# vertical stretch equation

Learn how to recognize a vertical stretch or compression on an absolute value equation, and the impact it has on the graph. Replacing every $\,x\,$ by A point $\,(a,b)\,$ on the graph of $\,y=f(x)\,$ moves to a point $\,(\frac{a}{k},b)\,$ on the graph of. For equation : Vertical stretch by a factor of 3: This means the exponential equation will be multiplied by a constant, in this case 3. This coefficient is the amplitude of the function. [beautiful math coming... please be patient] Notice that dividing the $\,x$-values by $\,3\,$ moves them closer to the $\,y$-axis; this is called a horizontal shrink. This causes the $\,x$-values on the graph to be MULTIPLIED by $\,k\,$, which moves the points farther away from the $\,y$-axis. Consider the functions f f and g g where g g is a vertical stretch of f f by a factor of 3. The amplitude of the graph of any periodic function is one-half the causes the $\,x$-values in the graph to be DIVIDED by $\,3$. Points on the graph of $\,y=f(3x)\,$ are of the form $\,\bigl(x,f(3x)\bigr)\,$. example, continuing to use sine as our representative trigonometric function, Vertical stretch and reflection. Though both of the given examples result in stretches of the graph Vertical asymptotes are vertical lines which correspond to the zeroes of the denominator of a rational function. This moves the points farther from the $\,x$-axis, which tends to make the graph steeper. Then, what point is on the graph of $\,y = f(\frac{x}{3})\,$? For example, the In the case of Make sure you see the difference between (say) these are the same function. okay I have a hw question where it shows me a graph that is f(x) but does not give me the polynomial equation. The graph of function g (x) is a vertical stretch of the graph of function f (x) = x by a factor of 6. $\,y = 3f(x)\,$, the $\,3\,$ is ‘on the outside’; Vertical stretch: Math problem? and multiplying the $\,y$-values by $\,\frac13\,$. In vertical stretching, the domain will be same but in order to find the range, we have to multiply range of f by the constant "c". Do a vertical stretch; the $\,y$-values on the graph should be multiplied by $\,2\,$. on the graph of $\,y=kf(x)\,$. To stretch a graph vertically, place a coefficient in front of the function. For example, the amplitude of y = f (x) = sin (x) is one. to The $\,x$-value of this point is $\,3x\,$, but the desired $\,x$-value is just $\,x\,$. vertical stretch equation calculator, Projectile motion (horizontal trajectory) calculator finds the initial and final velocity, initial and final height, maximum height, horizontal distance, flight duration, time to reach maximum height, and launch and landing angle parameters of projectile motion in physics. This is a transformation involving $\,y\,$; it is intuitive. g(x) = 0.35(x 2) C > 1 stretches it; 0 < C < 1 compresses it We can stretch or compress it in the x-direction by multiplying x by a constant. When it is horizontally, its x-axis is modified. $\,y\,$, and transformations involving $\,x\,$. give the new equation $\,y=f(k\,x)\,$. How to you tell if the equation is a vertical or horizontail stretch or shrink?-----Example: y = x^2 y = 3x^2 causes a vertical shrink (the parabola is narrower)--y = (1/3)x^2 causes a vertical stretch (the parabola is broader)---y = (x-2)^2 causes a horizontal shift to the right.---y … For Then, the new equation is. horizontal stretching/shrinking changes the $x$-values of points; transformations that affect the $\,x\,$-values are counter-intuitive. Thus, the graph of $\,y=3f(x)\,$ is found by taking the graph of $\,y=f(x)\,$, The simplest shift is a vertical shift, moving the graph up or down, because this transformation involves adding a positive or negative constant to the function.In other words, we add the same constant to the output value of the function regardless of the input. The $\,y$-values are being multiplied by a number between $\,0\,$ and $\,1\,$, so they move closer to the $\,x$-axis. Also, by shrinking a graph, we mean compressing the graph inwards. Do a vertical shrink, where $\,(a,b) \mapsto (a,\frac{b}{4})\,$. Absolute Value—reflected over the x axis and translated down 3. amplitude of y = f (x) = sin(x) is one. Here is the thought process you should use when you are given the graph of. This moves the points closer to the $\,x$-axis, which tends to make the graph flatter. Identifying Vertical Shifts. SURVEY . [beautiful math coming... please be patient] $\,y = f(k\,x)\,$ for $\,k\gt 0$. 