Quadratic functions can also be graphed using transformations of a basic parent function. The graph below shows an example where the house is made wider and shorter: \(\frac-5\) Graph Quadratic Functions As with translation, x-direction changes will seem backward. Changes in the x-direction are made by changing the input, which is the argument of the function. As with translation, changes in the y-direction are made by changing the output, which it the result of the whole function. We can also stretch out or compress our house by multiplying. We will discuss two types of reflections: reflections across the x-axis and reflections across the y-axis. Examples of this type of transformation are: translations, rotations, and reflections In other transformations, such as dilations, the size of the figure will change. retains its size and only its position is changed. We shift a function to the right by subtracting from the argument, and we shift to the left by adding to the argument. In geometry, a transformation is a way to change the position of a figure. Horizontal shifts seem backward, but it's only because we're affecting the input rather than the output of the function. credit (wikimedia/Conrad Poirier) A flat mirror produces an image called a reflection where everything is inverted left to right. Figure 1: The reflection of two people in a distorting mirror of Cartierville Belmont Park. When our x-value is \(4\), then our transformed function argument is \((4-3)\) and the function responds like the original function did when the argument was \(1\). Graph functions with stretches and shrinks. This is accomplished by making the function argument \((x-3)\). Why? If we want our reference point near the chimney to be located at \(x=4\), then we need an input of \(x=4\) in our transformed function to be treated as though we had put \(x=1\) into the original function. To make the house shift 3 units to the right, we need to subtract 3 from the input. Horizontal shifts are often less intuitive than vertical shifts. The x-axis corresponds to the input of the function, so if we wanted to shift the house to the right or left, we would need to change what is inside the function parentheses (the function argument). Since function outputs correspond to y-axis values, we can shift a function up by adding or or down by subtracting from the whole function. If we subtracted \(4\) from every output of the function, we would get \(H(x)-4\), which corresponds to the little blue house shifted downward. The house has a point near the chimney at the coordinate \((1,1)\). The graph below shows an example "function", \(H(x)\), that draws a little red house at the origin. There are three basic transformations that can be done to a shape:1. This section presents a simplified visual example of several ways to transoform functions: translation, compression, expansion, and reflection. Note: there are no method marks awarded for any transformation questions no matter how many marks are being awarded, they are all answer marks.ĭilation from the x-axis by a factor of aĭilation from the y-axis by a factor of aĮxample 3.So far we have worked with basic linear, quadratic, radical, exponential, and logarithmic functions, but these functions often appear in different forms. You will learn how to perform the transformations, and how to map one figure into another using these transformations. In methods we are generally concerned with - reflections in the x-axis, y-axis and line y=x Reflections simply flip the graph relative to a certain line. Graph functions using compressions and stretches. Determine whether a function is even, odd, or neither from its graph. Graph functions using reflections about the x -axis and the y -axis. All dilation factors are positive.Ī dilation from the y-axis by a factor larger than one ( > 1) will result in the graph being stretched away from the y-axis.Ī dilation from the y-axis by a factor less than one ( 1) will result in the graph being stretched away from the x-axis.Ī dilation from the x-axis by a factor less than one ( < 1) will result in the graph being compressed towards the x-axis. Highlights Learning Objectives In this section, you will: Graph functions using vertical and horizontal shifts. A transformation is a way of changing the position (and sometimes the size) of a shape. Translations are when we shift the entire graph left, right, up or down.ĭilations stretch or compress the graph. Translations and reflections are examples of transformations. There are three types of transformations which we will be dealing with: translations, dilations and reflections.
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