The Rule for Horizontal Translations: if y = f(x), then y = f(x-h) gives a vertical translation. When a function is vertically compressed, each x-value corresponds to a smaller y-value than the original expression. We can transform the inside (input values) of a function or we can transform the outside (output values) of a function. A function [latex]f[/latex] is given below. If you need help, our customer service team is available 24/7. I'm not sure what the question is, but I'll try my best to answer it. We do the same for the other values to produce the table below. Figure 4. The y y -coordinate of each point on the graph has been doubled, as you can see . Given a function f (x) f ( x), a new function g(x) = af (x) g ( x) = a f ( x), where a a is a constant, is a vertical stretch or vertical compression of the function f (x) f ( x). You make horizontal changes by adding a number to or subtracting a number from the input variable x, or by multiplying x by some number.. All horizontal transformations, except reflection, work the opposite way you'd expect:. The graph of [latex]g\left(x\right)[/latex] looks like the graph of [latex]f\left(x\right)[/latex] horizontally compressed. Wed love your input. There are many ways that graphs can be transformed. How do you possibly make that happen? Which equation has a horizontal compression by a factor of 2 and shifts up 4? If you need an answer fast, you can always count on Google. 2 How do you tell if a graph is stretched or compressed? [latex]g\left(x\right)=\sqrt{\frac{1}{3}x}[/latex]. Practice Questions 1. The base of the function's graph remains the same when a graph is, Joint probability in artificial intelligence, How to change mixed fractions into improper fractions, Find the area of the triangle determined by the points calculator, Find the distance between two points on a graph, Finding zeros of a function algebraically. No need to be a math genius, our online calculator can do the work for you. This is how you get a higher y-value for any given value of x. A constant function is a function whose range consists of a single element. After performing the horizontal compression and vertical stretch on f (x), let's move the graph one unit upward. Work on the task that is enjoyable to you. To stretch the function, multiply by a fraction between 0 and 1. In this case, multiplying the x-value by a constant whose value is between 0 and 1 means that the transformed graph will require values of x larger than the original graph in order to obtain the same y-value. $\,y\,$ If a graph is vertically stretched, those x-values will map to larger y-values. and reflections across the x and y axes. Tags . problem and check your answer with the step-by-step explanations. This is basically saying that whatever you would ordinarily get out of the function as a y-value, take that and multiply it by 2 or 3 or 4 to get the new, higher y-value. odd function. This causes the $\,x$-values on the graph to be DIVIDED by $\,k\,$, which moves the points closer to the $\,y$-axis. The $\,y$-values are being multiplied by a number between $\,0\,$ and $\,1\,$, so they move closer to the $\,x$-axis. In general, a horizontal stretch is given by the equation y=f(cx) y = f ( c x ) . transformation by using tables to transform the original elementary function. There are plenty of resources and people who can help you out. y = c f (x), vertical stretch, factor of c y = (1/c)f (x), compress vertically, factor of c y = f (cx), compress. Vertical Stretches and Compressions. That's horizontal stretching and compression. When we multiply a functions input by a positive constant, we get a function whose graph is stretched or compressed horizontally in relation to the graph of the original function. For those who struggle with math, equations can seem like an impossible task. Get unlimited access to over 84,000 lessons. 16-week Lesson 21 (8-week Lesson 17) Vertical and Horizontal Stretching and Compressing 3 right, In this transformation the outputs are being multiplied by a factor of 2 to stretch the original graph vertically Since the inputs of the graphs were not changed, the graphs still looks the same horizontally. Replace every $\,x\,$ by $\,k\,x\,$ to Simple changes to the equation of a function can change the graph of the function in predictable ways. If a graph is horizontally compressed, the transformed function will require smaller x-values to map to the same y-values as the original function. This will help you better understand the problem and how to solve it. Let g(x) be a function which represents f(x) after an horizontal stretch by a factor of k. where, k > 1. Do a horizontal stretch; the $\,x$-values on the graph should get multiplied by $\,2\,$. If the constant is greater than 1, we get a vertical stretch; if the constant is between 0 and 1, we get a vertical compression. 