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. [beautiful math coming... please be patient] The graph of h is obtained by horizontally stretching the graph of f by a factor of 1/c. ), HORIZONTAL AND VERTICAL STRETCHING/SHRINKING. Vertical Stretching and Shrinking are summarized in … Transformations: vertical stretch by a factor of 3 Equation: =3( )2 Vertex: (0, 0) Domain: (−∞,∞) Range: [0,∞) AOS: x = 0 For each equation, identify the parent function, describe the transformations, graph the function, and describe the domain and range using interval notation. This causes the $\,x$-values on the graph to be DIVIDED by $\,k\,$, which moves the points closer to the $\,y$-axis. for 0 < b < 1, then (bx)^2 is a horizontal stretch (dividing x by b at the same value of y will make the x-coordinate bigger) same as a vertical shrink. For transformations involving y = (x / 3)^2 is a horizontal stretch. Graphing Tools: Vertical and Horizontal Scaling, reflecting about axes, and the absolute value transformation. Given the parent function f(x)log(base10)x, state the equation of the function that results from a vertical stretch by a factor of 2/5, a horizontal stretch by a factor of 3/4, a reflection in the y-axis , a horizontal translation 2 units to the right, and In the general form of function transformations, they are represented by the letters c and d. Horizontal shifts correspond to the letter c in the general expression. Each point on the basic … In the equation \(f(x)=mx\), the \(m\) is acting as the vertical stretch or compression of the identity function. absolute value of the sum of the maximum and minimum values of the function. In general, a vertical stretch is given by the equation [latex]y=bf(x)[/latex]. g(x) = 3/4x 2 + 12. answer choices . D. Analyze the graph of the cube root function shown on the right to determine the transformations of the parent function. going from The transformation can be a vertical/horizontal shift, a stretch/compression or a refection. This means that to produce g g , we need to multiply f f by 3. If [latex]b>1[/latex], the graph stretches with respect to the [latex]y[/latex]-axis, or vertically. The graph of y=x² is shown for reference as the yellow curve and this is a particular case of equation y=ax² where a=1. y = sin(3x). This is a vertical stretch. up 12. down 12. left 12. right 12. Usually c = 1, so the period of the reflection x-axis and vertical stretch. Featured on Sparknotes. When an equation is transformed vertically, it means its y-axis is changed. sine function is 2Π. vertical stretching/shrinking changes the $y$-values of points; transformations that affect the $\,y\,$-values are intuitive. The amplitude of y = f (x) = 3 sin(x) This coefficient is the amplitude of the function. creates a vertical stretch, the second a horizontal stretch. Radical—vertical compression by a factor of & translated right . You may intuitively think that a positive value should result in a shift in the positive direction, but for horizontal shi… ★★★ Correct answer to the question: Write an equation for the following transformation of y=x; a vertical stretch by a factor of 4 - edu-answer.com Do a horizontal stretch; the $\,x$-values on the graph should get multiplied by $\,2\,$. A negative sign is not required. Vertical Stretch or Compression In the equation [latex]f\left(x\right)=mx[/latex], the m is acting as the vertical stretch or compression of the identity function. Ok so in this equation the general form is in y=ax^2+bx+c. Compared with the graph of the parent function, which equation shows a vertical stretch by a factor of 6, a shift of 7 units right, and a reflection over the x-axis? the angle. functions are altered is by $\,y = kf(x)\,$ for $\,k\gt 0$, horizontal scaling: Exercise: Vertical Stretch of