![]() 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. Each change has a specific effect that can be seen graphically. We can transform the inside (input values) of a function or we can transform the outside (output values) of a function. We now explore the effects of multiplying the inputs or outputs by some quantity. Is the function f ( s ) = s 4 + 3 s 2 + 7 f ( s ) = s 4 + 3 s 2 + 7 even, odd, or neither? Graphing Functions Using Stretches and CompressionsĪdding 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. For a function g ( x ) = f ( x ) + k, g ( x ) = f ( x ) + k, the function f ( x ) f ( x ) is shifted vertically k k units. In other words, we add the same constant to the output value of the function regardless of the input. 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. One simple kind of transformation involves shifting the entire graph of a function up, down, right, or left. There are systematic ways to alter functions to construct appropriate models for the problems we are trying to solve. One method we can employ is to adapt the basic graphs of the toolkit functions to build new models for a given scenario. Often when given a problem, we try to model the scenario using mathematics in the form of words, tables, graphs, and equations. Graphing Functions Using Vertical and Horizontal Shifts In this section, we will take a look at several kinds of transformations. In a similar way, we can distort or transform mathematical functions to better adapt them to describing objects or processes in the real world. But what happens when we bend a flexible mirror? Like a carnival funhouse mirror, it presents us with a distorted image of ourselves, stretched or compressed horizontally or vertically. When we tilt the mirror, the images we see may shift horizontally or vertically. The following steps will help to ensure a correct solution.We all know that a flat mirror enables us to see an accurate image of ourselves and whatever is behind us. ![]() It is often necessary to perform trasnformations in a certain order to guarantee the arrival at the correct graph. Expressions such as ( x + 3) and ( x - 4) can represent a horizontal shift, but expressions such as ( x 2 + 3) and ( x 3 - 4) cannot. Remember that a horizontal shift is associated with a change in the x-coordinate value (expressed as a linear expression - x with a power of 1). The parentheses were done first, then any multiplication/division, followed by any addition/subtraction. This pattern is similar to order of operations. There was a pattern to the order in which this problem was analyzed ( horizontal shift - vertical stretch - vertical shift). The subtraction of 1 indicated a vertical shift of one unit down. The parent has a slope of 1, whereas this new function will have a slope of 2. The multiplication of 2 indicates a vertical stretch of 2, which will cause to line to rise twice as fast as the parent function. This is a horizontal shift of three units to the left from the parent function. It can be seen that the parentheses of the function have been replaced by x + 3,Īs in f ( x + 3) = x + 3. The parent function is f ( x) = x, a straight line. Vertically oriented transformations and horizontally oriented transformations to not affect one another.Ĭonsider the problem f ( x) = 2( x + 3) - 1. ![]() If two or more of the transformations have a horizontal effect on the graph, the order of those transformations will most likely affect the graph.
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