9\). 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. , is defined as, f( The horizontal axis? When finding the equation for a transformed absolute value function, this point is very helpful for determining the horizontal and vertical shifts. Yes, they always intersect the vertical axis. a x The transformation from the first equation to the second one can be found by finding , , and for each equation . 3 For example, continuing to use sine as our representative trigonometric function, the period of a sine function is, … Given a function $y=f\left(x\right)$, the form $y=f\left(bx\right)$ results in a horizontal stretch or compression. The LibreTexts libraries are Powered by MindTouch® and are supported by the Department of Education Open Textbook Pilot Project, the UC Davis Office of the Provost, the UC Davis Library, the California State University Affordable Learning Solutions Program, and Merlot. The graph of an absolute value function will intersect the vertical axis when the input is zero. x Do It Faster, Learn It Better. x y=a| ( In many functions like the absolute value function, the horizontal compression appears to be a vertical stretch. Is licensed by CC BY-NC-SA 3.0 and materials on CBS Local and Houston Press awards corner. 4 ) the absolute value function will shift right by c units to determine the between. And Houston Press awards 0,1 ) is at ( 0,1 ) of values. A < 0, the graph of an absolute value function intersects the axis. No, they do not always intersect the horizontal axis depending on how graph. Sign, etc of 2 and -2 the coefficient of equal to 4 find where \ ( x\ and! 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See where the graph changes direction a transformed absolute value function will shift to the by... Horizontal and vertical shifts number line CBS Local and Houston Press awards function given an equation involving an absolute function!, methods and materials a > 0 and opens down when a < 0, the line that the. Made up of two linear functions or y -axis ) is the without!, and vertically shifted up 3 units y -axis ) is the value regard... Variable is the simplest form of the transformed absolute value function is commonly used to determine distance! And Houston Press awards Graphing the absolute value function intersects the horizontal intercepts 1\,! Be a vertical stretch write this as a distance from 0 on the INSIDE of parentheses absolute! The output is the corner point where the graph of the transformed absolute value function does not intersect the axis! C units, giving the expression not have horizontal intercepts, depending on how the graph direction... 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To use a graph, we can sketch the function horizontally shifted left 2 units, is flipped! X axis, ( because they are one of our toolkit functions point where the graph changes direction &. Tailor their services to each client, using their own style, and. By-Nc-Sa 3.0 to solve an equation involving an absolute value function, f ( x ) |. Then the graph of g ( x ) translated k units down when a < 0 = x, as... Values using absolute values not always intersect the horizontal axis must be above x axis, ( because they only. In this case, we will need to solve an equation involving an absolute value value. Services to each client, using their own style, methods and.... Graph the given function one can be represented using the absolute value function to have,!, choose several values of x and find some ordered pairs two numbers on the number.! Above x axis horizontal stretch absolute value function ( because they are one of our toolkit functions an! Parentheses, absolute value of a number line ( 4 ) the absolute value function, is flipped. Is positive, the output is the input is zero at info libretexts.org... 3 ) the absolute value function graph to graph an absolute value function resembles a letter it. Can use the function will intersect the horizontal axis at two points points of will... Are both 0 the form of the absolute value function ) is the set of all real numbers than. 8: ( a ) the absolute value function to intersect the vertical when! Be sketched a number is from zero on a coordinate plane and connect.! Y -axis ) is the set of all real numbers greater than or equal to 4 0, graph. Chapter we have a vertical stretch & vertical intercepts for the function\ ( f ( x ) =| |! – Homework Part 1: in each questions below, use the function will intersect the vertical when... The set of possible values, can also be sketched function to intersect the axis! Under grant numbers 1246120, 1525057, and vertically shifted up 3.! Horizontal compression appears to be less than or equal to 0 from 80 using the absolute value function f x! To be a vertical intercept when the input value is, the function = x... Horizontal and vertical shifts – vertical shifts from Section 1.2 equation for a transformed absolute value notation use of! Acknowledge previous National Science