There some more examples on this page: Even and Odd Functions Note if we reflect the graph in the x-axis, then the y-axis, we get the same graph. An odd function either passes through the origin (0, 0) or is reflected through the origin.Īn example of an odd function is f( x) = x 3 − 9 x This kind of symmetry is called origin symmetry. This time, if we reflect our function in both the x-axis and y-axis, and if it looks exactly like the original, then we have an odd function. Note if we reflect the graph in the y-axis, we get the same graph (or we could say it "maps onto" itself).Īn odd function has the property f( −x) = −f( x). The above even function is equivalent to: That is, if we reflect an even function in the y-axis, it will look exactly like the original.Īn example of an even function is f( x) = x 4 − 29 x 2 + 100 We say the reflection "maps on to" the original.Īn even function has the property f( −x) = f( x). But sometimes, the reflection is the same as the original graph. We really should mention even and odd functions before leaving this topic.įor each of my examples above, the reflections in either the x- or y-axis produced a graph that was different. Reflection in y-axis (green): f( −x) = −x 3 − 3 x 2 − x − 2 Even and Odd Functions Reflection in x-axis (green): − f( x) = − x 3 + 3 x 2 − x + 2 The green line also goes through 2 on the y-axis. Note that the effect of the "minus" in f( −x) is to reflect the blue original line ( y = 3 x + 2) in the y-axis, and we get the green line, which is ( y = −3 x + 2). Now, graphing those on the same axes, we have: Now for f(− x)į( −x) = −3 x + 2 (replace every " x" with a " −x"). What we've done is to take every y-value and turn them upside down (this is the effect of the minus out the front). Note that if you reflect the blue graph ( y = 3 x + 2) in the x-axis, you get the green graph ( y = −3 x − 2) (as shown by the red arrows). When you graph the 2 lines on the same axes, it looks like this: Our new line has negative slope (it goes down as you scan from left to right) and goes through −2 on the y-axis. going uphill as we go left to right) and y-intercept 2. You'll see it is a straight line, slope 3 (which is positive, i.e. If you are not sure what it looks like, you can graph it using this graphing facility. Let's see what this means via an example. Transformations are used to change the graph of a parent function into the graph of a more complex function.This mail came in from reader Stuart recently:Ĭan you explain the principles of a graph involving y = − f( x) being a reflection of the graph y = f( x) in the x-axis and the graph of y = f(− x) a reflection of the graph y = f( x) in the y-axis? Stretching a graph means to make the graph narrower or wider. They are caused by differing signs between parent and child functions.Ī stretch or compression is a function transformation that makes a graph narrower or wider. Reflections are transformations that result in a "mirror image" of a parent function. Reflecting a graph means to transform the graph in order to produce a "mirror image" of the original graph by flipping it across a line. All other functions of this type are usually compared to the parent function. Sketch the graph of each of the following transformations of y = xĪ stretch or compression is a function transformation that makes a graph narrower or wider, without translating it horizontally or vertically.įunction families are groups of functions with similarities that make them easier to graph when you are familiar with the parent function, the most basic example of the form.Ī parent function is the simplest form of a particular type of function. Graph each of the following transformations of y=f(x). Let y=f(x) be the function defined by the line segment connecting the points (-1, 4) and (2, 5).
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