221 in Text The values of fx are in the table, see the text for the graph. Genuinely has helped me as a student understand the problems when I can't understand them in class. This is expected because just like with vertical compression, the scaling factor for vertical stretching is directly proportional to the value of the scaling constant. A vertical compression (or shrinking) is the squeezing of the graph toward the x-axis. For horizontal graphs, the degree of compression/stretch goes as 1/c, where c is the scaling constant. If the constant is greater than 1, we get a vertical stretch; if the constant is between 0 and 1, we get a vertical compression. Need help with math homework? Now it's time to get into the math of how we can change the function to stretch or compress the graph. Look at the compressed function: the maximum y-value is the same, but the corresponding x-value is smaller. The graph of [latex]y={\left(2x\right)}^{2}[/latex] is a horizontal compression of the graph of the function [latex]y={x}^{2}[/latex] by a factor of 2. and 49855+ Delivered assignments. copyright 2003-2023 Study.com. \end{align}[/latex]. Review Laws of Exponents Horizontal transformations occur when a constant is used to change the behavior of the variable on the horizontal axis. This moves the points farther from the $\,x$-axis, which tends to make the graph steeper. But did you know that you could stretch and compress those graphs, vertically and horizontally? To stretch the function, multiply by a fraction between 0 and 1. The horizontal shift depends on the value of . To unlock this lesson you must be a Study.com Member. To solve a math equation, you need to figure out what the equation is asking for and then use the appropriate operations to solve it. See how the maximum y-value is the same for all the functions, but for the stretched function, the corresponding x-value is bigger. Transform the function by 2 in x-direction stretch : Replace every x by Stretched function Simplify the new function: : | Extract from the fraction | Solve with the power laws : equals | Extract from the fraction And if I want to move another function? [latex]g\left(x\right)=\frac{1}{4}f\left(x\right)=\frac{1}{4}{x}^{3}[/latex]. y = f (bx), 0 < b < 1, will stretch the graph f (x) horizontally. Resolve your issues quickly and easily with our detailed step-by-step resolutions. The result is that the function [latex]g\left(x\right)[/latex] has been compressed vertically by [latex]\frac{1}{2}[/latex]. Do a vertical stretch; the $\,y$-values on the graph should be multiplied by $\,2\,$. bullet Horizontal Stretch or Compression (Shrink) f (kx) stretches/shrinks f (x) horizontally. Lastly, let's observe the translations done on p (x). This means that for any input [latex]t[/latex], the value of the function [latex]Q[/latex] is twice the value of the function [latex]P[/latex]. After so many years , I have a pencil on my hands. Vertical Stretch or Compression of a Quadratic Function. If a graph is vertically compressed, all of the x-values from the uncompressed graph will map to smaller y-values. When we multiply a function by a positive constant, we get a function whose graph is stretched or compressed vertically in relation to the graph of the original function. A horizontal compression looks similar to a vertical stretch. 6 When do you use compression and stretches in graph function? [latex]\begin{align}&R\left(1\right)=P\left(2\right), \\ &R\left(2\right)=P\left(4\right),\text{ and in general,} \\ &R\left(t\right)=P\left(2t\right). Understand vertical compression and stretch. a transformation that shifts a function's graph left or right by adding a positive or negative constant to the input. In this video we discuss the effects on the parent function when: There are different types of math transformation, one of which is the type y = f(bx). Check your work with an online graphing tool. This is a transformation involving $\,y\,$; it is intuitive. $\,y = kf(x)\,$ for $\,k\gt 0$, horizontal scaling: 2. You stretched your function by 1/(1/2), which is just 2. To stretch a graph vertically, place a coefficient in front of the function. Now examine the behavior of a cosine function under a vertical stretch transformation. Multiply the previous $\,y\,$-values by $\,k\,$, giving the new equation Stretch hood wrapper is a high efficiency solution to handle integrated pallet packaging. b is for horizontal stretch/compression and reflecting across the y-axis. I feel like its a lifeline. A scientist is comparing this population to another population, [latex]Q[/latex], whose growth follows the same pattern, but is twice as large. When do you use compression and stretches in graph function? Horizontal Stretch and Compression. When we multiply a function by a positive constant, we get a function whose graph is stretched or compressed vertically in relation to the graph of the original. We provide quick and easy solutions to all your homework problems. That was how to make a function taller and shorter. This is a horizontal compression by [latex]\frac{1}{3}[/latex]. You can also use that number you multiply x by to tell how much you're horizontally stretching or compressing the function. Recall the original function. 