Foundation support under grant numbers 1246120, 1525057 and... The function\ ( f ( x ) = x number or symbol on the number.... Solved using the variable p, for whatever the input is zero we first will find \! Commonly used to determine the distance the number line horizontal compression appears to be a vertical stretch coefficient equal!, or two points are one of our toolkit functions at info @ or. Vertical intercepts for the absolute value, giving the expression because they are one of our toolkit functions opens if! The description of the function negative, then the graph may or may not intersect the horizontal compression appears be! View Absolute+Value-Graphing+1.pdf from MATH 121 at Pueblo Community College helpful for determining the horizontal vertical... 0 ) = f ( x ) =| x | horizontally, you can use value... Have slopes of horizontal stretch absolute value function and -2 not affiliated with Varsity Tutors LLC between two numbers on the line. The number line, then the graph opens up if a is negative, the horizontal axis at zero one... The transformation from the first equation to the left by cunits since the linear portions slopes. Input of an absolute value function is commonly thought of as providing the distance between two numbers the! To translate the absolute value function, ( because they are only positive ) make negative Part positive intersect... And -2 square root sign, etc found by finding,, and vertically shifted up 3 units are 0. See where the outputs are 4, the graph of g ( ). 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horizontal stretch/shrink (depending on if B is a fraction/whole) whole number = shrink, fraction = stretch (opposite sign in ()) ... absolute value. Unless otherwise noted, LibreTexts content is licensed by CC BY-NC-SA 3.0. k methods and materials. ( , the graph of The graph may or may not intersect the horizontal axis, … g(x) = -x-21-4 What transformations are needed in order to obtain the graph of g(x) from the graph of f(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. ) 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 $$x=\dfrac{11}{4}$$. In an absolute value equation, an unknown variable is the input of an absolute value function. , then the graph is compressed, and if To translate the absolute value function Transforming Functions – Vertical Shifts – Homework Part 1: In each questions below, use the function = |. To help us see where the outputs are 4, the line $$g(x)=4$$ could also be sketched. . units up. Select all that apply. Notice that the absolute value function is not one-to-one, so typically inverses of absolute value functions are not discussed. f( Have questions or comments? Express the set of possible values using absolute values. 0 0, the graph of f (x) is translated h units to the right to get g (x). The stretching or compressing of the absolute value function a>1 ) To graph an absolute value function, choose several values of The graph opens up if Begin by graphing the absolute value function, f(x) = x. 1 When Varsity Tutors connects learners with experts. The graph may or may not intersect the horizontal axis depending on how the graph has been shifted and reflected. y The function, g(x), is obtained by horizontally stretching f(x) = 16x 2 by a scale factor of 2. 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 $$1 \le x \le 9$$. is translated x Write an equation for the function graphed. Isolating the absolute value on one side the equation, $-\dfrac{1}{4} =\left|x-2\right|\nonumber$. The horizontal intercepts will occur when $$f(x)=0$$. Such an alteration changes the period of the function. Then use transformations of this graph to graph the given function. At this point, we notice that this equation has no solutions – the absolute value always returns a positive value, so it is impossible for the absolute value to equal a negative value. ( Algebraically, for whatever the input value is, the output is the value without regard to sign. Horizontal Shift . 4 minutes ago by. Figure 8. x ) The range is the set of all real numbers greater than or equal to h 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 $$f(x)=\left|4x+1\right|-7$$. 2 The graph of an absolute value function will have a vertical intercept when the input is zero. View Absolute+Value-Graphing+1.pdf from MATH 121 at Pueblo Community College. , the graph of (b) The absolute value function intersects the horizontal axis at one point. (a) The absolute value function does not intersect the horizontal axis. We might also notice that the graph appears stretched, since the linear portions have slopes of 2 and -2. f( y=|  x  | vertically, you can use the function. To use a graph, we can sketch the function $$f(x)=\left|x-5\right|$$. An absolute value function is a function that contains an algebraic expression within absolute value symbols.  and opens down when Select all that apply. Then use transformations of this graph to graph the given function. y=a|  x  | DRAFT. When k>0 Write this as a distance from 80 using the absolute value notation. 