![]() ![]() That it does that stretching so that we can match up to G of X? And the best way to do And so let's think about,Ĭan we multiply this times some scaling factor so ![]() 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. ![]() Take the negative of that to get to negative one. 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. if they multiplied inside the function by something larger than 1, or outside the function by something smaller than 1, then they did a function compression.įor instance, looking at y = x 2 − 4, you can see that multiplying outside the function doesn't change the location of the zeroes, but multiplying inside the function does.G can be thought of as a scaled version of F If they multiplied inside the function by something smaller than 1, or outside the function by something larger than 1, then they did a function stretch.If these points line up horizontally, then they multiplied inside the function. If these points line up vertically, then they multiplied outside of the function. If the y-intercepts remain the same, then they multiplied inside the function. If the x-intercepts remain the same, then they multiplied outside of the function. To figure out whether a graph is stretched or compressed, in comparison with the original graph, look at the max/min points and the x- and y-intercepts. How can you tell if a graph is stretched or compressed? Sometimes, though, it helps to look at the zeroes of the graph (if it has more than one) or turning points (that is, the max/min points). In my experience, it can feel just about impossible to discern whether a multiplication transformation involved multiplying inside or outside the function. The graphic below shows g( x) in blue, g(2 x) in dark green, and g(½ x) in dark red: Instead of multiplying by 2 and by ½ on the outside, I'll instead multiply on the inside that is, I'll multiply on the x that is the argument of the function.įor clarity, the two new functions are these: On the other hand, we can also multiply inside a function. This behavior - namely, that the x-intercepts don't move and the max/min points line up vertically - is the hallmark of multiplying a function on the outside by some number. All of the x-intercepts are the same, and the max/min points line up vertically (that is, the max/min points occur at the same x values, but at different y-value heights). And multiplying by ½ (which is smaller than 1) caused the highs and lows of the original graph to contract, drawing closer to the x-axis. Using the function g( x) = (2 − x)( x + 1)( x + 4), the graphic below shows g( x) in blue, 2 g( x) in dark green, and ½ g( x) in dark red:Īs you can see, multiplying on the outside of the function by 2 (which is larger than 1) caused the highs and lows of the original graph to go higher and lower. This fixedness of the intercepts makes sense, since x-intercepts are where the function is zero, and zero times the number is still gonna be zero. But - and this is important - the x-intercepts will remain the same. If you multiply a function by a number, then the function will get taller (if the number is greater than 1) or shorter (if the number is less than 1). The more important thing is to understand the difference between multiplying outside the function and multiplying inside the function.) What happens if you multiply a function by a number? (The terms "squeezed", "shortened", "compressed", "stretche", etc, seem not to have fixed meanings in this context, which makes this type of function transformation even more confusing. If the graph of the new function is shorter or narrower than the original, then the function has been compressed. If the graph of the new function is taller or wider than the original function's graph, then the function has been stretched. ![]()
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