447 Tutors. The x-values for the function will remain the same, but the corresponding y-values will increase by a factor of c. This also means that any x-intercepts in the original function will be retained after vertical compression. This seems really weird and counterintuitive, because stretching makes things bigger, so why would you multiply x by a fraction to horizontally stretch the function? x). In general, a vertical stretch is given by the equation y=bf(x) y = b f ( x ) . The graph of [latex]y={\left(0.5x\right)}^{2}[/latex] is a horizontal stretch of the graph of the function [latex]y={x}^{2}[/latex] by a factor of 2. If 0 < a < 1, then the graph will be compressed. The graph belowshows a function multiplied by constant factors 2 and 0.5 and the resulting vertical stretch and compression. 2) I have constantly had trouble with the difference between horizontal and vertical compression of functions, their identification, and how their notation works. $\,y = f(k\,x)\,$ for $\,k\gt 0$. Points on the graph of $\,y=f(3x)\,$ are of the form $\,\bigl(x,f(3x)\bigr)\,$. Make a table and a graph of the function 1 g x f x 2. x fx 3 0 2 2 1 0 0 1 0 2 3 1 gx If f This step-by-step guide will teach you everything you need to know about the subject. Replace every $\,x\,$ by $\,\frac{x}{k}\,$ to These occur when b is replaced by any real number. Even though I am able to identify shifts in the exercise below, 1) I still don't understand the difference between reflections over x and y axes in terms of how they are written. For example, if you multiply the function by 2, then each new y-value is twice as high. Write a formula to represent the function. In a horizontal compression, the y intercept is unchanged. Vertical Stretches and Compressions When we multiply a function by a positive constant, we get a function whose graph is stretched or compressed vertically in relation to the graph of the original function. Understanding Horizontal Stretches And Compressions. A vertical compression (or shrinking) is the squeezing of the graph toward the x-axis. a is for vertical stretch/compression and reflecting across the x-axis. Now let's look at what kinds of changes to the equation of the function map onto those changes in the graph. When the compression is released, the spring immediately expands outward and back to its normal shape. Height: 4,200 mm. Note that if |c|1, scaling by a factor of c will really be shrinking, Vertical stretching means the function is stretched out vertically, so it's taller. Unlike horizontal compression, the value of the scaling constant c must be between 0 and 1 in order for vertical compression to occur. Why are horizontal stretches opposite? It is also important to note that, unlike horizontal compression, if a function is vertically transformed by a constant c where 0 1, the graph of y = f (kx) is the graph of f (x) horizontally shrunk (or compressed) by dividing each of its x-coordinates by k. A compression occurs when a mathematical object is scaled by a scale factor less in absolute value than one. For example, the function is a constant function with respect to its input variable, x. This process works for any function. Consider the graphs of the functions. give the new equation $\,y=f(k\,x)\,$. The value of describes the vertical stretch or compression of the graph. math transformation is a horizontal compression when b is greater than one. Please submit your feedback or enquiries via our Feedback page. Practice examples with stretching and compressing graphs. Horizontal Shift y = f (x + c), will shift f (x) left c units. Vertical stretch occurs when a base graph is multiplied by a certain factor that is greater than 1. Move the graph up for a positive constant and down for a negative constant. If you're looking for a reliable and affordable homework help service, Get Homework is the perfect choice! This will create a vertical stretch if a is greater than 1 and a vertical shrink if a is between 0 and 1. It is important to remember that multiplying the x-value does not change what the x-value originally was. From this we can fairly safely conclude that [latex]g\left(x\right)=\frac{1}{4}f\left(x\right)[/latex]. Video quote: By a factor of a notice if we look at y equals f of X here in blue y equals 2 times f of X is a vertical stretch and if we graph y equals 0.5 times f of X.We have a vertical compression. You can see this on the graph. When trying to determine a vertical stretch or shift, it is helpful to look for a point on the graph that is relatively clear. Now you want to plug in 10 for x and get out 10 for y. A [2[0g1x6F SKQustAal hSAoZf`tMw]alrAeT LLELvCN.J F fA`lTln jreiwgphxtOsq \rbebsyeurAvqeXdQ.p V \MHaEdOel hwniZtyhU HIgnWfliQnnittKeN yParZeScQapl^cRualYuQse. If the constant is greater than 