For each equation, (a) give a table with at least three ordered p...

Question

For each equation, (a) give a table with at least three ordered pairs that are solutions, and (b) graph the equation. See Examples 3 and 4.

y = x³

Accepted Answer

Welcome back everyone. In this problem, we want to sketch the graph of the equation Y equals two X cubed by using a table of ordered pairs. And we are already given a graph on which we're going to sketch our equation. Now, before we do that, let's kind of think about the nature of our equation. So our equation is Y equals to X cubed. This is a cubic equation and recall that for a cubic equation. OK. And that means an equation that's written in the form Y equals A X cubed where A is some constant. Then when we plot the graph, we're expecting that it's going to look something like this. OK. And this graph would be a cubic. This is the graph of a cubic equation where its inflection point. That is where the graph changes its curvature is at the origin. So these are the points that we're expecting, we're expecting to get something that looks like that on our own graph. So we're going to see if we can substitute different values of X to figure out what our Y values would be so that we can plot our coordinates. Now, we're given two graphs. But in the first one, the values on the Y axis are bigger. In the second one, the values on the X axis are bigger. But most likely we're going to use our first graph because this is a cubic equation X is being cubed. That means for small values of X, you'll have larger values of Y. Now to sketch our graph, I'm going to choose the domain of X being between negative three and positive three. And I'm going to explain in a few why I chose that domain. But first, let's plot the points. So using that domain, we have our values of X and we know Y which equals to X cubed. So let me draw a little table here and we're going to try and find all the values of X between negative three. So that's negative three, negative two, negative 10 all the way up to positive three. OK? So that's 0123. So let's see if we can figure those values out. So first, let's start when X equals negative three, when X equals negative three, that means Y will be equal to two, multiplied by a negative three cubed, negative three cubed is negative 27 and two multiplied by a negative 27 equals negative 54. So we have our first card in it. Next, let's find it. When X equals, let me put a little line here. X equals negative two. No, Y equals two multiplied by a negative two cubed, negative two cubed is negative eight and two multiplied by negative eight equals negative 60. When X equals negative one, then Y equals two multiplied by a negative one cube. Negative one cube is negative one and two multiplied by negative one equals negative two. Then when X equals zero, Y equals two multiplied by zero cube which is two multiplied by zero, which of course equals zero. No, when X equals one, OK. When X equals one, then Y equals two multiplied by one cued which is two multiplied by one, which equals two. No notice that when X was equal to negative one, our answer is two multiplied by a negative one. But no, this is the positive version of what we would have had if it was negative one. So that means the same must apply for negative to a negative three. In other words, when X equals positive two, then Y equals two multiplied by a positive two cube which is two multiplied by a positive eight, which is a positive 16. Likewise, that tells us when X equals positive three, Y will be equal to two multiplied by positive 27 which is our positive 54. So now we have all of the values for X and Y and we can go ahead and plot on our graph. Now, the reason I chose the domain from negative 3 to 3, it all starts in the idea of the cube. When we look at our range on our graph, it ranged from 50 to negative 50. I knew that I didn't need all of the values all of the X values in our domain from negative 10 to 10 because they couldn't fit on our sketch for the graph. So I chose three because three cubed is 27 and multiplied by two equals 54 which would be beyond the scope of our graph for our Y values. So I only needed all of those values in between. And you can kind of use those ideas to help you uh when you're plotting a graph, you can use your number sense just to try and figure out how the domain and the range relate just so that you can, you know, save some time. Because if I were to go from negative 10 to 10, that would take like a really long time, right? So now we have these values we can plan so we can't plot negative three and negative 54 neither three and 54. We can indicate that it's going to be somewhere on our graph, but we won't know exactly where. So let's just put them outside the scope of our graph. And then let's actually plot the values that we can put on our graph. Know when X is negative tool I'm going to zoom in a little bit so that you can see the graph better, right? So when X equals negative two, then Y equals negative 16. It should probably be somewhere here, let me use a lighter. It's probably somewhere there when X equals negative one, Y equals negative two. OK? From somewhere here when X equals zero, Y equals zero, of course, it passes through our origin or inflection point when X equals one, Y equals two, when X equals two, Y equals eight, sorry, Y equals Y equals 16. And when XC equals three, Y equals 54 which we can see. So now we have our points, we can do our sketch. I hope mine looks good, but our sketch would probably look something like that. So this would be the graph of Y equals two XQ. Thanks a lot for watching everyone. I hope this video helped.

For each equation, (a) give a table with at least three ordered pairs that are solutions, and (b) graph the equation. See Examples 3 and 4. y = x³

Key Concepts

Ordered Pairs

Graphing Functions

Cubic Functions

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