When we have both vertical and horizontal reflections, it doesn’t matter which we perform first. On the other hand, if the negative is inside the function, then the input x-value switches sign, essentially reflecting every point over the y-axis. If the negative is outside the function, then the output y-value switches sign, essentially reflecting every point over the x-axis. The intuition behind reflections is that, depending where it is placed, the negative sign switches positive and negative values of the $x$ or $y$ variable. Reflections occur when a function is made negative – when the negative is outside the function, the reflection is over the y-axis and when the negative is inside the function, the reflection is over the x-axis. So it's gonna look something, something like that but the key issue and the reason why I'mĭrawing is so you can see that it looks like it'sīeing scaled vertically.When a function is reflected, it flips across one of the axes to become its mirror image. Let me make it at least lookĪ little bit more symmetric. It would make it look, it would make it look wider. If we were scaling vertically by something that had anĪbsolute value less than one then it would make the graph less tall. It's going to be stretchedĪlong the vertical axis. So our graph is now going to look, is now going to look like this. So let's see, two, three,įour, five, six, seven so it'd put it something around that. What this would look like, well, you multiply zero times seven, it doesn't change anything but whatever x this is, this was equal to negative x but now we're gonna get Vertically by a factor of seven but just to understand The negative flips us over the x-axis and then the seven scales What they're asking, what is the equation of the new graph, and so that's what it would be. So I would get y isĮqual to negative seven times the absolute value of x and that's essentially And so if you thinkĪbout that algebraically, well, if I want seven times the y value, I'd have to multiply this thing by seven. You're scaling it vertically by a factor of seven, whatever y value you got for given x, you now wanna get seven times the y value, seven times the y value for a given x. Vertically by a factor of seven and the way I view that is if So that's what reflectingĪcross the x-axis does for us but then they say scaled Once again, whatever absolute value of x was giving you before for given x, we now wanna get the negative of it. Is equal to the negative of the absolute value of x. In general, if you'reįlipping over the x-axis, you're getting the negative. So in general, what we are doing is we are getting the negative The absolute value of x but now we wanna flip across the x-axis and we wanna get the negative of it. The negative of that value associated with that corresponding x and so for example, this x, before, we would get

The absolute value of x and I would end up there but now we wanna reflect across the x-axis so we wanna essentially get So for example, if I have some x value right over here, before, I would take Now, let's think about theĭifferent transformations.

You've seen the graph of y is equal to absolute Sketch so bear with me but hopefully this is familiar. It's gonna have a slope of one and then for negative values, when you take the absolute value, you're gonna take the opposite. So for non-negative values of x, y is going to be equal to x. So let's say that's my x-axis and that is my y-axis. We can all together visualize what is going on. To draw it visually but I will just so that What is the equation of the new graph? So pause the video and see The graph of y is equal to absolute value of x is reflected across the x-axis and then scaled verticallyīy a factor of seven.

Reflection on y axis function