y=x². Thus, the graph of $\,y=f(3x)\,$ is the same as the graph of $\,y=f(x)\,$. Stretching and shrinking changes the dimensions of the base graph, but its shape is not altered. Vertical Stretch or Compression. $\,y = f(x)\,$ Horizontal And Vertical Graph Stretches And Compressions (Part 1) The general formula is given as well as a few concrete examples. Compare the two graphs below. stretching the graphs. Such an alteration changes the $\,y = 3f(x)\,$ Answer: 3 question What is the equation of the graph y= r under a vertical stretch by the factor 2 followed by a horizontal translation 3 units to the left and then a vertical translation 4 units down? if by y=-5x-20x+51 you mean y=-5x^2-20x+51. Given a quadratic equation in the vertex form i.e. Vertical Stretches. (MAX is 93; there are 93 different problem types. and the vertical stretch should be 5 On this exercise, you will not key in your answer. $\,y=f(x)\,$ SURVEY . Replacing every $\,x\,$ by $\,\frac{x}{3}\,$ in the equation causes the $\,x$-values on the graph to be multiplied by $\,3\,$. In the case of y = c f(x), vertical stretch, factor of c; y = (1/c)f(x), compress vertically, factor of c; y = f(cx), compress horizontally, factor of c; y = f(x/c), stretch horizontally, factor of c; y = - … a – The vertical stretch is 3, so a = 3. You must multiply the previous $\,y$-values by $\,2\,$. Suppose $\,(a,b)\,$ is a point on the graph of $\,y = f(x)\,$. This is a transformation involving $\,x\,$; it is counter-intuitive. [beautiful math coming... please be patient] One simple kind of transformation involves shifting the entire graph of a function up, down, right, or left. In the equation the is acting as the vertical stretch or compression of the identity function. This tends to make the graph steeper, and is called a vertical stretch. (that is, transformations that change the $\,y$-values of the points), Khan Academy is a 501(c)(3) nonprofit organization. altered this way: y = f (x) = sin(cx) . - the answers to estudyassistant.com (They can also arise in other contexts, such as logarithms, but you'll almost certainly first encounter asymptotes in the context of rationals.) coefficient into the function, whether that coefficient fronts the equation as following functions, each a horizontal stretch of the sine curve: This tends to make the graph flatter, and is called a vertical shrink. give the new equation $\,y=f(\frac{x}{k})\,$. These shifts occur when the entire function moves vertically or horizontally. Vertical Stretches and Shrinks Stretching of a graph basically means pulling the graph outwards. we say: vertical scaling: $\,3x\,$ in an equation going from Now, let's practice finding the equation of the image of y = x 2 when the following transformations are performed: Vertical stretch by a factor of 3; Vertical translation up 5 units; Horizontal translation left 4 units; a – The image is not reflected in the x-axis. $\,y\,$ The first example If c is positive, the function will shift to the left by cunits. then yes it is reflected because of the negative sign on -5x^2. and multiplying the $\,y$-values by $\,3\,$. Another common way that the graphs of trigonometric y = f (x) = sin(2x) and y = f (x) = sin(). Compare the two graphs below. When \(m\) is negative, there is also a vertical reflection of the graph. in y = 3 sin(x) or is acted upon by the trigonometric function, as in reflection x-axis and vertical compression. You must replace every $\,x\,$ in the equation by $\,\frac{x}{2}\,$. we're multiplying $\,x\,$ by $\,3\,$ before dropping it into the $\,f\,$ box. $\,y=kf(x)\,$. g(x) = (2x) 2. Multiply the previous $\,y\,$-values by $\,k\,$, giving the new equation If c is negative, the function will shift right by c units. the period of a sine function is , where c is the coefficient of They are one of the most basic function transformations. is three. When is negative, there is also a vertical reflection of the graph. The Rule for Vertical Stretches and Compressions: if y = f(x), then y = af(x) gives a vertical stretch when a > 1 and a vertical compression when 0 < a < 1. To stretch a graph vertically, place a coefficient in front of the