vertical and horizontal stretch and compression

When by either f (x) or x is multiplied by a number, functions can "stretch" or "shrink" vertically or horizontally, respectively, when graphed. Resolve your issues quickly and easily with our detailed step-by-step resolutions. Given a function [latex]f\left(x\right)[/latex], a new function [latex]g\left(x\right)=f\left(bx\right)[/latex], where [latex]b[/latex] is a constant, is a horizontal stretch or horizontal compression of the function [latex]f\left(x\right)[/latex]. 4 How do you know if its a stretch or shrink? How can we locate these desired points $\,\bigl(x,f(3x)\bigr)\,$? To visualize a horizontal compression, imagine that you push the graph of the function toward the y axis from both the left and the right hand side. From this we can fairly safely conclude that [latex]g\left(x\right)=\frac{1}{4}f\left(x\right)[/latex]. A vertical compression (or shrinking) is the squeezing of the graph toward the x-axis. We can graph this math We will compare each to the graph of y = x2. Identify the vertical and horizontal shifts from the formula. Thus, the graph of $\,y=3f(x)\,$ is found by taking the graph of $\,y=f(x)\,$, Horizontal transformations of a function. We do the same for the other values to produce the table below. That was how to make a function taller and shorter. When trying to determine a vertical stretch or shift, it is helpful to look for a point on the graph that is relatively clear. Thats what stretching and compression actually look like. If a1 , then the graph will be stretched. Vertical compression means the function is squished down, Find circumference of a circle calculator, How to find number of employees in a company in india, Supplements and complements word problems answers, Explorations in core math grade 7 answers, Inverse normal distribution calculator online, Find the area of the region bounded calculator, What is the constant term in a linear equation, Match each operation involving f(x) and g(x) to its answer, Solving exponential equations module 1 pg. is a vertical stretch (makes it narrower) is a vertical compression (makes it wider) Vertical Stretch: Stretched. Find the equation of the parabola formed by compressing y = x2 vertically by a factor of 1/2. In this lesson, you learned about stretching and compressing functions, vertically and horizontally. y = f (x - c), will shift f (x) right c units. Writing and describing algebraic representations according to. The $\,y$-values are being multiplied by a number between $\,0\,$ and $\,1\,$, so they move closer to the $\,x$-axis. This transformation type is formally called, IDEAS REGARDING HORIZONTAL SCALING (STRETCHING/SHRINKING). As a member, you'll also get unlimited access to over 84,000 2) I have constantly had trouble with the difference between horizontal and vertical compression of functions, their identification, and how their notation works. Step 2 : So, the formula that gives the requested transformation is. You must replace every $\,x\,$ in the equation by $\,\frac{x}{2}\,$. A horizontal compression looks similar to a vertical stretch. 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? To compress the function, multiply by some number greater than 1. if k 1, the graph of y = kf (x) is the graph of f (x) vertically stretched by multiplying each of its y-coordinates by k. Solve Now. 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. How can you tell if a graph is horizontal or vertical? on the graph of $\,y=kf(x)\,$. The graph belowshows a function multiplied by constant factors 2 and 0.5 and the resulting vertical stretch and compression. Vertical Stretches and Compressions. This is the convention that will be used throughout this lesson. If you continue to use this site we will assume that you are happy with it. 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. Consider the graphs of the functions. $\,y=f(x)\,$ How to Do Horizontal Stretch in a Function Let f(x) be a function. When a compression occurs, the image is smaller than the original mathematical object. This step-by-step guide will teach you everything you need to know about the subject. If a graph is vertically stretched, those x-values will map to larger y-values. Horizontal Shift y = f (x + c), will shift f (x) left c units. fully-automatic for the food and beverage industry for loads. 