Linear Absolute Value & Quadratic Functions Review Worksheet

2.5: Accented Value Functions

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And then far in this chapter we accept been studying the beliefs of linear functions. The Absolute Value Part is a piecewise-defined function made up of two linear functions. The name, Absolute Value Part, should exist familiar to you from Department one.2. In its basic form\(f(x)=\left|x\correct|\) it is i of our toolkit functions.

Definition: Absolute Value Part

The absolute value function can exist defined every bit

\[f(10)=\left|ten\right|=\left\{\begin{assortment}{ccc} {x} & {if} & {x\ge 0} \\ {-x} & {if} & {x<0} \end{array}\right.\]

The absolute value function is commonly used to determine the distance betwixt ii numbers on the number line. Given two values a and b, then \(\left|a-b\right|\) will requite the altitude, a positive quantity, betwixt these values, regardless of which value is larger.

Instance \(\PageIndex{1}\)

Describe all values, \(x\), inside a distance of iv from the number five.

Solution

We desire the altitude between \(x\) and v to be less than or equal to 4. The distance can be represented using the absolute value, giving the expression

\[\left|10-five\right|\le iv\nonumber \]

Case \(\PageIndex{2}\)

A 2010 poll reported 78% of Americans believe that people who are gay should exist able to serve in the US military, with a reported margin of mistake of 3% (http://www.pollingreport.com/civil.htm, retrieved Baronial 4, 2010). The margin of error tells us how far off the bodily value could be from the survey value (Technically, margin of error usually means that the surveyors are 95% confident that actual value falls inside this range.). Express the set of possible values using absolute values.

Solution

Since we want the size of the difference between the bodily percentage, \(p\), and the reported percentage to be less than iii%,

\[\left|p-78\right|\le 3\nonumber \]

Practise \(\PageIndex{one}\)

Students who score within 20 points of 80 will laissez passer the test. Write this as a distance from 80 using the absolute value notation.

Answer

Using the variable p, for passing, \(\left|p-80\right|\le twenty\)

Of import Features

The most significant feature of the absolute value graphAbsolute Value Functions:Graphing is the corner point where the graph changes direction. When finding the equation for a transformed absolute value function, this betoken is very helpful for determining the horizontal and vertical shifts.

Example \(\PageIndex{3}\)

Write an equation for the function graphed.

Solution

The basic absolute value office changes direction at the origin, so this graph has been shifted to the correct 3 and down 2 from the basic toolkit role.

We might too notice that the graph appears stretched, since the linear portions have slopes of 2 and -2. From this information we could write the write the equation in two ways:

\(f(ten)=2\left|x-3\right|-two\), treating the stretch as a vertical stretch

\(f(10)=\left|two(10-three)\right|-2\), treating the stretch as a horizontal pinch

Annotation that these equations are algebraically equivalent – the stretch for an absolute value function can be written interchangeably as a vertical or horizontal stretch/pinch.

If you had not been able to make up one's mind the stretch based on the slopes of the lines, you can solve for the stretch factor by putting in a known pair of values for 10 and f(x)

\[f(x)=a\left|x-3\right|-2\nonumber \] At present substituting in the point (one, ii)

\[\begin{array}{l} {two=a\left|1-3\correct|-2} \\ {four=2a} \\ {a=2} \stop{array}\nonumber \]

Exercise \(\PageIndex{2}\)

Given the description of the transformed absolute value function write the equation. The absolute value function is horizontally shifted left 2 units, is vertically flipped, and vertically shifted up 3 units.

Reply

\[f(ten)=-\left|x+2\correct|+three\nonumber \]

The graph of an absolute value function will accept a vertical intercept when the input is zero. The graph may or may not have horizontal intercepts, depending on how the graph has been shifted and reflected. It is possible for the absolute value function to have zero, one, or two horizontal intercepts.

Zero horizontal intercepts I Two

To find the horizontal intercepts, nosotros volition need to solve an equation involving an accented value.

Notice that the accented value function is non one-to-i, and so typically inverses of absolute value functions are not discussed.

Solving Accented Value Equations

Absolute Value Functions:Solving

To solve an equation like \(8=\left|2x-vi\correct|\), we tin can notice that the accented value will be equal to eight if the quantity within the accented value were 8 or -8. This leads to two different equations we tin can solve independently:

\[2x - 6 = 8\text{ or }2x - 6 = -8\nonumber \]

\[2x = 14\quad 2x = -ii\nonumber\]

\[ten = vii\quad x = -1\nonumber\]

solutions to absolute value equations

An equation of the class \(\left|A\correct|=B\), with \(B\ge 0\), volition have solutions when

\[A=B\text{ or }A=-B\nonumber\]

Example \(\PageIndex{four}\)

Find the horizontal intercepts of the graph of \(f(x)=\left|4x+i\correct|-vii\)

Solution

The horizontal intercepts will occur when \(f(x)=0\). Solving,

\[0=|4x+one|-vii\nonumber \] Isolate the absolute value on one side of the equation

\[7=|4x+ane|\nonumber \] At present we can break this into 2 divide equations:

\[7 = 4x + 1\quad -vii = 4x + ane\nonumber\]

\[6 = 4x\text{ or }-8 = 4x\nonumber\]

\[10 = \dfrac{6}{4} = \dfrac{three}{2}\quad x = \dfrac{-8}{4} = -2\nonumber\]

The graph has two horizontal intercepts, at \(x=\dfrac{3}{2}\) and \(x = -2\)

Example \(\PageIndex{5}\)

Solve \(i=iv\left|x-2\right|+two\)

Solution

Isolating the absolute value on one side the equation,

\[1=4\left|x-2\right|+2\nonumber\]

\[-1=4\left|ten-two\correct|\nonumber\]

\[-\dfrac{1}{iv} =\left|x-ii\right|\nonumber\]

At this point, nosotros observe that this equation has no solutions – the absolute value always returns a positive value, so information technology is impossible for the absolute value to equal a negative value.