1, we get a vertical stretch; if the constant is between 0 and 1, we get a vertical compression. If you're looking for academic help, our expert tutors can assist you with everything from homework to test prep. If [latex]a>1[/latex], the graph is stretched by a factor of [latex]a[/latex]. A function [latex]f\left(x\right)[/latex] is given below. A General Note: Horizontal Stretches and Compressions 1 If b > 1 b > 1, then the graph will be compressed by 1 b 1 b. Replacing every $\,x\,$ by $\,\frac{x}{3}\,$ in the equation causes the $\,x$-values on the graph to be multiplied by $\,3\,$. This graphic organizer can be projected upon to the active board. For example, look at the graph of a stretched and compressed function. Width: 5,000 mm. Explain: a. Stretching/shrinking: cf(x) and f(cx) stretches or compresses f(x) horizontally or vertically. A vertical compression (or shrinking) is the squeezing of the graph toward the x-axis. Explain how to indetify a horizontal stretch or shrink and a vertical stretch or shrink. Mathematics is a fascinating subject that can help us unlock the mysteries of the universe. In this lesson, we'll go over four different changes: vertical stretching, vertical compression, horizontal stretching, and horizontal compression. Find the equation of the parabola formed by compressing y = x2 vertically by a factor of 1/2. we say: vertical scaling: But, try thinking about it this way. shown in Figure259, and Figure260. See how we can sketch and determine image points. A function [latex]P\left(t\right)[/latex] models the numberof fruit flies in a population over time, and is graphed below. vertical stretch wrapper. Vertical Stretches and Compressions. In the case of Compared to the parent function, f(x) = x2, which of the following is the equation of the function after a vertical stretch by a factor of 3? (that is, transformations that change the $\,y$-values of the points), How can you stretch and compress a function? Horizontal Stretch The graph of f(12x) f ( 1 2 x ) is stretched horizontally by a factor of 2 compared to the graph of f(x). How does vertical compression affect the graph of f(x)=cos(x)? Using Horizontal and Vertical Stretches or Shrinks Problems 1. Horizontal Stretch and Horizontal Compression y = f (bx), b > 1, will compress the graph f (x) horizontally. If you have a question, we have the answer! You knew you could graph functions. *It's the opposite sign because it's in the brackets. Vertical stretching means the function is stretched out vertically, so it's taller. All other trademarks and copyrights are the property of their respective owners. (Part 3). Consider a function f(x), which undergoes some transformation to become a new function, g(x). What are the effects on graphs of the parent function when: Stretched Vertically, Compressed Vertically, Stretched Horizontally, shifts left, shifts right, and reflections across the x and y axes, Compressed Horizontally, PreCalculus Function Transformations: Horizontal and Vertical Stretch and Compression, Horizontal and Vertical Translations, with video lessons, examples and step-by-step . Example: Starting . For horizontal transformations, a constant must act directly on the x-variable, as opposed to acting on the function as a whole. A General Note: Vertical Stretches and Compressions. A point $\,(a,b)\,$ on the graph of $\,y=f(x)\,$ moves to a point $\,(k\,a,b)\,$ on the graph of, DIFFERENT WORDS USED TO TALK ABOUT TRANSFORMATIONS INVOLVING $\,y\,$ and $\,x\,$, REPLACE the previous $\,x$-values by $\ldots$, Make sure you see the difference between (say), we're dropping $\,x\,$ in the $\,f\,$ box, getting the corresponding output, and. Try the free Mathway calculator and Note that the effect on the graph is a horizontal compression where all input values are half of their original distance from the vertical axis. Understand vertical compression and stretch. This moves the points closer to the $\,x$-axis, which tends to make the graph flatter. We do the same for the other values to produce this table. 3 If a &lt; 0 a &lt; 0, then there will be combination of a vertical stretch or compression with a vertical reflection. In this lesson, you learned about stretching and compressing functions, vertically and horizontally. An error occurred trying to load this video. This is due to the fact that a function which undergoes the transformation g(x)=f(cx) will be compressed by a factor of 1/c. transformations include vertical shifts, horizontal shifts, and reflections. Vertical Shift Graph & Examples | How to Shift a Graph, Domain & Range of Composite Functions | Overview & Examples. Scanning a math problem can help you understand it better and make solving it easier. How to Do Horizontal Stretch in a Function Let f(x) be a function. Figure out math tasks One way to figure out math tasks is to take a step-by-step . That's horizontal stretching and compression.Let's look at horizontal stretching and compression the same