The graph is then translated right 4 units and up 3 units. The graph is then stretched by a factor of 2. The graph is reflected across the x-axis. The graphs below illustrate the above sequence of combined transformations. Horizontal stretch/compression/reflection by a factor of B Vertical stretch/compression/reflection by a factor of A In an expression such as A* f( B*( x - C) + D, the transformations can be applied in the following order, although several other possible orderings will give the same results: The vertical reflection and the vertical stretch commute with each other, but not with the vertical shift. In the example above, the horizontal shift commutes with all three of the other transformations. The order of the transformations is not necessarily commutative. Shifted upward 3 units because 3 is added to the basic function Shifted right 4 because 4 is subtracted from x. Vertically stretched by a factor of 2 because basic function is multiplied by 2. Reflected about the x-axis because a negative multiplies the basic function. That is, the graph is compressed horizontally by a factor ofĬombines several transformations of the graph of Notice that in Y 2 the x-values that produce the same y-values are half those in Y 1. The table and diagram below illustrate how the five special points of the basic sine wave are transformed. Notice that the graph of y = sin 2 x is compressed horizontally and one period is completed in half the original period. Set the style of the graph of Y 1 = sin(X) to thick. Now display the graph of y = sin 2 x and compare it to the graph of y = sin x. It is often useful to identify the five special points on the sine wave whose x-values are The domain of the sine function is the set of all real numbers. That is, the y-values rise to 1 unit above the centerline and fall to 1 unit below the centerline. That is, the y-values repeat over x-intervals of length 2 The graph is periodic and has a period of 2 The graph of y = sin x has the characteristics listed below. The graphs of y = sin x and y = sin 2 x better illustrate the effect of multiplying x by a constant.Ĭheck that your calculator is in radian mode and set Y 1 = sin(X) and display the graph in a In many functions like the absolute value function, the horizontal compression appears to be a vertical stretch.įrom the table it appears that the values of Y 2 are twice that of the values of Y 1. 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. Left 3 units, which may be opposite to the direction you expected.ģ.2.2 Use your calculator to experiment and find the function that will shift The fixed output of y = 0 was produced by x = 0 in Y 1 and was produced by x = -3 in Y 2. With the basic absolute value function and compare the x-values that produce a specific y-value.ĭisplay the table of values for Y 1 = abs(X) and Y 2 = abs(X + 3). When describing a horizontal shift, it is helpful to see which x-values produce the same y-value. It should seem reasonable to conclude that when you add a constant to the x-values, the transformation will be a horizontal shift. Recall that when a constant was added to the y-values, the transformation was a vertical shift. We will now examine the effect of adding to (or subtracting from) the x-values before applying the function. In each case, we described how the transformed graph's y-values were related to the corresponding y-values of the original function. We have examined vertical transformations that were created by stretching or shrinking vertically, by reflecting across the x-axis, or by shifting upward or downward. The table and diagram below illustrate the vertical shift of 3 units upward.Īnd describe the transformations. The transformed graph is a vertical shift of Notice that the values in Y 2 are three more than the values in Y 1. Explore the effect of adding three to the absolute value function.Įnter Y 1 = abs(x) and Y 2 = abs(x) + 3 in the Y= editor. This lesson will introduce translations, a third type of transformation, and discuss the effects of combining several types of transformations.Ī translation sends all points of a graph the same distance in the same direction.Ī vertical translation, or vertical shift, moves every point on a graph up or down the same distance. Lesson 3.1 discussed two types of transformations: stretches and reflections. Lesson 3.2: Translations and Combined Transformations Module 3 - Functions and Transformations - Lesson 2 Module 3 - Functions and Transformations
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