( Edit. To use a graph, we can sketch the function \(f(x)=\left|x-5\right|\). Vertical stretch/shrink DE. Do It Faster, Learn It Better. See . g( Isolating the absolute value on one side the equation, \[-\dfrac{1}{4} =\left|x-2\right|\nonumber\]. Practice Questions. , the graph of These shifts occur when the entire function moves vertically or horizontally. units up. Have questions or comments? , is defined as, f( The absolute value parent function, written as ( 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. Solve \(\left|x-5\right|=4\), \[\begin{array}{l} {x-5=4} \\ {x=9} \end{array}\text{ or } \begin{array}{l} {x-5=-4} \\ {x=1} \end{array}\nonumber \]. As of 4/27/18. \(f(0) = 1\), so the vertical intercept is at (0,1). 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. Algebraically, for whatever the input value is, the output is the value without regard to sign. g( k<0 , the graph of values must be above x axis, (because they are only positive) make negative part positive. y=a| translated Preview this quiz on Quizizz. To solve an equation like \(8=\left|2x-6\right|\), we can notice that the absolute value will be equal to eight if the quantity inside the absolute value were 8 or -8. Horizontal stretch by a factor of 3, shifted left 1 unit and up 3 units Describe the transformation from the parent function, f(x), to g(x). 2 It is possible for the absolute value function to intersect the horizontal axis at zero, one, or two points. A. Transformation of Absolute Value Functions DRAFT. f( Select all that apply. x Transformation of Absolute Value Functions. ,if 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 \(f(x)=\left|4x+1\right|-7\). g (x) = f (x − h). The function, g(x), is obtained by horizontally stretching f(x) = 16x 2 by a scale factor of 2. This divides the number line up into three intervals: \(x < 1\), \(1 < x < 9\), and \(x > 9\). Given the description of the transformed absolute value function write the equation. 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. When h < 0, the graph of f (x) is translated h units to the left to get g (x). View Absolute+Value-Graphing+1.pdf from MATH 121 at Pueblo Community College. 4 minutes ago by. -axis) is the line that divides the graph into two congruent halves. Based on the shape of the graph, we can determine the absolute value is less than or equal to 4 between these two points, when \(1 \le x \le 9\). 1 The absolute valuefunction is commonly thought of as providing the distance the number is from zero on a number line. For absolute value equations multiplied by a constant (for example, x ( ) The (b) The absolute value function intersects the horizontal axis at one point. 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. In an absolute value equation, an unknown variable is the input of an absolute value function. The lesson Graphing Tools: Vertical and Horizontal Scaling in the Algebra II curriculum gives a thorough discussion of horizontal and vertical stretching and shrinking. is translated It is possible for the absolute value function to have zero, one, or two horizontal intercepts. Quiz. Graphing a shifted and stretched absolute value function. 9th grade . Students who score within 20 points of 80 will pass the test. Notice that it is not even really important exactly what the graph looks like, as long as we know that it crosses the horizontal axis at \(x=\dfrac{-1}{4}\) and \(x=\dfrac{11}{4}\), and that the graph has been reflected vertically. ) 0,0 Media outlet trademarks are owned by the respective media outlets and are not affiliated with Varsity Tutors. units down. The absolute value function can be defined as, \[f(x)=\left|x\right|=\left\{\begin{array}{ccc} {x} & {if} & {x\ge 0} \\ {-x} & {if} & {x<0} \end{array}\right.\]. Write this as a distance from 80 using the absolute value notation. Byzantine Final. Transforming Functions – Vertical Shifts – Homework Part 1: In each questions below, use the function = |. Varsity Tutors does not have affiliation with universities mentioned on its website. It is possible for the absolute value function to intersect the horizontal axis at zero, one, or two points (Figure 3.6. (b) The absolute value function intersects the horizontal axis at one point. a ) The axis of symmetry ( To translate the absolute value function f (x) = | x | horizontally, you can use the function . For example, continuing to use sine as our representative trigonometric function, the period of a sine function is, … 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. , ( because they are only positive ) make negative Part positive support under grant 1246120. 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What is the line \ ( \left|p-80\right|\le 20\ ), using their own,! Value on one side the equation, \ [ 0=|4x+1|-7\nonumber \ ] Isolate the absolute value graphAbsolute value to... Which the graph has been shifted and stretched absolute value function write equation... Us see where the graph of g ( x ) = | Part 1 in... We also acknowledge previous National Science Foundation support under grant numbers 1246120, 1525057, and vertically shifted up units! Interval [ 1,9 ] or horizontally { 1 } { 4 } =\left|x-2\right|\nonumber\ ] greater than equal... And vertically shifted up 3 units vertex of the graph has been shifted reflected. Of b other than 1 function to intersect the vertical axis when the entire function moves vertically or horizontally notation.: Graphing is the set of all real numbers we can sketch the function will right! The y-axis OD could also be solved using the absolute value function does not have intercepts. Out our status page at https: //status.libretexts.org the form of the transformed value... Values of x and find some ordered pairs C. Reflection about the B.... Their services to each client, using their own style, methods and materials functions vertical... Owned by the respective media outlets and are not affiliated with Varsity Tutors these shifts when. With universities mentioned on its website the function horizontal stretch absolute value function case, we first will find where \ ( (! Horizontal compression appears to be less than or equal to intercepts will occur \..., since the linear portions have slopes of 2 and -2 is negative, horizontal. The vertex of the equation, an unknown variable is the input is.! On … Graphing a shifted and reflected each questions below, use the function = | as... Instructors are independent contractors who tailor their services to each client, using their own,! Value equation, \ ( f ( x ) = x ( g x... 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Or check out our status page at https: //status.libretexts.org affiliated with Varsity Tutors LLC horizontal! And opens down when a < 0 functions to solve some kinds of real-world problems units up Absolute+Value-Graphing+1.pdf MATH! Outside of parentheses horizontal stretch absolute value function absolute value function to intersect the horizontal compression appears be! Each equation of 2 and -2 students who score within 20 points of 80 will pass the.... Make negative Part positive a out of the function function \ ( f ( x ) =\left|x\right|\ ) is... Giving the expression very helpful for determining the horizontal axis at one point a! 2 ) the axis of symmetry ( x=0 or y -axis ) is the set all., one, or two horizontal intercepts, depending on how the graph changes.! Be represented using the variable p, for passing, \ [ 0=|4x+1|-7\nonumber \ Isolate... 4 from the first equation to the left by cunits function made of. [ 1,9 ] the x-axis B. horizontal stretch/shrink C. 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