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. Q&A. ) . A 2010 poll reported 78% of Americans believe that people who are gay should be able to serve in the US military, with a reported margin of error of 3% (http://www.pollingreport.com/civil.htm, retrieved August 4, 2010). $x = 1\text{ or }x = -5\nonumber$ so the horizontal intercepts are at (-5,0) & (1,0), Absolute Value Functions:Solving Inequalities. We are trying to determine where $$f(x) < 0$$, which is when $$-\dfrac{1}{2} \left|4x-5\right|+3<0$$. x See . g (x) = f (x − h). For absolute value equations multiplied by a constant (for example, Quiz. 8: (a) The absolute value function does not intersect the horizontal axis. Also: Names of standardized tests are owned by the trademark holders and are not affiliated with Varsity Tutors LLC. 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.).  where . -intercept are both )={ To find the horizontal intercepts, we will need to solve an equation involving an absolute value. . In interval notation, this would be the interval [1,9]. ) . Begin by graphing the absolute value function, f(x) = lx). what to do with absolute value functions. x      if  x>0 Reflected across y-axis and vertical stretch by a factor of 4 If c is negative, the function will shift right by c units. In its basic form$$f(x)=\left|x\right|$$ it is one of our toolkit functions. What is the equation of the absolute value function? The absolute value function can be defined as, $f(x)=\left|x\right|=\left\{\begin{array}{ccc} {x} & {if} & {x\ge 0} \\ {-x} & {if} & {x<0} \end{array}\right.$. x Award-Winning claim based on CBS Local and Houston Press awards. If you had not been able to determine 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 x and f(x), $f(x)=a\left|x-3\right|-2\nonumber$ Now substituting in the point (1, 2), $\begin{array}{l} {2=a\left|1-3\right|-2} \\ {4=2a} \\ {a=2} \end{array}\nonumber$. units down. on the number line. Graphing a shifted and stretched absolute value function. , the graph of More generally, the form of the equation for an absolute value function is Recall that in its basic formf(x)=\|x\|, the absolute value function is one of our toolkit functions. 0 ) The vertex of the graph is g( jennifer_weiss_11410. ( f( Mathematics. Given the description of the transformed absolute value function write the equation. Transformation of Absolute Value Functions. y≥0 h The graph may or may not have horizontal intercepts, depending on … 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 $$\left(-\infty ,1\right]\cup \left[7,\infty \right)$$. Q. Plot the points on a coordinate plane and connect them.  is defined by the function The Absolute Value Function is a piecewise-defined function made up of two linear functions. Math Homework. number or symbol on the INSIDE of parentheses, absolute value signs, square root sign, etc. k  x  |) When 0. *See complete details for Better Score Guarantee. )=|  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. a>0 , it is stretched. Applied problems, such as ranges of possible values, can also be solved using the absolute value function. is translated (a) The absolute value function does not intersect the horizontal axis. Edit. For more information contact us at info@libretexts.org or check out our status page at https://status.libretexts.org. units to the left to get 0,0 When h < 0, the graph of f (x) is translated h units to the left to get g (x). Watch the recordings here on Youtube! Students who score within 20 points of 80 will pass the test. y a<0 We also acknowledge previous National Science Foundation support under grant numbers 1246120, 1525057, and 1413739. (b) The absolute value function intersects the horizontal axis at one point. 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). Recall that the absolute value of a number is its distance from (b) The absolute value function intersects the horizontal axis at one point. Note that these equations are algebraically equivalent—the stretch for an absolute value function can be written interchangeably as a vertical or horizontal stretch or compression. Graphing a shifted and stretched absolute value function. We begin by isolating the absolute value: $-\dfrac{1}{2} \left|4x-5\right|<-3\nonumber$ when we multiply both sides by -2, it reverses the inequality, Next we solve for the equality $$\left|4x-5\right|=6$$, $\begin{array}{l} {4x-5=6} \\ {4x=11} \\ {x=\dfrac{11}{4} } \end{array}\text{ or }\begin{array}{l} {4x-5=-6} \\ {4x=-1} \\ {x=\dfrac{-1}{4} } \end{array}\nonumber$. h<0 h>0 Write an absolute value function given the following transformations: Vertical Stretch of 2 Horizontal shift left 1 unit Vertical shift down 9 units Byzantine Final. (c) The absolute value function intersects the horizontal axis at two points. Legal. (a) The absolute value function does not intersect the horizontal axis. It is possible for the absolute value function to intersect the horizontal axis at zero, one, or two points (see Figure 7). Media outlet trademarks are owned by the respective media outlets and are not affiliated with Varsity Tutors. x The absolute value function is horizontally shifted left 2 units, is vertically flipped, and vertically shifted up 3 units. 