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. y f x have the same x-coordinates but different y-coordinates.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? Any point on a line of reflection is an invariant point.

Invariant Point: a point on a graph that remains unchanged after a transformation is applied to it. What is the image of (- 6 3 after a reflection over the line Y? What is an invariant point example? Reflecting the point over the y-axis, the point will move to the right into Quadrant IV, where x is positive and y is negative, so the point will be (6, -3). The point (-6, -3) is located in Quadrant III. What is the image of (- 6 3 after a reflection over the y-axis? If you reflect over the line y = -x, the x-coordinate and y-coordinate change places and are negated (the signs are changed). When you reflect a point across the line y = x, the x-coordinate and y-coordinate change places. You are expected to identify invariant points. When we transform a shape – using translations, reflections, rotations, enlargements, or some combination of those 4, there are sometimes points on the shape that end up in the same place that they started. Solved example to find the reflection of a point in the y-axis: Find the points onto which the points (11, -8), (-6, -2) and (0, 4) are mapped when reflected in the y-axis. (iv) The image of the point (-6, 5) in the y-axis is the point (-(-6), 5) i.e., (6, 5). What is the image of − 6 5 after a reflection over the y-axis?

Reflection over y axis