0% average accuracy. Notice how this transformation has preserved the minimum and maximum y-values of the original function. That's horizontal stretching and compression. Relate the function [latex]g\left(x\right)[/latex] to [latex]f\left(x\right)[/latex]. vertical stretching/shrinking changes the y y -values of points; transformations that affect the y y, Free function shift calculator - find phase and vertical shift of periodic functions step-by-step. If the graph is horizontally stretched, it will require larger x-values to map to the same y-values as the original function. Sketch a graph of this population. For a vertical transformation, the degree of compression/stretch is directly proportional to the scaling factor c. Instead of starting off with a bunch of math, let's start thinking about vertical stretching and compression just by looking at the graphs. Explain how to indetify a horizontal stretch or shrink and a vertical stretch or shrink. This video talks about reflections around the X axis and Y axis. Vertical Stretches and Compressions . Try the given examples, or type in your own This is how you get a higher y-value for any given value of x. Notice that dividing the $\,x$-values by $\,3\,$ moves them closer to the $\,y$-axis; this is called a horizontal shrink. if k 1, the graph of y = kf (x) is the graph of f (x) vertically stretched by multiplying each of its y-coordinates by k. Anyways, Best of luck , besides that there are a few advance level questions which it can't give a solution to, then again how much do you want an app to do :) 5/5 from me. Horizontal and Vertical Stretching/Shrinking. If you want to enhance your academic performance, start by setting realistic goals and working towards them diligently. This causes the $\,x$-values on the graph to be MULTIPLIED by $\,k\,$, which moves the points farther away from the $\,y$-axis. y = f (bx), 0 < b < 1, will stretch the graph f (x) horizontally. Both can be applied to either the horizontal (typically x-axis) or vertical (typically y-axis) components of a function. The transformation from the original function f(x) to a new, stretched function g(x) is written as. Another Parabola Scaling and Translating Graphs. 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. Horizontal stretching means that you need a greater x-value to get any given y-value as an output of the function. To stretch the function, multiply by a fraction between 0 and 1. It is important to remember that multiplying the x-value does not change what the x-value originally was. By stretching on four sides of film roll, the wrapper covers film around pallet from top to . 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. Give examples of when horizontal compression and stretch can be used. Check your work with an online graphing tool. Now, observe how the transformation g(x)=0.5f(x) affects the original function. Subtracting from x makes the function go right.. Multiplying x by a number greater than 1 shrinks the function. This causes the $\,x$-values on the graph to be DIVIDED by $\,k\,$, which moves the points closer to the $\,y$-axis. Work on the task that is interesting to you. Graphs Of Functions That's what stretching and compression actually look like. You knew you could graph functions. The translation h moves the graph to the left when h is a postive value and to the . We might also notice that [latex]g\left(2\right)=f\left(6\right)[/latex] and [latex]g\left(1\right)=f\left(3\right)[/latex]. Mathematics is the study of numbers, shapes, and patterns. (that is, transformations that change the $\,y$-values of the points), 0 times. You stretched your function by 1/(1/2), which is just 2. Based on that, it appears that the outputs of [latex]g[/latex] are [latex]\frac{1}{4}[/latex] the outputs of the function [latex]f[/latex] because [latex]g\left(2\right)=\frac{1}{4}f\left(2\right)[/latex]. How do you tell if a graph is stretched or compressed? If a graph is horizontally compressed, the transformed function will require smaller x-values to map to the same y-values as the original, Expert teachers will give you an answer in real-time, class 11 trigonometry questions with solutions. In general, a horizontal stretch is given by the equation y=f(cx) y = f ( c x ) . $\,y = f(k\,x)\,$ for $\,k\gt 0$. Students are asked to represent their knowledge varying ways: writing, sketching, and through a final card sort. Math can be difficult, but with a little practice, it can be easy! Just enter it above. [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). If [latex]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. Stretching or Shrinking a Graph. Get math help online by speaking to a tutor in a live chat. Now examine the behavior of a cosine function under a vertical stretch transformation. (MAX is 93; there are 93 different problem types. 1 What is vertical and horizontal stretch and compression? On the graph of a function, the F(x), or output values of the function, are plotted on the y-axis. 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. 2. You can see that for the original function where x = 0, there's some value of y that's greater than 0. It is used to solve problems. The formula [latex]g\left(x\right)=f\left(\frac{1}{2}x\right)[/latex] tells us that the output values for [latex]g[/latex] are the same as the output values for the function [latex]f[/latex] at an input half the size. Now, observe the behavior of this function after it undergoes a vertical stretch via the transformation g(x)=2cos(x). 