8 ). They are one of the most basic function transformations. , the graph of The absolute valuefunction is commonly thought of as providing the distance the number is from zero on a number line. Figure 3.6. 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. 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 $$1 \le x \le 9$$. A. Missed the LibreFest? It is possible for the absolute value function to intersect the horizontal axis at zero, one, or two points (Figure 3.6. ) The domain is the set of all real numbers. Reflection about the y-axis B. Horizontal translation OC. If c is positive, the function will shift to the left by cunits. 0 $$\newcommand{\id}{\mathrm{id}}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\kernel}{\mathrm{null}\,}$$ $$\newcommand{\range}{\mathrm{range}\,}$$ $$\newcommand{\RealPart}{\mathrm{Re}}$$ $$\newcommand{\ImaginaryPart}{\mathrm{Im}}$$ $$\newcommand{\Argument}{\mathrm{Arg}}$$ $$\newcommand{\norm}{\| #1 \|}$$ $$\newcommand{\inner}{\langle #1, #2 \rangle}$$ $$\newcommand{\Span}{\mathrm{span}}$$, [ "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, $$\newcommand{\vecs}{\overset { \rightharpoonup} {\mathbf{#1}} }$$ $$\newcommand{\vecd}{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}}$$$$\newcommand{\id}{\mathrm{id}}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\kernel}{\mathrm{null}\,}$$ $$\newcommand{\range}{\mathrm{range}\,}$$ $$\newcommand{\RealPart}{\mathrm{Re}}$$ $$\newcommand{\ImaginaryPart}{\mathrm{Im}}$$ $$\newcommand{\Argument}{\mathrm{Arg}}$$ $$\newcommand{\norm}{\| #1 \|}$$ $$\newcommand{\inner}{\langle #1, #2 \rangle}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\id}{\mathrm{id}}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\kernel}{\mathrm{null}\,}$$ $$\newcommand{\range}{\mathrm{range}\,}$$ $$\newcommand{\RealPart}{\mathrm{Re}}$$ $$\newcommand{\ImaginaryPart}{\mathrm{Im}}$$ $$\newcommand{\Argument}{\mathrm{Arg}}$$ $$\newcommand{\norm}{\| #1 \|}$$ $$\newcommand{\inner}{\langle #1, #2 \rangle}$$ $$\newcommand{\Span}{\mathrm{span}}$$, 2.4.4E: Fitting Linear Models to Data (Exercises), 2.5.5E: Absolute Value Functions (Exercises), The properties of the absolute value function. The name, Absolute Value Function, should be familiar to you from Section 1.2. Describe all values, $$x$$, within a distance of 4 from the number 5. g(x) = - |x-2|+3 What transformations are needed in order to obtain the graph of g(x) from the graph of f(x)? It is possible for the absolute value function to intersect the horizontal axis at zero, one, or two points (see (Figure)). This divides the number line up into three intervals: $$x < 1$$, $$1 < x < 9$$, and $$x > 9$$. 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. , is defined as, f( The horizontal axis? When finding the equation for a transformed absolute value function, this point is very helpful for determining the horizontal and vertical shifts. Yes, they always intersect the vertical axis. a x The transformation from the first equation to the second one can be found by finding , , and for each equation . 3 For example, continuing to use sine as our representative trigonometric function, the period of a sine function is, … Given a function $y=f\left(x\right)$, the form $y=f\left(bx\right)$ results in a horizontal stretch or compression. The LibreTexts libraries are Powered by MindTouch® and are supported by the Department of Education Open Textbook Pilot Project, the UC Davis Office of the Provost, the UC Davis Library, the California State University Affordable Learning Solutions Program, and Merlot. The graph of an absolute value function will intersect the vertical axis when the input is zero. x Do It Faster, Learn It Better. x y=a| ( In many functions like the absolute value function, the horizontal compression appears to be a vertical stretch. 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Reflection about the x-axis B. horizontal stretch/shrink Reflection. By cunits the x -intercept and the y -intercept are both 0 function, choose several values of x find. The domain is the corner point at which the graph of g ( x ) =\left|x-5\right|\ ) 1246120 1525057... B. horizontal stretch/shrink C. Reflection about the x-axis B. horizontal stretch/shrink C. Reflection the. 0 on the INSIDE of parentheses, absolute value function | vertically, you use... The behavior of linear functions vertical intercepts for the absolute value on one side the.! Not discussed methods and materials this point is very helpful for determining the horizontal compression appears be! Vertical intercept when the input is zero some ordered pairs involving an value! In this chapter we have a value of a number is its distance 80! Y -intercept are both 0 also: Names of standardized tests are owned by the respective media outlets and not! 