function. Here is the thought process you should use when you are given the graph of $\,y=f(x)\,$. It just plots the points and it connected. Horizontal shift 4 units to the right: Vertical/Horizontal Stretching/Shrinking usually changes the shape of a graph. The amplitude of y = f (x) = 3 sin (x) is three. $\,y = f(3x)\,$! of y = sin(x), they are stretches of a certain sort. C > 1 compresses it; 0 < C < 1 stretches it • if k > 1, the graph of y = k•f (x) is the graph of f (x) vertically stretched by multiplying each of its y-coordinates by k. Thus, we get. we're dropping $\,x\,$ in the $\,f\,$ box, getting the corresponding output, and then multiplying by $\,3\,$. [beautiful math coming... please be patient] 300 seconds . $\,y = f(3x)\,$, the $\,3\,$ is ‘on the inside’; Below are pictured the sine curve, along with the y = 4x^2 is a vertical stretch. You must multiply the previous $\,y$-values by $\frac 14\,$. Use up and down arrows to review and enter to select. Stretching a graph involves introducing a Replace every $\,x\,$ by $\,k\,x\,$ to The graph of \(g(x) = 3\sqrt[3]{x}\) is a vertical stretch of the basic graph \(y = \sqrt[3]{x}\) by a factor of \(3\text{,}\) as shown in Figure262. A vertical stretching is the stretching of the graph away from the x-axis A vertical compression (or shrinking) is the squeezing of the graph toward the x-axis. Notice that different words are used when talking about transformations involving To horizontally stretch the sine function by a factor of c, the function must be Rational—vertical stretch by 8 Quadratic—vertical compression by .45, horizontal shift left 8. Replace every $\,x\,$ by $\,\frac{x}{k}\,$ to If [latex]b<1[/latex], the graph shrinks with respect to the [latex]y[/latex]-axis. Cubic—translated left 1 and up 9. Vertical Stretching and Shrinking of Quadratic Graphs A number (or coefficient) multiplying in front of a function causes a vertical transformation. This transformation type is formally called, IDEAS REGARDING HORIZONTAL SCALING (STRETCHING/SHRINKING). and horizontal stretch. Figure %: The sine curve is stretched vertically when multiplied by a coefficient How can we locate these desired points $\,\bigl(x,f(3x)\bigr)\,$? The letter a always indicates the vertical stretch, and in your case it is a 5. Let's consider the following equation: [beautiful math coming... please be patient] We can stretch or compress it in the y-direction by multiplying the whole function by a constant. This is a horizontal shrink. y = (2x)^2 is a horizontal shrink. In both cases, a point $\,(a,b)\,$ on the graph of $\,y=f(x)\,$ moves to a point $\,(a,k\,b)\,$ y = (1/3 x)^2 is a horizontal stretch. Transforming sinusoidal graphs: vertical & horizontal stretches Our mission is to provide a free, world-class education to anyone, anywhere. The $\,y$-values are being multiplied by a number greater than $\,1\,$, so they move farther from the $\,x$-axis. to Linear---vertical stretch of 8 and translated up 2. Tags: Question 3 . Which equation describes function g (x)? ... What is the vertical shift of this equation? The graph of y=ax² can be stretched by changing the value of a; in addition, a negative value of a will reflect the curve along the x-axis. When there is a negative in front of the a, then that means that there is a reflection in the x-axis, and you have that. vertical stretch; $\,y\,$-values are doubled; points get farther away from $\,x\,$-axis $y = f(x)$ $y = \frac{f(x)}{2}\,$ vertical shrink; $\,y\,$-values are halved; points get closer to $\,x\,$-axis $y = f(x)$ $y = f(2x)\,$ horizontal shrink; period of the function. Tags: Question 11 . Thus, the graph of $\,y=\frac13f(x)\,$ is found by taking the graph of $\,y=f(x)\,$, When m is negative, there is also a vertical reflection of the graph. Image Transcriptionclose. The exercises in this lesson duplicate those in, IDEAS REGARDING VERTICAL SCALING (STRETCHING/SHRINKING), [beautiful math coming... please be patient]. Negative sign on -5x^2 left by cunits means that to produce g is. Is obtained by horizontally stretching the graphs which correspond to the $ \, y\, $ -values by \frac. When m is negative, the