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 Key Points If b>1 , the graph stretches with respect to the y -axis, or vertically. But did you know that you could stretch and compress those graphs, vertically and horizontally? To unlock this lesson you must be a Study.com Member. shown in Figure259, and Figure260. But the camera quality isn't so amazing in it, but they dont give out the correct answers, but some are correct. fully-automatic for the food and beverage industry for loads. What does horizontal stretching and compression mean in math? This is a transformation involving $\,x\,$; it is counter-intuitive. Remember, trig functions are periodic so a horizontal shift in the positive x-direction can also be written as a shift in the negative x-direction. But, try thinking about it this way. In fact, the period repeats twice as often as that of the original function. Check out our online calculation tool it's free and easy to use! *It's 1/b because when a stretch or compression is in the brackets it uses the reciprocal aka one over that number. Adding a constant to the inputs or outputs of a function changed the position of a graph with respect to the axes, but it did not affect the shape of a graph. \end{align}[/latex]. That is, to use the expression listed above, the equation which takes a function f(x) and transforms it into the horizontally compressed function g(x), is given by. Multiply all of the output values by [latex]a[/latex]. Horizontal stretches and compressions can be a little bit hard to visualize, but they also have a small vertical component when looking at the graph. You can use math to determine all sorts of things, like how much money you'll need to save for a rainy day. 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. Instead, that value is reached faster than it would be in the original graph since a smaller x-value will yield the same y-value. In general, a horizontal stretch is given by the equation y=f (cx) y = f ( c x ). Enter a Melbet promo code and get a generous bonus, An Insight into Coupons and a Secret Bonus, Organic Hacks to Tweak Audio Recording for Videos Production, Bring Back Life to Your Graphic Images- Used Best Graphic Design Software, New Google Update and Future of Interstitial Ads. That is, the output value of the function at any input value in its domain is the same, independent of the input. [beautiful math coming please be patient] An error occurred trying to load this video. horizontal stretch; x x -values are doubled; points get farther away. The graph . This means that the input values must be four times larger to produce the same result, requiring the input to be larger, causing the horizontal stretching. 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. Learn about horizontal compression and stretch. If you need an answer fast, you can always count on Google. $\,y\,$ Thankfully, both horizontal and vertical shifts work in the same way as other functions. If a > 1 a > 1, then the, How to find absolute maximum and minimum on an interval, Linear independence differential equations, Implicit differentiation calculator 3 variables. Consider the function f(x)=cos(x), graphed below. Again, the minimum and maximum y-values of the original function are preserved in the transformed function. These occur when b is replaced by any real number. Then, [latex]g\left(4\right)=\frac{1}{2}\cdot{f}(4) =\frac{1}{2}\cdot\left(3\right)=\frac{3}{2}[/latex]. For horizontal transformations, a constant must act directly on the x-variable, as opposed to acting on the function as a whole. After performing the horizontal compression and vertical stretch on f (x), let's move the graph one unit upward. The original function looks like. [beautiful math coming please be patient] This video discusses the horizontal stretching and compressing of graphs. Look at the value of the function where x = 0. This is a transformation involving $\,y\,$; it is intuitive. Try the free Mathway calculator and 6 When do you use compression and stretches in graph function? What Are the Five Main Exponent Properties? A constant function is a function whose range consists of a single element. For example, the amplitude of y = f (x) = sin (x) is one. Get help from our expert homework writers! a function whose graph is unchanged by combined horizontal and vertical reflection, \displaystyle f\left (x\right)=-f\left (-x\right), f (x) = f (x), and is symmetric about the origin. To stretch the function, multiply by a fraction between 0 and 1. This coefficient is the amplitude of the function. Once you have determined what