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Chapter 2: Problem 34

Use transformations to graph the functions. $$ n(x)=3|x| $$

Short Answer

Expert verified
Graph of n(x) = 3|x| is a vertically stretched V-shaped absolute value function.

Step by step solution

01

- Understand the parent function

The parent function for this exercise is the absolute value function, given by f(x) = |x|. The graph of this function is a V-shaped curve that opens upwards.
02

- Identify the transformation

The given function n(x) = 3|x| includes a vertical stretch by a factor of 3. In other words, each point on the graph of |x| is moved 3 times farther away from the x-axis.
03

- Apply the transformation

To graph n(x) = 3|x|, take each y-coordinate of the parent function |x| and multiply it by 3. For example, if x = 1, then |1| = 1, which becomes 3 脳 1 = 3 on the transformed graph.
04

- Sketch the graph

Draw the V-shaped curve with the transformation applied. This means the standard V-shaped graph of |x| will be stretched vertically, making the graph steeper.

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Key Concepts

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

Parent Function
In mathematics, the term 'parent function' refers to the simplest form of a given family of functions. It serves as the baseline or reference for other transformations. For example, the parent function of linear functions is f(x) = x. The importance of recognizing the parent function lies in using it as the starting point for graphing transformations.
In our exercise, the parent function is the absolute value function, represented as f(x) = |x|. This forms a V-shaped graph that opens upwards.
Absolute Value Function
The absolute value function is a special type of piecewise function often depicted as f(x) = |x|. This function represents the distance of a number on the number line from zero, regardless of its direction. The absolute value function transforms all inputs to their non-negative equivalents. For instance:
  • |3| = 3
  • |-4| = 4
Graphically, this function creates a distinct V shape. The vertex of the V is at the origin (0,0), and the arms of the V extend symmetrically along the x-axis.
Vertical Stretch
A vertical stretch occurs when all y-values of a function are multiplied by a constant greater than 1. This transformation makes the graph steeper as it pulls points away from the x-axis.
In the given function n(x) = 3|x|, the absolute value function |x| undergoes a vertical stretch by a factor of 3. To apply this transformation, multiply the y-coordinate of each point on the graph of the parent function |x| by 3. For example, if the point on |x| is (1,1), it transforms to (1,3) on n(x) = 3|x|.
Graph Transformation
Graph transformation involves altering the graph of a function in various ways, such as shifting, reflecting, stretching, or compressing. Understanding these transformations helps in sketching complex functions easily.
In our case, we take the graph of the parent function |x| and apply a vertical stretch. This process includes:
  • Identifying the parent function (f(x) = |x|)
  • Recognizing the transformation (vertical stretch by a factor of 3)
  • Applying the transformation to each point (multiplying y-coordinates by 3)
  • Drawing the new graph with these transformed points
The resulting graph for n(x) = 3|x| is a steeper V-shaped curve compared to the parent function |x|.

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