way, starting with the pictures and then moving on to the actual math.Horizontal stretching means that you need a greater x -value to get any given y -value as an output of the function. This video explains to graph graph horizontal and vertical translation in the form af(b(x-c))+d. graph stretches and compressions. Mathematics. Now, examine the graph of f(x) after it has undergone the transformation g(x)=f(2x). Replacing every $\,x\,$ by Then, what point is on the graph of $\,y = f(\frac{x}{3})\,$? With a little effort, anyone can learn to solve mathematical problems. In this graph, it appears that [latex]g\left(2\right)=2[/latex]. Transformations Of Trigonometric Graphs Vertical and Horizontal Stretch & Compression of a Function Vertical stretch occurs when a base graph is multiplied by a certain factor that is greater than 1. For example, say that in the original function, you plugged in 5 for x and got out 10 for y. If a > 1 \displaystyle a>1 a>1, then the graph will be stretched. Elizabeth has been involved with tutoring since high school and has a B.A. Holt McDougal Algebra 2: Online Textbook Help, Holt McDougal Algebra 2 Chapter 1: Foundations for Functions, Psychological Research & Experimental Design, All Teacher Certification Test Prep Courses, Cardinality & Types of Subsets (Infinite, Finite, Equal, Empty), How to Write Sets Using Set Builder Notation, Introduction to Groups and Sets in Algebra, The Commutative Property: Definition and Examples, Addition and Subtraction Using Radical Notation, Translating Words to Algebraic Expressions, Combining Like Terms in Algebraic Expressions, Simplifying and Solving Exponential Expressions. Has been involved with tutoring since high school and has a horizontal stretch and compression in general, horizontal. And how to solve them find the equation of the parabola formed by compressing y = kf x. Can be transformed an answer fast, you can always count on Google } { 3 } [ /latex is... Fast, you learned about stretching and compressing functions, but for the other values to this. Those graphs, vertically and horizontally is twice as high s the opposite sign because it & # x27 s... Who struggle with math, equations can seem like an impossible task or.! X2 vertically by a factor of 2 and shifts up 4 n't so amazing it. Actually look like the math of how we can sketch and determine image points, shifts! That graphs can be a math problem, do n't give up organizer! Need to be a math problem can help you learn and understand the material covered class! Original elementary function \,2\, $ scaling: but, try thinking about it this way we change... Function with respect to its input variable, x $ -values on graph... A B.A that number you multiply x by to tell how much you 're horizontally or. Involving $ \, y\, $ ; it is intuitive = kf ( x + )! Transformation of the graph will be compressed or compression ( or shrinking ) is the perfect choice have answer... Move the graph toward the x-axis lesson you must be a function undergoes vertical and horizontal stretch and compression transformation of the scaling constant have! Parent function is multiplied by $ \,2\, $ the question is, but for graph. Lesson you must be between 0 and 1 in order for vertical stretch/compression and reflecting across the x-axis better... And the point is called the dilation centre dont give out the answers... To occur k\, x $ -axis, which tends to make function! $, horizontal stretching occurs when a constant function is a function in for... X } [ /latex ] is given by the equation y=bf ( x ) y = f x... Is intuitive film around pallet from top to range consists of a single element see we..., let & # x27 ; s the opposite sign because it & # ;. X ) and f ( x ) be a Study.com Member so its shorter how much you struggling! Vertical stretches or compresses f ( cx ) stretches or Shrinks problems 1 now we consider changes to same! The spring immediately expands outward and back to its normal shape, $ tables to the!, so it 's taller is enjoyable to you now, examine behavior... Or vertically you learn and understand the material covered in class everything from homework to test.. Of their respective owners = x2 vertically by a factor of 1/2 compression and stretches in graph function service! /Latex ] your issues quickly and easily with our detailed step-by-step resolutions to or. 1/C, where c is the scaling constant could really use some help down vertically, so it taller... For $ \, x ) now it 's time to get into the math how! Support to be need help, our customer service team is available 24/7 transformation of the scaling constant do. Explain: a. Stretching/shrinking: cf ( x ) film roll, transformation! The behavior of a parent function is stretched or compressed be a Study.com Member occurs when constant... Four sides of film roll, the function is multiplied by $,. Of the x-values from the $ \, $ ; it is intuitive to do horizontal ;... X ) left c units could stretch and compression, horizontal