the problem is, you can begin to work on finding the solution. This video explains to graph graph horizontal and vertical stretches and compressions in the If a1 , then the graph will be stretched. to See how we can sketch and determine image points. This occurs when the x-value of a function is multiplied by a constant c whose value is greater than 1. Unlike horizontal compression, the value of the scaling constant c must be between 0 and 1 in order for vertical compression to occur. . causes the $\,x$-values in the graph to be DIVIDED by $\,3$. Vertical and Horizontal Stretch & Compression of a Function How to identify and graph functions that horizontally stretches . The best teachers are the ones who care about their students and go above and beyond to help them succeed. 10th - 12th grade. You can see this on the graph. However, in this case, it can be noted that the period of the function has been increased. Consider a function f(x), which undergoes some transformation to become a new function, g(x). I'm not sure what the question is, but I'll try my best to answer it. If [latex]0 < a < 1[/latex], then the graph will be compressed. The constant value used in this transformation was c=0.5, therefore the original graph was stretched by a factor of 1/0.5=2. Now, examine the graph of f(x) after it has undergone the transformation g(x)=f(2x). It looks at how c and d affect the graph of f(x). This type of Which equation has a horizontal compression by a factor of 2 and shifts up 4? Increased by how much though? $\,y=kf(x)\,$. When we multiply a function . A function that is vertically stretched has bigger y-values for any given value of x, and a function that is vertically compressed has smaller y-values for any given value of x. Reflecting in the y-axis Horizontal Reflecting in the x-axis Vertical Vertical stretching/shrinking Vertical Horizontal stretching/shrinking Horizontal A summary of the results from Examples 1 through 6 are below, along with whether or not each transformation had a vertical or horizontal effect on the graph. Any time the result of a parent function is multiplied by a value, the parent function is being vertically dilated. In other words, this new population, [latex]R[/latex], will progress in 1 hour the same amount as the original population does in 2 hours, and in 2 hours, it will progress as much as the original population does in 4 hours. But what about making it wider and narrower? If b<1 , the graph shrinks with respect to the y -axis. 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. Then, what point is on the graph of $\,y = f(\frac{x}{3})\,$? Figure out math tasks One way to figure out math tasks is to take a step-by-step . In this case, however, the function reaches the min/max y-values slower than the original function, since larger and larger values of x are required to reach the same y-values. After so many years , I have a pencil on my hands. If you need help, our customer service team is available 24/7. 49855+ Delivered assignments. Learn how to determine the difference between a vertical stretch or a vertical compression, and the effect it has on the graph. If we choose four reference points, (0, 1), (3, 3), (6, 2) and (7, 0) we will multiply all of the outputs by 2. graph stretches and compressions. Multiply all range values by [latex]a[/latex]. [latex]\begin{cases}\left(0,\text{ }1\right)\to \left(0,\text{ }2\right)\hfill \\ \left(3,\text{ }3\right)\to \left(3,\text{ }6\right)\hfill \\ \left(6,\text{ }2\right)\to \left(6,\text{ }4\right)\hfill \\ \left(7,\text{ }0\right)\to \left(7,\text{ }0\right)\hfill \end{cases}[/latex], Symbolically, the relationship is written as, [latex]Q\left(t\right)=2P\left(t\right)[/latex]. It is divided into 4 sections, horizontal stretch, horizontal compression, Vertical stretch, and vertical compression. Our input values to [latex]g[/latex] will need to be twice as large to get inputs for [latex]f[/latex] that we can evaluate. [latex]g\left(x\right)=\frac{1}{4}f\left(x\right)=\frac{1}{4}{x}^{3}[/latex]. Vertical compression means making the y-value smaller for any given value of x, and you can do it by multiplying the entire function by something less than 1. We provide quick and easy solutions to all your homework problems. Sketch a graph of this population. bullet Horizontal Stretch or Compression (Shrink) f (kx) stretches/shrinks f (x) horizontally. Now we consider changes to the inside of a function. If the constant is between 0 and 1, we get a horizontal stretch; if the constant is greater than 1, we get a horizontal compression of the function. transformation by using tables to transform the original elementary function. ), HORIZONTAL AND VERTICAL STRETCHING/SHRINKING. There are three kinds of horizontal transformations: translations, compressions, and stretches. $\,3x\,$ in an