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. |y| = x

Accepted Answer

Hello there. Today we're going to solve the following practice problem together. So first off, let us read the problem and highlight all the key pieces of information that we need to use in order to solve this problem. Sketch the graph of the absolute value of Y minus 3 equals X and explain why it is not a graph of a function of X. Awesome. So it appears for this particular problem, we're asked to solve for two separate answers. Firstly, we're asked to create a sketch of this graph of the absolute value of Y minus 3 equals X. That's our first answer, so we have to create a sketch of this graph. And our second answer is we're trying to explain why this particular function is not a graph of a function of X. So we're trying to explain why it's not a graph of the function of X. So that's our second answer that we're ultimately trying to solve for. So with that said, First off, let us rewrite the absolute value equation. So let us recall and note that the absolute value equation can be rewritten as two separate equations. So let's say for our first equation, we'll call it 1. We can say that Y minus 3. is equal to X, and let us also say for our second function, which we'll call number 2, we can say that Y minus 3 is equal to negative X. So now we need to rearrange this equation to isolate and solve for Y in both cases. So our 1st 1st equation will be Y equals X + 3, whereas our second equation will be Y equals negative X + 3. So now we need to identify the shape of our graph. So let us recall and note looking at our two different expressions here for our two different functions. The two equations will represent straight lines because they're linear in nature. So let us recall a note that Y equals x. Plus 3, we'll have a slope of 1 and we'll pass through, and let's make a note of this, it'll pass through 0.3, and for our second equation, Y equals negative X + 3. It has a slope of -1, and it will pass through 0.3 also. And since the absolute value ensures that X is always non-negative, that will mean as a result that X will have to be greater than or equal to 0. And with that said, the graph will consist of two lines that form a V shape that opens to the right, and the vertex of the graph will be at 0.3. And the two lines will extend infinitely and it will both extend infinitely. In the positive direction and it, so it's gonna be infinitely. Expanding in both the positive and negative Y directions. That's very important to note. So we now need to find a point on the line, Y equals X + 3. So in order to do that, let us state that X is equal to 2, so we could say that Y is equal to 2. Plus 3, and when we plug that into our calculator or we do the math in our head, we know that 2 + 3 is simply going to be equal to 5. So that will mean that our coordinate point will be 2.5 is our value for X is 2, and the value for Y is 5. So let us find a point on the line for Y equals x + 3. So let us also say that X equals 2 in this particular case. So Y is going to be equal to -2. Plus, and once again, we're plugging in a value negative 2 for X, so plus 3 is simply gonna be equal to 1. So that will mean for this particular case, Our coordinate point is going to be 2.1. Fantastic. So now we need to plot the points 0.3, 2.1, and 2.5 are three different coordinate points. So let's, I'll box these in green, these will help us find our final answer. So those are three different coordinate points that we need to plot. So we now need to draw a line from 0.3, which passes through 2.1 to help us graph Y equals X + 3, and then another line from 0.3, which passes through 2.5 to help us graph our function of Y equals X + 3. And when we do that, so I'm, I've gone ahead and used graphing software to digitally graph our final graph here using our functions here. And as you can see, when we plot our two lines, we have our V shape pointing to the right. And as we could see, and I will highlight our points in green, we have our vertex at 03. We also have our point at 21. And we also have our point at 25. And we now in order to help us explain why it's not a function of X, so a function must pass the vertical line test, which means that for every value of X, there should be only one corresponding value of Y. However, in this particular graph, when we do our vertical line test, so I'm going to use a red dotted line to do our vertical line test. As you can see, when we do our vertical line test. It will intersect two points on our two different points on our two separate graphs here. So we need to note that when X is greater than 0, there are 2 corresponding values of Y for each X, one from each line. So that will mean, without a doubt that for our explanation it is not a function of X because a single X value corresponds to two different y values which fails the vertical line test, and those are our final answers. Hooray, we did it. Thank you so much for watching. Hopefully that helped, and I can't wait to see you in the next video. Bye.

Functions and Graphs

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

a. |y| = x

Key Concepts

Definition of a Function

Vertical Line Test

Absolute Value Functions

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