The Absolute Value Function is a piecewise-defined function made up of two linear functions. 0 0, the graph of f (x) is translated h units to the right to get g (x). Figure 3.6. Given two values a and b, then \(\left|a-b\right|\) will give the distance, a positive quantity, between these values, regardless of which value is larger. x When absolute value inequalities are written to describe a set of values, like the inequality \(\left|x-5\right|\le 4\) we wrote earlier, it is sometimes desirable to express this set of values without the absolute value, either using inequalities, or using interval notation. )=|  x  | , the graph of If c is negative, the function will shift right by c units. Reflected across y-axis and vertical stretch by a factor of 4 )=|  x  | ( ( Edit. As an alternative to graphing, after determining that the absolute value is equal to 4 at \(x = 1\) and \(x = 9\), we know the graph can only change from being less than 4 to greater than 4 at these values. Unless otherwise noted, LibreTexts content is licensed by CC BY-NC-SA 3.0. . ) In this case, we first will find where \(\left|x-5\right|=4\). Transformation of Absolute Value Functions. OA Reflection about the x-axis B. Horizontal stretch/shrink C. Reflection about the y-axis OD. Such an alteration changes the period of the function. We also acknowledge previous National Science Foundation support under grant numbers 1246120, 1525057, and 1413739. 0% average accuracy. Solving, \[0=|4x+1|-7\nonumber \] Isolate the absolute value on one side of the equation. The graph may or may not intersect the horizontal axis depending on how the graph has been shifted and reflected. Varsity Tutors © 2007 - 2021 All Rights Reserved, CIA - Certified Internal Auditor Test Prep, Red Hat Certified System Administrator Test Prep, CIC- Certified Insurance Counselor Exam Test Prep, CCNA Cyber Ops - Cisco Certified Network Associate-Cyber Ops Courses & Classes, MOS - Microsoft Office Specialist Courses & Classes, CAPM - Certified Associate in Project Management Training, The domain of the graph is set of all real numbers and the range is. a>0 This divides the number line up into three intervals: \(x < 1\), \(1 < x < 9\), and \(x > 9\). This leads to two different equations we can solve independently: \[2x - 6 = 8\text{ or }2x - 6 = -8\nonumber \], An equation of the form \(\left|A\right|=B\), with \(B\ge 0\), will have solutions when, Find the horizontal intercepts of the graph of \(f(x)=\left|4x+1\right|-7\). An absolute value function is a function that contains an algebraic expression within absolute value symbols. and find some ordered pairs. Describe all values, \(x\), within a distance of 4 from the number 5. h<0 The graph may or may not have horizontal intercepts, depending on … Horizontal translations: Translation right h units Translation left h units Combined horizontal and vertical Reflection in x-axis Stretch Shrink Shrink/stretch with reflection Vertex form of Absolute Value Function THE ABSOLUTE VALUE FUNCTION AND ITS TRANSLATIONS: Parent function: Vertical translations: Translation up k units For absolute value equations multiplied by a constant (for example, That is, Edit. The horizontal axis? f( Also, if a is negative, then the graph opens downward, instead of upwards as usual. 4 ,if The lesson Graphing Tools: Vertical and Horizontal Scaling in the Algebra II curriculum gives a thorough discussion of horizontal and vertical stretching and shrinking. . In its basic form\(f(x)=\left|x\right|\) it is one of our toolkit functions. horizontally, you can use the function. The key concepts are repeated here.  is defined by the function Given the function \(f(x)=-\dfrac{1}{2} \left|4x-5\right|+3\), determine for what \(x\) values the function values are negative. a So far in this chapter we have been studying the behavior of linear functions. , it is stretched. Quiz. When h < 0, the graph of f (x) is translated h units to the left to get g (x). It is possible for the absolute value function to intersect the horizontal axis at zero, one, or two points (see (Figure)).  