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Gwendolyn C.

July 15, 2021

Formula for distance

Cornell University

University of Michigan - Ann Arbor

Hope College

University of Winnipeg

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welcome to our algebra preview focusing on plotting in Cartesian coordinates. The reason we're gonna go over this even though it may seem fun, amazing, trivial is that it is absolutely a wonderful tool for visualizing what's happening in physics. So, um, let's get started, then. Remember the Cartesian coordinates. That means we're using the standard X and Y axes is some people sometimes refer to it. Okay. And what we're gonna do here is we're gonna always say that the X axis, the horizontal axis is our independent variable, while the Y axis are vertical, axis is going to be our dependent variable. So the way we might write that in physics might be, for example, V of tea is equal to weigh. Might say fee not plus acceleration, times time. And we'll talk more about this equation when we get into Kim Dramatic section. Um, but what this means is that we need to be able to look at this and dry on a Cartesian plot to really understand what's going on. So let's throw on some numbers that might seem a little bit more familiar. Say we have, uh, actually, let's just say we're gonna do a plot of F of X is equal to X. Okay, so remember ffx here we're just substituting in for why goes on the y Axis X and since it's always equal to it at X equals zero f of X equals zero at X equals one f of X equals one X equals two f x equals two and so on and so forth. So what we get is actually a straight line a straight as I can draw it. Um, that extends in both directions. Okay, Um and this is part of a standard set of straight lines that we get from an equation that generally has written something like this. Why equals m x plus b where again we're using. Why on our vertical axis? And in this case, we have called M Slope and B is referred to as the Y intercept the reason for that being and notice if X equals zero i e. Here on the y axis, then why is equal to be, which means that B is the amount at which we cross the y axis. So, for example, say we have ah function F of X is equal to x minus one Okay, so f of X versus X. Okay, X minus one. That means that X equals zero f of X equals negative one. So we're here, then we go over one and up one, and we are there, so we're going to get in. Fact, this line is parallel to the first one I drew. It's just shifted down by one. On the other hand, if I were toe, multiply it there. Say we have a function f of X equals two X minus one. Now, when I put ffx on my Y axis and X on my X axis, then I'm going to still pass through negative one X equals zero. But when I move over one I'm also going to go up to. So I'm gonna end up here, so I'm gonna have a line this'll way. So it has an intersection point with the previous line I drew over here, but it has a different slope, so it's not going to run parallel to either of the previous lines that I've drawn. Now we could have a more complicated situation where instead of just a linear equation, we have something like f X equals X squared. Now Some of you may remember that immediately what it looks like, but just to plug it in. Remember, if X equals zero, then ffx would be zero here. X equals one. FX will be one if X equals negative one. We also get one, because negative one squared is one negative. Two squared is gonna be 1234 and two squared is also gonna be four. And we end up with this. Nice what we call parabolic arc that's drawn in here. Okay? And we can continue and, uh, increase the complexity here. But the basic ideas you plug in numbers for the independent variable on the right hand side, solve for what the dependent variable is. And then plot that in terms of your ex. Why coordinates? Or in this case, what we're writing is X f of X coordinates, and this is how you plot in Cartesian coordinates.

University of North Carolina at Chapel Hill

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