This is what flips it over the x-axis, and then multiplying it by this fraction that has an absolute value less than one, this is actually stretching it wider. A function g (x) represents a vertical compression of f (x. Given a function f (x), we can formalize compressing and stretching the graph of f (x) as follows: A function g (x) represents a vertical stretch of f (x) if g (x) c f (x) and c > 1. When we were saying we were scaling it, we're If we multiply a function by a coefficient, the graph of the function will be stretched or compressed. G of X is equal to negativeġ/4 times X squared. So I'm feeling really good that this is the equation of G of X. When X is equal to four,įour squared is 16. Here 'cause it looks like this is sitting on our graph as well. Two squared is four, times negative 1/4 is indeedĮqual to negative one. When X is equal to one, let me do this in another color, when X is equal to one, then one squared times negative 1/4, well that does indeed look Zero, well this is still all gonna be equal to Our green function, and if I multiply it by 1/4, that seems like it will Notice that the vertical strech has moved the sides closer together or made the interior angle smaller while the vertical compression has moved the sides farther apart or made the interior angle larger. Okay, well let's up take to see if we could take It has the parent function in purple, a vertical strech in red, and a vertical compression in blue. Well negative one is 1/4 of negative four, so that's why I said Well we want that when X is equal to two to be equal to negative one. When X is equal to two I get to negative four. So how did you get 1/4? Well I looked at when X is equal to two. So in that case, we're gonna have Y is equal to not just negative X squared, but negative 1/4 X squared. See if we scale by 1/4, does that do the trick? Scale by 1/4. When X is equal to two Y is equal to negative four. So you could say G of two is negative one. When X is equal to two, Y is equal to negative one on G of X. Our original function, f (x) - the red one, is taller than the new function, c f (x) - the green one.For the new functions to be shorter, our c value needs to be 0 < c < 1. Here that at the point two comma negative one, sits on G of X. Stretch vs Shrink: Vertical Stretch vs Vertical Shrink (Compression) On the top left we have a vertical shrink or compression. This is to pick a point that we know sits on G of X,Īnd they in fact give us one. That it does that stretching so that we can match up to G of X? And the best way to do At first the two functions might look like two parabolas.If you graph by hand, or if you set your calculator to sequential mode (and not simultaneous), you can see that the graph of y -x 3 is in fact a reflection of y x 3 over the x-axis. And so let's think about,Ĭan we multiply this times some scaling factor so Sketch a graph of y x 3 and y -x 3 on the same axes. G of X also seems to be stretched in the horizontal direction. Whatever X is, you square it, and then you take the negative of it, and you see that that willįlip it over the x-axis. Whatever the X is, you square it, and then you take the negative of it. To the negative of F of X, or we could say Y is equal So this green function right over here is going to be Y is equal Getting before for a given X, we would now get the opposite So as we just talk throughĪs we're trying to draw this flipped over version, whatever Y value we were Y when is X is equal to negative two instead of Y being equal to four, it would now be equal to negative four. We will choose the points (0, 1) and (1, 2). Let us follow two points through each of the three transformations. ![]() ![]() Take the negative of that to get to negative one. We can sketch a graph by applying these transformations one at a time to the original function. Instead of squaring one and getting one, you then But when X is equal to negative one, instead of Y being equal to one, it'd now be equal to negative one. ![]() Instead when X is equal to zero, Y is still gonna be equal to zero. So first let's flip over, flip over the x-axis. We might appreciate is that G seems not only toīe flipped over the x-axis, but then flipped overĪnd then stretched wider. So like always, pause this video and see if you can do it on your own. This means that the input values must be four times larger to produce the same result, requiring the input to be larger, causing the horizontal stretching.G can be thought of as a scaled version of F Because f\left(x\right) ends at \left(6,4\right) and g\left(x\right) ends at \left(2,4\right), we can see that the x\textx\right). The graph of g\left(x\right) looks like the graph of f\left(x\right) horizontally compressed.
0 Comments
Leave a Reply. |
AuthorWrite something about yourself. No need to be fancy, just an overview. ArchivesCategories |