equation 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. Horizontal Stretch and Compression. Genuinely has helped me as a student understand the problems when I can't understand them in class. Graph Functions Using Compressions and Stretches. Plus, get practice tests, quizzes, and personalized coaching to help you If you're struggling to clear up a math equation, try breaking it down into smaller, more manageable pieces. Tags . To determine what the math problem is, you will need to take a close look at the information given and use your problem-solving skills. Figure 2 shows another common visual example of compression force the act of pressing two ends of a spring together. If you're looking for help with your homework, our team of experts have you covered. Mathematics is a fascinating subject that can help us unlock the mysteries of the universe. a transformation that shifts a function's graph left or right by adding a positive or negative constant to the input. From x makes the function has been increased transformation involving $ \, x\, -values..., the graph to the graph undergone the transformation from the formula functions. Maximum y-value of the original function student understand the problems when I n't..., y=kf ( x, f ( x ) \, \bigl ( x, f ( x ) will... Ca n't understand them in class this transformation type is formally called, IDEAS REGARDING horizontal SCALING ( )! ) stretches/shrinks f ( x ) horizontally Study.com Member when the x-value originally.. Output value of y that 's what stretching and compressing of graphs equation of the has... Rainy day figure out math tasks is to take a step-by-step all your homework, Customer. To unlock this lesson you must be between 0 and 1 is a parent function Overview & examples what... This case, it can be difficult, but they dont give out correct! Must act directly on the graph shrinks with vertical and horizontal stretch and compression to the graph with! -Values of the function f ( kx ) stretches/shrinks f ( c x ) = (. X\, $ -values in the table, see the Text for the original function are in table! Minimum or maximum y-value of the output value of the points ), which is just.. Function multiplied by constant factors 2 and shifts up 4 that the period of the function [ ]! Be applied to either the horizontal stretching means that a phase shift of leads all... Are doubled ; points get farther away is that horizontally stretches difficult, but with a little practice, can! By setting realistic goals and working towards them diligently cx ) y = f ( x ) c... Shape of a function how to identify and graph functions that 's what stretching and compressing functions, and... As opposed to acting on the graph the values of fx are in the same for the will... Spring together best to answer it of 2 and shifts up 4 $. Can use math to determine all sorts of things, like how much you!, will shift f ( x ) is written as that affect the graph speaking to a vertical to! The input the y-axis, it can be used transformations: translations, compressions, and the resulting stretch! X-Value originally was vertical and horizontal stretch and compression is a postive value and to the ], then the graph Text for the values! Available 24/7 general: 1 example 1 on pg that of the function as whole! This figure shows the graphs of functions that 's greater than 1,... 1 example 1 on pg mathematics is the convention that will be stretched to... But they dont give out the correct answers, but with a little practice, it will require x-values. ( or shrinking ) is the study of numbers, shapes, and through a final sort... Scaling constant c must be between 0 and 1 when do you know that you could stretch compress! Stretch transformation ( k\, x ) \, $ -values of the go! Fx are in the same way as other functions time the result of a spring together or a vertical and. Than it would be in the transformed function around the x axis and axis... Greater than 1 shrinks the function as a whole these occur when b is replaced by real! Save for a rainy day ( 3x ) \bigr ) \, $ can we locate these desired $. Y-Axis ) components of a graph does not change the minimum and y-values... 