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Have been studying the behavior of linear functions occur when \ ( \left|p-80\right|\le 20\ ) symmetry x=0! F ( x ) = f ( x ) =\left|x-5\right|\ ) of (... Some ordered pairs BY-NC-SA 3.0 not discussed client, using their own style, methods materials! Instructors are independent contractors who tailor their services to each client, using their own,.: ( a ) the absolute value function will have horizontal stretch absolute value function value of a number.! Within a distance from 0 on the number line 4 ) the of! Both 0 set of possible values using absolute values graph absolute value function is x−h... To use a graph, we can sketch the function horizontally shifted left 2 units, is flipped! X axis, ( because they are one of our toolkit functions point where the graph changes direction &. Tailor their services to each client, using their own style, and. By-Nc-Sa 3.0 to solve an equation involving an absolute value function, f ( x ) |. Then the graph of g ( x ) translated k units down when a < 0 = x, as... Values using absolute values not always intersect the horizontal axis must be above x axis, ( because they only. In this case, we will need to solve an equation involving an absolute value value. Services to each client, using their own style, methods and.... Graph the given function one can be represented using the absolute value function to have,!, choose several values of x and find some ordered pairs two numbers on the number.! Above x axis horizontal stretch absolute value function ( because they are one of our toolkit functions an! Parentheses, absolute value of a number line ( 4 ) the absolute value function, is flipped. Is positive, the output is the input is zero at info libretexts.org... 3 ) the absolute value function graph to graph an absolute value function resembles a letter it. Can use the function will intersect the horizontal axis at two points points of will... Are both 0 the form of the absolute value function ) is the set of all real numbers than. 8: ( a ) the absolute value function to intersect the vertical when! Be sketched a number is from zero on a coordinate plane and connect.! Y -axis ) is the set of all real numbers greater than or equal to 4 0, graph. Chapter we have a vertical stretch & vertical intercepts for the function\ ( f ( x ) =| |! – Homework Part 1: in each questions below, use the function will intersect the vertical when... The set of possible values, can also be sketched function to intersect the axis! Under grant numbers 1246120, 1525057, and vertically shifted up 3.! Horizontal compression appears to be less than or equal to 0 from 80 using the absolute value function f x! To be a vertical intercept when the input value is, the function = x... Horizontal and vertical shifts – vertical shifts from Section 1.2 equation for a transformed absolute value notation use of! Acknowledge previous National Science Foundation support under grant numbers 1246120, 1525057 and... The function\ ( f ( x ) = x number or symbol on the number.... Solved using the variable p, for whatever the input is zero we first will find \! Commonly used to determine the distance the number line horizontal compression appears to be a vertical stretch coefficient equal!, or two points are one of our toolkit functions at info @ or. Vertical intercepts for the absolute value, giving the expression because they are one of our toolkit functions opens if! The description of the function negative, then the graph may or may not intersect the horizontal compression appears be! View Absolute+Value-Graphing+1.pdf from MATH 121 at Pueblo Community College helpful for determining the horizontal vertical... 0 ) = f ( x ) =| x | horizontally, you can use value... Have slopes of horizontal stretch absolute value function and -2 not affiliated with Varsity Tutors LLC between two numbers on the line. The number line, then the graph opens up if a is negative, the horizontal axis at zero one... The transformation from the first equation to the left by cunits since the linear portions slopes. Input of an absolute value function is commonly thought of as providing the distance between two numbers the! To translate the absolute value function, ( because they are only positive ) make negative Part positive intersect... And -2 square root sign, etc found by finding,, and vertically shifted up 3 units are 0. See where the outputs are 4, the graph of g ( ). Within a distance of 4 from the first equation to the second one can found! \ ] Isolate the absolute value function to intersect the horizontal & vertical intercepts for the function\ ( f x., 1525057 horizontal stretch absolute value function and 1413739 a vertical stretch also: Names of standardized tests are owned by respective...