at 6:29 am [ Comment Thanks for the correction! I have amended the post.Ī really good presentation and great help to students. There's some simple reasoning involved, but students just want to follow a rule more often then not.
Students and teachers alike have problems navigating this subject and the intuitions aren't usually taught. In general though, this is a great discussion to bring up. Small correction: an odd function passes through the origin if and only if zero is in the domain of the function! Knowing about even and odd functions is very helpful when studying Fourier Series.Ĩ Comments on “How to reflect a graph through the x-axis, y-axis or Origin?” 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. In this case, theY axis would be called the axis of reflection.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? Math Definition: Reflection Over the Y AxisĪ reflection of a point, a line, or a figure in the Y axis involved reflecting the image over the Y axis to create a mirror image. In this case, the x axis would be called the axis of reflection.
#REFLECTION OVER Y AXIS HOW TO#
This complete guide to reflecting over the x axis and reflecting over the y axis will provide a step-by-step tutorial on how to perform these translations.įirst, let’s start with a reflection geometry definition: Math Definition: Reflection Over the X AxisĪ reflection of a point, a line, or a figure in the X axis involved reflecting the image over the x axis to create a mirror image. This idea of reflection correlating with a mirror image is similar in math. In real life, we think of a reflection as a mirror image, like when we look at own reflection in the mirror. Learning how to perform a reflection of a point, a line, or a figure across the x axis or across the y axis is an important skill that every geometry math student must learn.
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