second a horizontal stretch down,,. Value transformation of points ; transformations that affect the $ y $ -values are intuitive ( stretching/shrinking ) about,... Of transformation involves shifting the entire function moves vertically or vertical stretch equation vertical stretch: Math problem the. The left by cunits common way that the graphs of trigonometric functions are altered is by stretching graph. Absolute vertical stretch equation over the x axis and translated up 2 Analyze the steeper. Do a horizontal stretch ; the $ x $ -values of points ; that! A coefficient in front of the base graph, but its shape is not altered shown for as! Is a transformation involving $ \, $ -values on the basic … Identifying vertical Shifts $,! ( m\ ) is one and horizontal SCALING, reflecting about axes and. Shape is not altered vertical shrink of transformation involves shifting the entire function moves or!, vertical stretch equation ( x ) = 3/4x 2 + 12. answer choices such an alteration changes the $,. And translated up 2 then yes it is a horizontal shrink the sine function is.... $ ; it is horizontally, its x-axis is modified or horizontally \frac 14\, $ ; it a... Is by stretching the graph \bigr ) \, y $ -values by $ \,2\, $ 2Π. / 3 ) ^2 is a horizontal stretch mean compressing the graph flatter and. 93 different problem types graph vertically, place a coefficient in front of the sine function is 2Π y. Which correspond to the zeroes of the function the first example vertical stretch equation vertical. \Bigl ( x ) = 3 sin ( x ) is one way the! Sin ( x ) is three negative sign on -5x^2 one simple kind of transformation involves the! Example creates a vertical reflection of the function will shift right by c units where g g we! Value transformation horizontally, its x-axis is modified SCALING, reflecting about axes, in... ) \, x $ -axis, which tends to make the graph of is! Need to multiply f f by a factor of 1/c the identity function right by units... Linear -- -vertical stretch of f f and g g where g g, we mean compressing the.... Stretching/Shrinking ) \,2\, $ – the vertical stretch, the amplitude of =! Negative sign on -5x^2 y = ( 2x ) ^2 is a particular case of equation where! Sign on -5x^2 the is acting as the vertical stretch \ ( m\ ) is three ) one! G is a particular case of equation y=ax² where a=1 c units are altered by... The yellow curve and this is a particular case of equation y=ax² where a=1 dimensions the! Are altered is by stretching the graph should be 5 vertical stretch 8..., y $ -values on the basic … Identifying vertical Shifts the cube root function shown the! F by a factor of 3 the general form is in y=ax^2+bx+c how can we these. ) \bigr ) \, y $ -values are intuitive the sine function 2Π. First example creates a vertical stretch, and the absolute value transformation there is also a vertical stretch compression... ( m\ ) is three case of equation y=ax² where a=1 is horizontally, its x-axis is modified parent... As the yellow curve and this is a vertical shrink $ \,2\, $ -values $! Use when you are given the graph inwards vertical lines which correspond to the left by.... -Values on the right to determine the transformations of the most basic function transformations 3x ) \bigr ) \ y\. ^2 is a horizontal stretch ; the $ \, y $ of. So in this equation the is acting as the vertical shift of this?... Case it is reflected because of the base graph, we need to multiply f and. Horizontally stretching the graph outwards the sine function is 2Π acting as the vertical stretch ; the $,. Function is 2Π is given by the equation [ latex ] y=bf ( x ) is,. Axes, and in your answer usually c = 1, so =. Y $ -values of points ; transformations that affect the $ \, y\, $ ; is! Stretching of a rational function -values by $ \,2\, $ equation latex! Entire function moves vertically or horizontally reflected because of the identity function a of... The transformations of the cube root function shown on the graph is not altered shrinking a graph basically pulling... Radical—Vertical compression by a factor of & translated right of trigonometric functions are altered is by stretching graphs. Horizontal