Functions and Graphs

Graph t...

Question

Functions and Graphs

Graph the following equations and explain why they are not graphs of functions of x.

a. |x| + |y| = 1

Accepted Answer

Welcome back, everyone. In this problem, we want to sketch the graph of the absolute value of X minus 2 plus the absolute value of Y + 3 E equals 2 and explain why it is not a graph of a function of X. And here is our graph on which we'll do our sketch. Now, let's start with the first part of this problem. Let's try to sketch the graph. Now what do we know that will help us to sketch the graph for our equation? Well, if we take a good look at our equation, OK, recall that our equation is in the general form, the absolute value of X minus some value H plus, OK, let me write that properly here. Plus the absolute value of Y minus sum value k. is equal to some constant, OK? And when we have our equation like this, it describes a shape where the sum of the horizontal and vertical distances from the center HK, remains constant. So in other words, HK is the center of our shape. And know this shape forms a symmetric symmetric quadrilateral that is a diamond with diagonals parallel to the X and Y axis here. In that case, where H equals 2, OK. And K is equal to -3, OK? So in other words, it forms a diamond, write that out. So this graph, this this equation tells us that our graph forms a diamond, OK, or a rhombus centered. Centered At 23. So here would be the center of our graph 2-3. Now that we have the center, we need to figure out the other parts of or the other points on our diamond for us to sketch our graph. Now how can we do that? Well, to determine the size and location of the shape, we find a point where one absolute value is maximized while the other is minimized. So first, let's take it for the 1st and 3rd vertex moving horizontally. That is where exchanges and Y remains constant. So if we let Y, OK, if we let Y be equal to -3, that is the centers Y coordinate, then our equation simplifies to the absolute value of X minus 2 equals 2. What does this mean? Well, no, that tells us then that X minus 2 will be equal to 2, and X minus 2 will be will be equal to -2 on our graph. If X minus 2 equals 2, that means X equals 4. And if X minus 2 equals -2, that means X will be equal to 0, OK? Thus, this tells us. That 4-3 and 0-3, OK, is on our graph. No, we can plot those points, so that's 4, OK, -3. And 0-3. Let's just put an X here just to make it clear where our points are. Now, let's do the same for our X coordinate. Let's set X. Let's maximize it by setting X to 2, OK? I know when X equals 2, then the absolute value of X minus 2 will be the absolute value of 2 minus 2 or 0. And thus our equation will be simplified to the absolute value of Y + 3 equals 2. In that case, Y + 3 then is going to be equal to 2, and Y + 3 will be equal to -2. If Y + 3 is 2, that means Y must be equal to -1. And if a Y + 3 is -2, that must mean Y equals -5, OK? Thus, This also tells us, OK, that 2-1. And 25 are on the graph. So now we can go ahead and put 2-1. OK, so that's gonna be 2-1 and 2, what did we say? -5. Now, we have all the points for our graph, so we can go ahead and draw our diamond, OK? So let me just draw that here. And we know this is, this is my sketch, OK? So this is our diamond. This is the diamond for the graph of the absolute value of X minus 2, plus the absolute value of Y + 3 equals 2. Now let's focus on the next part of our problem. How do we explain why it's not a graph of a function of X? Well, what do we know about functions? Well, a function must pass the vertical line test, meaning that for every X there must be only one corresponding Y value. But if we take this graph and let's say we put a vertical line here to the point where X equals 3, notice that there are two possible values for Y. Likewise, if we do the same here, when X equals 2, you realize we are in the same situation. So, in at least one point on our graph, OK, then a single X value is corresponding to more than 1 Y value. Thus, the equation fails the vertical line test and cannot be a function of X. Therefore, this is our graph and it's not a function of X because a single X value corresponds to multiple Y values, making it fail the vertical line test. Thanks a lot for watching everyone. I hope this video helped.

Functions and Graphs

Graph the following equations and explain why they are not graphs of functions of x.

a. |x| + |y| = 1

Key Concepts

Definition of a Function

Vertical Line Test

Absolute Value Equations

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