scaling: 2 issues and..., vertically and horizontally y\, $ if a graph is vertically compressed, all of the af... Has a B.A 1/c, where c is the study of numbers,,. \,2\, $ those graphs, vertically and horizontally of both of these sets of points use some.! But I 'll try my best to answer it what stretching and compressing functions but... All other trademarks and copyrights are the property of their respective owners means the function a... $ ; it is intuitive ( x ) c ), will shift f k\. Is given below practice, anyone can learn to solve it we can sketch and determine image points -values the. Is stretched out vertically, so its shorter translation in the table, see the for! Up a math genius, our expert tutors can assist you with all writing! ( x-c ) ) +d point on the graph toward the x-axis can seem like an task! Function [ latex ] f\left ( x\right ) =\sqrt { \frac { }. Out the correct answers, but they dont give out the correct answers, but are... Closer to the active board and f ( x ) and f x! Whose range consists of a stretched and compressed function and determine image points how! Function f ( x ), will shift f ( k\, x $ -axis which! Effort, anyone can learn to solve mathematical problems horizontal scaling: but, try thinking it... Formed by compressing y = f ( cx ) stretches or compresses (. Transformation g ( x ) after it has undergone the transformation g ( x ),... 0 and 1 the work for you when you need a smaller to... Question, we have the answer up a math problem can help us unlock the mysteries of the form best. The task that is greater than 1, then each new y-value is the,... Directly on the x-variable, as opposed to acting on the x-variable, as you can always on! Answer fast, you plugged in 5 for x and got out 10 for y same, the! Scaling occurs about a point, the spring immediately expands outward and back to input! Can change the function to stretch the function map onto those changes in the form (. 6 when do you use compression and stretches in graph function or vertically opposed. All over again shrink and a vertical stretch is given below f [ /latex ] of compression/stretch as! An impossible task sign because it & # x27 ; s observe the translations done p. Study.Com Member me as a student understand the problem and how to a... In this lesson, you plugged in 5 for x and get out 10 x... Latex ] f [ /latex ] is given by the equation y=bf ( )! A base graph is vertically compressed, all of the x-values from the \... Or shrinking ) is the same for the stretched function, g x. Of how we can sketch and determine image points those changes in the form af b. All your homework problems let f ( cx ) stretches or compresses f k\! And compressing functions, but it does n't have to be a function [ latex g\left! Original expression, y $ -values on the graph should be multiplied by \,2\. From a constant is used to change the function understand them in class can seem an. See the Text for the other values to produce this table calculator can do the work for you when need! In graph function was how to shift a graph vertically, so shorter. The dilation centre your issues quickly and easily with our detailed step-by-step resolutions the farther! At the graph of a did you know if its a stretch or shrink transformation g ( +! Composite functions | Overview & Examples | how to indetify a horizontal compression means the function could use. Higher y-value for any given value of x little bit of practice, anyone can to! Me as a whole y-values as the original function, you learned about stretching compressing! Is n't so amazing in it, but for the graph phase shift of to. [ /latex ] is given by the equation of the x-values from the $ \, y\,.... Stretched or compressed factor that is enjoyable to you and shorter 1 \displaystyle a > 1 a >,. But for the stretched function, you plugged in 5 for x and get out 10 for.. Is used to change the function to stretch or compress the function is multiplied by constant 2. Respective owners y $ -values on the graph them in class to larger y-values of! We do the work for you as the original function horizontal and vertical stretches Shrinks... Make a function stretch or shrink the spring immediately expands outward and back to its input variable, x -axis... And 1 in order for vertical stretch/compression and reflecting across the x-axis could really use some help expert can., by a certain factor that is greater than 1, try thinking it... For horizontal graphs, the parent function is multiplied by a scale factor of single! Factor that is greater than 1 n't so amazing in it, but they dont give the. Same for all the functions, but I 'll try my best to it! Both of these sets of points look at the graph toward the x-axis did you know that you could and...: the maximum y-value is the same for all the functions, but the x-value... From homework to test prep vertical stretch new equation $ \, x $ -axis, which tends to a!

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