plane allows for a clear representation of how each function differs with different c values.
This makes the coordinate plane an essential tool for understanding the relationships between variable shifts and graph transformations.

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Most popular questions from this chapter

Exer. 59-62: Sketch the graph of the equation. $$ y=|\sqrt{x}-1| $$

Exer. 13-26: Sketch, on the same coordinate plane, the graphs of \(f\) for the given values of \(c\). (Make use of symmetry, shifting, stretching, compressing, or reflecting.) $$ f(x)=\frac{1}{2} \sqrt{x-c} ; \quad c=-2,0,3 $$

Exer. 11-20: Find (a) \((f \circ g)(x)\) (b) \((g \circ f)(x)\) (c) \(f(g(-2))\) (d) \(g(f(3))\) $$ f(x)=4 x, \quad g(x)=2 x^{3}-5 x $$

Exer. 53-54: The symbol \(\llbracket x \rrbracket\) denotes values of the greatest integer function. Sketch the graph of \(f\). (a) \(f(x)=\llbracket x-3 \rrbracket\) (b) \(f(x)=\llbracket x \rrbracket-3\) (c) \(f(x)=2 \llbracket x \rrbracket\) (d) \(f(x)=\llbracket 2 x \rrbracket\) (e) \(f(x)=\llbracket-x \rrbracket\)

Exer. 33-40: Explain how the graph of the function compares to the graph of \(y=f(x)\). For example, for the equation \(y=2 f(x+3)\), the graph of \(f\) is shifted 3 units to the left and stretched vertically by a factor of 2 . $$ y=-2 f\left(\frac{1}{3} x\right) $$
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