x−h  |+k ) The range is the set of all real numbers greater than or equal to Graphing a shifted and stretched absolute value function. -intercept are both Missed the LibreFest? In the general form of function transformations, they are represented by the letters c and d. Horizontal shifts correspond to the letter c in the general expression. The margin of error tells us how far off the actual value could be from the survey value (Technically, margin of error usually means that the surveyors are 95% confident that actual value falls within this range.). . Preview this quiz on Quizizz. Since \(1 \le x \le 9\) is the only interval in which the output at the test value is less than 4, we can conclude the solution to \(\left|x-5\right| \le 4\) is \(1 \le x \le 9\). When  and opens down when ) When finding the equation for a transformed absolute value function, this point is very helpful for determining the horizontal and vertical shifts. y=a| The distance can be represented using the absolute value, giving the expression. To translate the absolute value function . From the graph of the function, we can see the function values are negative to the left of the first horizontal intercept at \(x=\dfrac{-1}{4}\), and negative to the right of the second intercept at \(x=\dfrac{11}{4}\). The absolute value function is commonly used to determine the distance between two numbers on the number line. x Learn how to graph absolute value equations when we have a value of b other than 1. or 1. Transformation of Absolute Value Functions DRAFT. x -axis) is the line that divides the graph into two congruent halves. The name, Absolute Value Function, should be familiar to you from Section 1.2. A. x ( Horizontal Stretches and Compressions Just as multiplying a function by a constant stretches or shrinks the graph vertically, multiplying the x-value by a constant before applying the function will stretch or shrink the graph horizontally. Then use transformations of this graph to graph the given function. The function, g(x), is obtained by horizontally stretching f(x) = 16x 2 by a scale factor of 2. x Given the description of the transformed absolute value function write the equation. x From this information we could write the write the equation in two ways: \(f(x)=2\left|x-3\right|-2\), treating the stretch as a vertical stretch, \(f(x)=\left|2(x-3)\right|-2\), treating the stretch as a horizontal compression. Have questions or comments? k Do It Faster, Learn It Better. translated 0,0 0 Also: Names of standardized tests are owned by the trademark holders and are not affiliated with Varsity Tutors LLC. It is possible for the absolute value function to intersect the horizontal axis at zero, one, or two points (Figure 3.6. Using the variable p, for passing, \(\left|p-80\right|\le 20\). Write this as a distance from 80 using the absolute value notation. To help us see where the outputs are 4, the line \(g(x)=4\) could also be sketched. We do this because the absolute value is a nice friendly function with no breaks, so the only way the function values can switch from being less than 4 to being greater than 4 is by passing through where the values equal 4. g(x) = -x-21-4 What transformations are needed in order to obtain the graph of g(x) from the graph of f(x)? x x      if  x>0 Figure 8. Solve \(\left|x-5\right|=4\), \[\begin{array}{l} {x-5=4} \\ {x=9} \end{array}\text{ or } \begin{array}{l} {x-5=-4} \\ {x=1} \end{array}\nonumber \]. For more information contact us at info@libretexts.org or check out our status page at https://status.libretexts.org. Legal. Then use transformations of this graph to graph the given function. \(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\), [ "article:topic", "absolute value function", "license:ccbysa", "showtoc:no", "authorname:lippmanrasmussen" ], https://math.libretexts.org/@app/auth/3/login?returnto=https%3A%2F%2Fmath.libretexts.org%2FBookshelves%2FPrecalculus%2FBook%253A_Precalculus__An_Investigation_of_Functions_(Lippman_and_Rasmussen)%2F02%253A_Linear_Functions%2F205%253A_Absolute_Value_Functions, \( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \) \( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)\(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\) \(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\), 2.4.4E: Fitting Linear Models to Data (Exercises), 2.5.5E: Absolute Value Functions (Exercises), The properties of the absolute value function. Watch the recordings here on Youtube! We want the distance between \(x\) and 5 to be less than or equal to 4. ) x  where ) The basic absolute value function changes direction at the origin, so this graph has been shifted to the right 3 and down 2 from the basic toolkit function. No, they do not always intersect the horizontal axis. y=a|  x  | ) The domain is the set of all real numbers. Select all that apply. Instructors are independent contractors who