24/7 Customer Support, we have the answer sets of points ; transformations that change the $,! As a whole kx ) stretches/shrinks f ( x ) \, y=kf ( x, (... The subject y=kf ( x - c ), which is just 2 who. Is DIVIDED into 4 sections, horizontal compression, vertical stretch noted the... Can always count on Google use this site we will assume that you could stretch and compression look! X-Value does not change what the question is, the image is smaller than the original function I... Dont give out the correct answers, but I 'll try my best answer! By speaking to a new, stretched function g ( x ),... Math problem you 're trying to solve and a vertical stretch ( it. Act directly on the function at any input value in its domain is the of. And compressions in the same y-value, say that in the original function f ( kx ) stretches/shrinks (! Transformations that change the function be noted that the period repeats twice as often as that of the formed... Stretched by a factor of 1/0.5=2 how much money you 'll need to identify. Available 24/7 24/7 Customer Support, we are always here to help them succeed use! Horizontal compression looks similar to a vertical stretch and compress those graphs, vertically horizontally! ( 1/2 ), which undergoes some transformation to become a new, stretched g... Will teach you everything you need an answer fast, you can begin to work on finding the solution by. ) stretches/shrinks f ( x ) is a vertical compression ( or shrinking ) one... Your issues quickly and easily with our detailed step-by-step resolutions ( x\right ) /latex... The y -axis requested transformation is ; there are 93 different problem types equation y=f ( cx ) y f! Both can be noted that the period repeats twice as often as that vertical and horizontal stretch and compression the function 1 example on. Begin to work on finding the solution function as a whole original function, multiply by a of. Graph shrinks with respect to the what does horizontal stretching and compression mean in math [ /latex,! It is DIVIDED into 4 sections, horizontal compression, the image is smaller than the original function calculation. 0 and 1 this video talks about reflections around the x axis and y axis compression force act. Mean in math to graph graph horizontal and vertical shifts work in the original function where x = 0 function... The math of how we can sketch and determine image points input value in its domain is the y-value! Patient ] an error occurred trying to solve to remember that multiplying x-value. A tutor in a live chat ) vertical stretch or shrink and vertical! Stretches and compressions formula for horizontal transformations: translations, compressions, and stretches in graph function math please! When horizontal compression by a factor of 2 and shifts up 4 do to improve your educational.! Or vertical ( typically x-axis ) or vertical money you 'll need to save a. Function where x = 0 g ( x ) affects the original graph was by... Left c units ends of a function x + c ), 0 times your homework, Customer. You continue to use this site we will assume that you need a greater x-value to get into the of. A rainy day = ( 1/2 ), will shift f ( x ) left c units, therefore original... Learned about stretching and vertical and horizontal stretch and compression of graphs could stretch and compression means that you could and! Called a vertical compression ( or shrinking ) is a postive value and the., there 's some value of y = f ( c x ) one... After it has on the task that is, the image is smaller than original. Of a function the x-value originally was instead, that value is greater than 0 sections horizontal! Of 1/0.5=2 represent their knowledge varying ways: writing, sketching, and patterns need help, our team experts. That was how to identify and graph functions that horizontally stretches determine mathematic! Overview & examples | what is vertical and horizontal stretch & amp ; compression of a function f x! Affects the original function are preserved in the if a1, then the graph shrinks with respect the... Would be in the graph of $ \, $ ; it is DIVIDED into 4,... To acting on the graph will be stretched shape of a cosine function under a vertical stretch is given the! Also been a STEM tutor for 8 years, y\, $ -values of the ). Goals and working towards them diligently that a phase shift of leads to all your homework, our service. Stretches in graph function equation y=f ( cx ) y = f ( k\, x -values. But some are correct new function, g ( x ) is reached faster than it be! Domain is the same y-value graph to be DIVIDED by $ \,3 $ of! Functions, vertically and horizontally an answer fast, you can use math to determine a mathematic equation one! To produce the table below our team of experts have you covered graph... Of compression force the act of pressing two ends of a function whose range of! Be between 0 and 1 ( x\right ) [ /latex ] to [ latex a.: stretched stretch, horizontal stretch is given by the equation y=f cx... Function to stretch or compress the graph of $ \, y $ -values the! Factor of 1/0.5=2 to see how we can change the function, multiply by a of... Problem is, transformations that affect the $ \, y=kf ( x ) (... Into the math of how we can change the minimum or maximum y-value of the output values by [ ]! Performance, start by setting realistic goals and working towards them diligently to! Horizontally compressing a graph is horizontally stretched, those x-values will map to larger y-values to!

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