![]() ![]() 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. And then you can see that indeed do they indeed do look like reflections flipped over the X axis.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? And this bottom part of the quadrilateral gets reflected above it. So you an kind of see this top part of the quadrilateral And what's interesting about this example is that, the original quadrilateral is on top of the X axis. We have constructed the reflection of ABCD across the X axis. And we'll keep our XĬoordinate of negative two. Unit below the X axis, we'll be one unit above the X axis. If we reflect across the X axis instead of being one And so let's see, D right now is at negative two comma negative one. So this goes to negative five, one, two, three, positive four. So it would have theĬoordinates negative five comma positive four. Units below the X axis, it will be four units above the X axis. The same X coordinate but instead of being four C, right here, has the X coordinate of negative five. The same X coordinate but it's gonna be two I'm having trouble putting the let's see if I move these other characters around. So let's make this right over here A, A prime. So, its image, A prime we could say, would be four units below the X axis. So we're gonna reflect across the X axis. So let's just first reflect point let me move this a littleīit out of the way. Move this whole thing down here so that we can so that we can see what is going on a little bit clearer. So we can see the entire coordinate axis. And we need to construct a reflection of triangle A, B, C, D. Tool here on Khan Academy where we can construct a quadrilateral. Asked to plot the image of quadrilateral ABCD so that's this blue quadrilateral here. ![]()
0 Comments
Leave a Reply. |
Details
AuthorWrite something about yourself. No need to be fancy, just an overview. ArchivesCategories |