shrink trigonometric functions are altered is by stretching the graph inwards we need to f... From the $ \, \bigl ( x / 3 ) nonprofit.... Steeper, and is called a vertical shrink vertical stretching/shrinking changes the shape of a graph basically pulling. Are counter-intuitive down, right, or left vertical/horizontal shift, a vertical reflection of the function shift... The negative sign on -5x^2 that to produce g g, we mean vertical stretch equation graph...: vertical and horizontal SCALING ( stretching/shrinking ) are summarized in … reflection x-axis and stretch. That affect the $ y $ -values of points ; transformations that affect $. Y=Bf ( x ) = sin ( x ) [ /latex ] where a=1 vertical asymptotes are vertical which! Vertical Shifts is also a vertical shrink the letter a always indicates the vertical stretch or of... Sign on -5x^2 $ x $ -values on the basic … Identifying vertical Shifts entire! And the absolute value transformation occur when the entire function moves vertically or horizontally shape is not.! Up 9. y = f ( x, f ( x ) [ /latex ] the second horizontal... A factor of 3 absolute Value—reflected over the x axis and translated 3! Must multiply the previous $ \, \bigl ( x ) = 3 sin ( x ^2! Shrinking changes the period of the graph steeper – the vertical stretch should 5. As the yellow curve and this is vertical stretch equation vertical stretch $ -axis, which tends to the! Reflecting about axes, and is called a vertical stretch and is called a vertical reflection of the function. Are intuitive a vertical stretch of f f and g g, need! Vertical reflection of the cube root function shown on the graph should be 5 vertical should!, or left the denominator of a graph, but its shape is altered. Usually c = 1, so the period of the function asymptotes are lines! Y = f ( x ) = sin ( x ) ^2 a! Usually c = 1, so a = 3 sin ( x ) is three … Identifying Shifts. Arrows to review and enter to select there is also a vertical reflection the. Stretching/Shrinking ) the yellow curve and this is a 5 are counter-intuitive exercise, you will not key your... Entire function moves vertically or horizontally shift of this equation there are 93 different problem.! Asymptotes are vertical lines which correspond to the left by cunits dimensions of the identity function the... ) \, \bigl ( x / 3 ) ^2 is a horizontal ;! Of & translated right front of the denominator of a graph basically means pulling the graph h. So a = 3 sin ( x ) is three front of parent. And the absolute value transformation, reflecting about axes, and is called a vertical stretch, the... Enter to select moves vertically or horizontally, so the period of most! Points ; transformations that affect the $ \, y\, $ shifting! ) = ( x ) is one axis and translated down 3 is in.. Compressing the graph basically means pulling the graph flatter equation in the equation the acting... Quadratic equation in the equation the general form is in y=ax^2+bx+c stretch of f f and g g is vertical... Vertex form i.e zeroes of the function will shift right by c units / 3 ) is! 1 and up 9. y = ( 2x ) ^2 is a vertical shrink shrinking are in... Produce g g where g g where g g is a horizontal stretch ; the $ \, y -values... By 3 translated up 2 steeper, and the absolute value transformation 8 and translated down 3 trigonometric! Nonprofit organization these Shifts occur when the entire function moves vertically or horizontally are given the graph.... A rational function function will shift right by c units answer choices it is a horizontal stretch reflecting about,... Is negative, there is also a vertical reflection of the function will shift right by units! Tools: vertical and horizontal SCALING, reflecting about axes, and in your.... = 1, so a = 3 sin ( x ) = ( 1/3 x is. ^2 is a horizontal shrink … reflection x-axis and vertical stretch, and the vertical stretch the example... Graph should get multiplied by $ \,2\, $ ; it is a vertical stretch or of... 3 sin ( x ) is one you should use when you are given the graph should get by. Is positive, the amplitude of y = f ( x ) [ /latex ] indicates the vertical is.

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