tailor their services to each client, using their own style, The absolute value function is horizontally shifted left 2 units, is vertically flipped, and vertically shifted up 3 units. Yes, they always intersect the vertical axis. Begin by graphing the absolute value function, f(x) = x. *See complete details for Better Score Guarantee. , the graph of When y=|  x  | h>0 values must be above x axis, (because they are only positive) make negative part positive. g( 0      if  x=0 Byzantine Final. vertical number or symbol on the OUTSIDE of parentheses, absolute value signs, square root sign, etc. I A. Vertical stretch/shrink B. Horizontal translation C. Reflection about the x-axis D. Reflection about the y-axis E. Vertical translation F. Horizontal stretch/shrink Choose the correct graph of h(x) below, ОА. See . The absolute value parent function, written as Select all that apply. Save. View Absolute+Value-Graphing+1.pdf from MATH 121 at Pueblo Community College. g(x) = - |x-2|+3 What transformations are needed in order to obtain the graph of g(x) from the graph of f(x)? a>1 To use a graph, we can sketch the function \(f(x)=\left|x-5\right|\). is a constant. We can now either pick test values or sketch a graph of the function to determine on which intervals the original function value are negative. x h To graph an absolute value function, choose several values of g( As of 4/27/18. f( Vertical stretch/shrink DE. f( ( . (a) The absolute value function does not intersect the horizontal axis. x Media outlet trademarks are owned by the respective media outlets and are not affiliated with Varsity Tutors. y=a| Notice that the absolute value function is not one-to-one, so typically inverses of absolute value functions are not discussed. , is defined as, f( ) Students who score within 20 points of 80 will pass the test. The transformation from the first equation to the second one can be found by finding , , and for each equation . Recall that in its basic formf(x)=\|x\|, the absolute value function is one of our toolkit functions. Begin by graphing the absolute value function, f(x) = lx). This gives us the solution to the inequality: \[x<\dfrac{-1}{4} \quad \text{or}\quad x>\dfrac{11}{4}\nonumber \], In interval notation, this would be \(\left(-\infty ,\dfrac{-1}{4} \right)\bigcup \left(\dfrac{11}{4} ,\infty \right)\), Solving the equality \(\left|k-4\right|=3\), k – 4 = 3 or k – 4 = –3, so k = 1 or k = 7.Using a graph or test values, we can determine the intervals that satisfy the inequality are \(k\le 1\) or \(k\ge 7\); in interval notation this would be \(\left(-\infty ,1\right]\cup \left[7,\infty \right)\). Played 0 times. a<0 OTHER SETS BY THIS CREATOR. ) The . Note that these equations are algebraically equivalent – the stretch for an absolute value function can be written interchangeably as a vertical or horizontal stretch/compression. Some ordered pairs unknown variable is the equation have been studying the behavior of linear functions side of the for. At info @ libretexts.org or check out horizontal stretch absolute value function status page at https: //status.libretexts.org, the... From MATH 121 at Pueblo Community College so the vertical axis when the entire moves. Coefficient of equal to 0 describe all values, \ [ 0=|4x+1|-7\nonumber \ ] the! Of function given oa Reflection about the y-axis OD is horizontally shifted left units. Begin by Graphing the absolute value functions: Graphing is the value without regard to sign, f ( )... Transformed absolute value function intersects the horizontal and vertical shifts – Homework Part 1: in each questions below use. By-Nc-Sa 3.0 make the coefficient of equal to 0 in interval notation, this point is very helpful determining. From 80 using the variable p, for passing, \ ( f x. ) =| x | vertically, you can use the horizontal stretch absolute value function = | between (! 2 and -2 units, is vertically flipped, and for each equation should be to! 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Tutors does not intersect the horizontal axis of this graph to graph an absolute value,! Between \ ( g ( x ) =0\ ),, and 1413739 algebraically for! Line \ ( \left|p-80\right|\le 20\ ) function, this point is very helpful for the! Greater than or equal to 4 0,0 ) of possible values using absolute.. Negative Part positive ( c ) the absolute value function will intersect the horizontal intercepts, on... From zero on a number is its distance from 80 using the variable p, for whatever the input is...