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Chapter 2: Problem 11

Use transformations of graphs to sketch the graphs of \(y_{1}, y_{2},\) and \(y_{3}\) by hand. Check by graphing in an appropriate viewing window of your calculator. $$y_{1}=|x|, \quad y_{2}=|x-3|, \quad y_{3}=|x+3|$$

Short Answer

Expert verified
Graphs of \(y_2\) and \(y_3\) are horizontal shifts of \(y_1\). \(y_2\) shifts right by 3 units, and \(y_3\) shifts left by 3 units.

Step by step solution

01

Understanding the Parent Function

The parent function given is \(y_1 = |x|\), which is the absolute value function. This graph is V-shaped, with its vertex at the origin (0,0). The graph opens upwards and is symmetrical about the y-axis.
02

Graphing the Parent Function

Start by plotting \(y_1=|x|\). The key points on this graph are (0,0), (1,1), (-1,1), (2,2), and (-2,2). Join these points forming a V-shape, with the vertex at the origin.
03

Shifting the Graph Horizontally to Graph y2

To graph \(y_2 = |x-3|\), recognize that this is a horizontal shift of the parent function. The graph shifts 3 units to the right. So, the vertex of this graph moves from (0,0) to (3,0). Plot the points (3,0), (4,1), (2,1), (5,2), and (1,2) to produce the V-shape shifted to the right.
04

Shifting the Graph Horizontally to Graph y3

For \(y_3 = |x+3|\), this transformation shifts the graph of the absolute value function 3 units to the left. Translate the vertex from (0,0) to (-3,0). Plot the points (-3,0), (-2,1), (-4,1), (-1,2), and (-5,2) to reflect this leftward shift.
05

Verifying with a Calculator

Use a graphing calculator to plot all three functions \(y_1 = |x|\), \(y_2 = |x-3|\), and \(y_3 = |x+3|\). Ensure they match the hand-drawn graphs, with the shifts noted for \(y_2\) and \(y_3\). The calculator will confirm if the transformations were done accurately.

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Key Concepts

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

Understanding the Absolute Value Function
The absolute value function is often represented as \( f(x) = |x| \). This function creates a V-shaped graph on a coordinate plane. It has a vertex, which is the turning point where the graph changes direction. The vertex of the basic absolute value function is at the origin, notably at the coordinates (0,0). The arms of this V-shape extend upwards equally in both directions along the y-axis, creating a symmetrical appearance.
  • V-shape graph.
  • Vertex at the point (0,0).
  • Symmetrical about the y-axis.
These characteristics make the absolute value function easy to identify. Being familiar with the shape and symmetry of this function lays the groundwork for understanding more complex graph transformations, such as shifts and reflections.
Understanding Horizontal Shifts in Graphs
Horizontal shifts are a common and fundamental type of graph transformation. They involve shifting the graph left or right across the x-axis without altering its shape or orientation. This shift is dictated by a constant added to or subtracted from the variable x.
In our exercise, observe how the absolute value function \( |x| \) is shifted:
  • For \( y_2 = |x-3| \), the graph shifts 3 units to the right. This is because subtracting a number moves the graph right.
  • For \( y_3 = |x+3| \), the graph shifts 3 units to the left, as adding a number moves the graph left.
These horizontal shifts affect only the x-coordinates of the function's graph, moving the entire structure parallel to the x-axis. The vertex of the absolute value function thus moves horizontally to new positions based on the shift applied.
The Role of the Parent Function
Every transformation begins with a parent function, the simplest form of a function's family. In this context, the parent function is the absolute value function \( y_1 = |x| \).The role of the parent function is crucial:
  • It serves as a reference point. You start with the basic shape.
  • Transformations like shifts, stretches, or reflections are then applied to create new functions while maintaining the core characteristics.
Using \( y_1 = |x| \) as the parent, we introduced horizontal shifts to create \( y_2 = |x-3| \) and \( y_3 = |x+3| \). Understanding the parent function helps in visualizing how new graphs will look after transformations are applied. By mastering the parent function, understanding any derived function becomes significantly easier.

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