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Chapter 2: Problem 26

Use translations to graph the given functions. $$ m(x)=|x+1| $$

Short Answer

Expert verified
Translate the graph of \( f(x) = |x| \) 1 unit to the left to obtain \( m(x) = |x+1| \).

Step by step solution

01

Identify the Parent Function

The parent function for the given function is the absolute value function, written as \( f(x) = |x| \). This is a V-shaped graph that opens upwards with its vertex at the origin (0,0).
02

Determine the Transformation

The given function is \( m(x) = |x+1| \). This can be seen as a horizontal translation of the parent function \( f(x) = |x| \). Since the term inside the absolute value is \( x+1 \), the graph is translated 1 unit to the left.
03

Apply the Translation

To graph \( m(x) = |x+1| \), take the parent function \( f(x) = |x| \) and shift it 1 unit to the left. This means the new vertex of the translated function will be at (-1, 0).
04

Plot the Translated Graph

Plot the new vertex at (-1,0). Since this is a translation and the shape of the graph does not change, draw the V-shaped graph opening upwards with the vertex now at (-1,0). The slopes of the lines forming the V are still 1 and -1.

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Key Concepts

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

Parent Function
A parent function is a basic function that is not transformed. It is the simplest form of a graph in a family of functions. For the given function, the parent function is the absolute value function, denoted as \( f(x) = |x| \). This function is key to understanding how other functions in this family behave.
The graph of the absolute value parent function is a V-shaped curve that opens upward.
The vertex, or the point where the graph changes direction, is at the origin (0,0). When we talk about transforming this graph, we refer to shifts or changes applied to the parent function.
Absolute Value Function
An absolute value function returns the distance of a number from zero on a number line, disregarding any negative signs. In graphical terms, this function forms a V-shape because it 鈥渂ounces鈥� off the x-axis at the vertex.
Mathematically, it is written as \( f(x) = |x| \). This tells us the output is always non-negative, producing values like 0, 1, 2, no matter the input.
For instance, both \( |3| \) and \( |-3| \) equal 3.
This property is why the graph is symmetric across the y-axis.
Horizontal Translation
Horizontal translation involves shifting the entire graph of a function left or right. For the function \( m(x) = |x+1| \), the graph of the parent function \( f(x) = |x| \) is shifted left by 1 unit.
This happens because of the transformation inside the absolute value: \( x+1 \). The general rule for horizontal translation is:
  • If the transformation is \( |x+c| \), shift the graph left by \( c \) units.
  • If the transformation is \( |x-c| \), shift the graph right by \( c \) units.

Applying this to our example, the graph moves left 1 unit, causing the vertex to shift from (0,0) to (-1,0). This process does not change the shape of the graph, it only changes its position on the xy-plane.

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Most popular questions from this chapter

Refer to the functions \(r, p,\) and \(q .\) Evaluate the function and write the domain in interval notation. \(r(x)=-3 x \quad p(x)=x^{2}+3 x \quad q(x)=\sqrt{1-x}\) $$(r \cdot q)(x)$$

A function is given. (See Examples \(4-5)\) a. Find \(f(x+h)\). b. Find \(\frac{f(x+h)-f(x)}{h}\). $$f(x)=5 x+9$$

Provide an informal explanation of a relative maximum.

In computer programming the greatest integer function is sometimes called the "floor" function. Programmers also make use of the "ceiling" function which returns the smallest integer not less than \(x .\) For example: ceil( 3.1\()=4\). For Exercises \(115-116\), evaluate the floor and ceiling functions for the given value of \(x\). Floor \((x)\) is the greatest integer less than or equal to \(x\). Ceil \((x)\) is the smallest integer not less than \(x\). a. floor (5.5) b. floor (-0.1) c. floor (-2) d. ceil(5.5) e. ceil (-0.1) f. \(\operatorname{ceil}(-2)\)

a. Given \(m(x)=4 x^{2}+2 x-3,\) find \(m(-x)\). b. Find \(-m(x)\). c. Is \(m(-x)=m(x)\) ? d. Is \(m(-x)=-m(x)\) ? e. Is this function even, odd, or neither?
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