Practise \(\PageIndex{three}\)

Find the horizontal & vertical intercepts for the function\(f(ten)=-\left|x+2\right|+iii\)

Answer

\(f(0) = one\), and then the vertical intercept is at (0,1).

\(f(ten)= 0\) when

\[-\left|x+ii\correct|+iii=0\nonumber \]

\[\left|x+2\right|=3\nonumber \]

\[ten+2=3\text{ or }ten+2=-3\nonumber\]

\[x = ane\text{ or }x = -5\nonumber \] so the horizontal intercepts are at (-v,0) & (i,0)

Solving Absolute Value Inequalities

Accented Value Functions:Solving Inequalities

When absolute value inequalities are written to depict a gear up of values, similar the inequality \(\left|x-5\right|\le 4\) we wrote before, information technology is sometimes desirable to express this set of values without the absolute value, either using inequalities, or using interval notation.

We will explore 2 approaches to solving accented value inequalities:

1. Using the graph
2. Using test values

Instance \(\PageIndex{half-dozen}\)

Solve \(\left|x-five\right|\le iv\)

Solution

With both approaches, we will need to know first where the corresponding equality is truthful. In this example, we get-go will observe where \(\left|x-5\right|=4\). We do this considering the absolute value is a nice friendly function with no breaks, so the only way the role values can switch from being less than 4 to being greater than 4 is past passing through where the values equal iv. Solve \(\left|x-5\correct|=four\),

\[\begin{assortment}{l} {x-5=4} \\ {x=9} \end{assortment}\text{ or } \begin{assortment}{l} {x-5=-4} \\ {x=ane} \stop{array}\nonumber \]

To use a graph, nosotros can sketch the office \(f(x)=\left|x-5\right|\). To help u.s. see where the outputs are iv, the line \(g(x)=4\) could also be sketched.

On the graph, we can see that indeed the output values of the absolute value are equal to 4 at \(10 = 1\) and \(x = 9\). Based on the shape of the graph, we can make up one's mind the absolute value is less than or equal to four betwixt these 2 points, when \(1 \le x \le 9\). In interval notation, this would be the interval [1,9].

As an alternative to graphing, subsequently determining that the absolute value is equal to 4 at \(10 = 1\) and \(ten = 9\), we know the graph tin but modify from beingness less than iv to greater than four at these values. This divides the number line up into three intervals: \(x < 1\), \(1 < x < ix\), and \(ten > ix\). To determine when the part is less than iv, nosotros could pick a value in each interval and see if the output is less than or greater than 4.

Since \(1 \le ten \le 9\) is the but interval in which the output at the test value is less than 4, we can conclude the solution to \(\left|ten-5\right| \le 4\) is \(i \le x \le nine\).

Example \(\PageIndex{7}\)

Given the function \(f(x)=-\dfrac{1}{ii} \left|4x-5\correct|+3\), determine for what \(x\) values the function values are negative.

Solution

Nosotros are trying to determine where \(f(ten) < 0\), which is when \(-\dfrac{1}{2} \left|4x-5\right|+3<0\). We brainstorm by isolating the accented value:

\[-\dfrac{ane}{2} \left|4x-5\right|<-3\nonumber\] when we multiply both sides past -2, it reverses the inequality

\[\left|4x-5\right|>vi\nonumber \]

Next we solve for the equality \(\left|4x-5\right|=six\)

\[\brainstorm{assortment}{l} {4x-v=6} \\ {4x=11} \\ {x=\dfrac{11}{iv} } \finish{array}\text{ or }\begin{array}{fifty} {4x-5=-6} \\ {4x=-1} \\ {x=\dfrac{-ane}{4} } \end{array}\nonumber \]

We tin can now either choice test values or sketch a graph of the part to determine on which intervals the original function value are negative. Discover that it is non even really important exactly what the graph looks like, as long as we know that it crosses the horizontal axis at \(x=\dfrac{-1}{iv}\) and \(10=\dfrac{11}{4}\), and that the graph has been reflected vertically.

From the graph of the function, nosotros tin can see the function values are negative to the left of the first horizontal intercept at \(x=\dfrac{-1}{4}\), and negative to the correct of the second intercept at \(x=\dfrac{11}{4}\). This gives usa the solution to the inequality:

\[x<\dfrac{-1}{iv} \quad \text{or}\quad x>\dfrac{xi}{4}\nonumber \]

In interval notation, this would be \(\left(-\infty ,\dfrac{-1}{iv} \right)\bigcup \left(\dfrac{11}{iv} ,\infty \correct)\)

Practise \(\PageIndex{four}\)

Solve\(-ii\left|k-4\right|\le -six\)

Respond

\[-2\left|k-iv\right|\le -half dozen\nonumber \]

\[\left|1000-4\right|\ge 3\nonumber \]

Solving the equality \(\left|yard-four\correct|=3\), yard – 4 = iii or grand – 4 = –iii, so grand = i or k = seven.Using a graph or test values, we tin can decide the intervals that satisfy the inequality are \(k\le i\) or \(1000\ge 7\); in interval notation this would be \(\left(-\infty ,ane\right]\cup \left[7,\infty \right)\)

Important Topics of this Section

• The properties of the absolute value function
• Solving accented value equations
• Finding intercepts
• Solving absolute value inequalities

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Source: https://math.libretexts.org/Bookshelves/Precalculus/Book%3A_Precalculus__An_Investigation_of_Functions_(Lippman_and_Rasmussen)/02%3A_Linear_Functions/205%3A_Absolute_Value_Functions

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