![]() The following steps make graphing transformations so easier. But when a graph is given, graphing the function transformation is sometimes difficult. Identifying the transformation by looking at the original and transformed graphs is easy because just by looking at the graph, we can say that the graph moves up by 2 units or left by 3 units, etc. We can describe the transformations of functions by using the above tricks also. Thus, g(x) is obtained from f(x) by horizontal translation by 5 units to the left, vertical dilation by a scale factor of 3, reflection about the x-axis, and vertical translation by 2 units up. i.e., vertical translation by 2 units up. Finally, it changes into -3 f(x + 5) + 2.i.e., vertical dilation by a scale factor of 3. i.e., horizontal translation by 5 units to the left. For example, if the question is what is the effect of transformation g(x) = - 3f(x + 5) + 2 on y = f(x), then first observe the sequence of operations that had to be applied on f(x) to get g(x) and then use the above rules to define the transformations. We can use the above rules to describe any function transformation. If the minus sign is inside the bracket, it is with respect to the y-axis and if the minus sign is outside the bracket, it is with respect to the x-axis. ![]() Just in case of reflection, it is just the opposite of the first and second tricks here.For example, f(2x) is a horizontal dilation and 2 f(x) is a vertical dilation. If some number is being multiplied/ divided, then its related to "dilation".For example, f(x + 2) is a horizontal translation and f(x) + 2 is a vertical translation. If some number is being added/ subtracted, then its related to "translation".Let us tabulate all function transformation rules together. So far we have understood the types of transformations of functions and how do addition/subtraction/multiplication/division of a number and the multiplication of a minus sign would reflect a graph. For example, the point (1, 1) (on the original graph) corresponds to (1, 3) on the new graph. In the following graph, the original function y = x 3 is stretched vertically by a scale factor of 3 to give the transformed function graph y = 3x 3. Every old y-coordinate is multiplied by k to find the new y-coordinate. In this dilation, there will be changes only in the y-coordinates but there won't be any changes in the x-coordinates. It changes a function y = f(x) into the form y = f(kx), with a scale factor '1/k', parallel to the x-axis. The horizontal dilation (also known as horizontal scaling) of a function either stretches/shrinks the curve horizontally. Similarly, if it is dilated parallel to the y-axis, all the y-values are increased by the same scale factor. If a graph undergoes dilation parallel to the x-axis, all the x-values are increased by the same scale factor. Here, the original function y = x 2 (y = f(x)) is moved to 2 units up to give the transformed function y = x 2 + 2 (y = f(x) + 2).Ī dilation is a stretch or a compression. if k if k 0, then the function moves up by 'k' units.if k > 0, then the function moves to the left side by 'k' units.This changes a function y = f(x) into the form y = f(x ± k), where 'k' represents the horizontal translation. In this translation, the function moves to the left side or right side. There are two types of translations of functions. Let us learn each of these function transformations in detail.Ī translation occurs when every point on a graph (representing a function) moves by the same amount in the same direction. 'd' represents the vertical translation.'c' represents the horizontal translation.Any minus sign multiplies means that it is a reflection. Also, note that addition/subtraction indicates translation and multiplication/division represents dilation. ![]() Note that all outside numbers (that are outside the brackets) represent vertical transformations and all inside numbers represent horizontal transformations. Here, a, b, c, and d are any real numbers and they represent transformations. In math words, the transformation of a function y = f(x) typically looks like y = a f(b(x + c)) + d. Transformationįlips the curve and produces the mirror image. ![]() We can see what each of these transformations of functions mean in the table below. There are mainly three types of function transformations:Īmong these transformations, only dilation changes the size of the original shape but the other two transformations change the position of the shape but not the size of the shape. Notice the output values for \(g(x)\) remain the same as the output values for \(f(x)\), but the corresponding input values, \(x\) for \(g\), have shifted to the right by 3.A function transformation either "moves" or "resizes" or "reflects" the graph of the parent function. The result is that the function \(g(x)\) has been shifted to the right by 3. ![]()
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