Graph each inequality. x^2 + (y + 3)^2 ≤ 16

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Accepted Answer

Hello everybody. I have everything right today, today we're going to look at this question that states draw the graph for the given inequality where our inequality is X squared plus open parentheses, Y plus one closed parentheses squared is less than or equal to nine. And I'll label this as equation number one. Now, the first thing I'll do in trying to solve this inequality is I'll just change the less than or equal to symbol for an equal sign. So I'll have X squared plus open parentheses, Y plus one closed parentheses squared is equal to nine. Now, we should know for the sake of this question where we have a less than or equal to sign, we should know that this is a non strict inequality. Therefore, we will have solid lines, a, a solid line in our graph. And this will be more clear once we actually get into graphing non strict inequalities are inequalities that have less than or equal to or greater than or equal to. So if you have that or equal to, that is a non strict inequality. If it was just less than, or just greater than those are strict inequalities and you would use a dash line for the graph. So moving on, we have the X squared uh or we have our inequality now with an equal sign. So before continuing with that equation, I'm going to identify the parent function and I'll label this as equation number two and our parent function would be X squared plus Y squared equals nine. Now, this is just the basic way of writing our equation number one, the plus one inside of the parentheses is just a, a type of transformation that we use for that equation. So if we take that transformation out, we have our parent function X squared plus Y squared equals nine. Now this means that the F of X or I'm sorry. The equation number one graph is the equation number two graph shifting one unit down. So what that plus one inside of the parentheses does is that it shifts the parent functions graph by one unit down. So let's focus on the parent function. Now, the parent function or equation number two was X squared plus Y squared equals R squared, right? That is the basic or the general formula here. And that is the formula for a circle. So we have X squared, if we apply that to our equation number two, we'll have X squared plus Y squared equals if we have nine, our R squared would be three squared because three squared equals nine. So we have that the equation number two graph is a circle with center at the origin and a radius of three units. So now we know what our parent function looks like. And we had said that our current inequality or equation number one was just the parent function shifted one unit down. So now we know exactly what our inequality would look like as a graph. So now we'll test the point in this case, I'll test the origin zero comma zero and we do this because we have an inequality. So we, we're going to have to shade in a region of our graph as the answer because inequalities don't have just one answer. It's anything in this case, it would be anything that's less than or equal to nine. So those are many answers. So we have to shade in a region that would fit that. So I'm going to test the 0.0 comma zero, meaning I'll plug in zero for our X and zero for our Y in our equation number one. So we'll have that zero squared plus open parentheses, zero plus one, closed parentheses. Squared is less than equal to nine, which means that we have zero plus one is less than or equal to nine because zero squared is zero and zero plus one is one, squared is one. So zero plus one is one. So we have that one is less than or equal to nine, which is true. Therefore, the 0.0 comma zero will be part of the shaded region. So now we can go ahead and graph, we have enough information to complete our graph. So we draw a Y axis and an X axis. And I'll draw, I'll label different points on each axis. On the Y axis. I will label one, 234 and five and same thing on the negative side, I'll do negative and negative five. And on the X axis, I'll do the same up to five. So I'll have one, 234 and five on the positive side. And on the negative side, we'll have negative 1234 and negative five. So now I have to think about my parent function and how we changed it. So we said that we started with a circle. So our, our graph will be a circle. Now we started with a center at the origin and now we're shifting it one unit down. So if we moved, we, we were at the origin and now we move one unit down, we'll be at a Y of negative one. So have a 10.0 comma negative one, then we had a radius of three. So we will have, if our origin is at zero, comma negative one to the right, we move three. So we'll have 12 and three. So we'll have a point at three comma negative one. And that would be for the radius on the right side, for the radius to, to the left side would be the same thing but negative. So we'll have negative three comma negative one and we do the same thing on the Y axis. So if we have a center now at zero comma negative one, we go up three units. So we'll be at zero, comma 00, comma one and end at zero comma two. And we do the same thing on the negative side of the axis, we'll be at zero comma negative 20, comma negative three N zero, comma negative four is where we will end up. So now we can connect all our points. And because we did this hand written, it's not gonna look like a perfect graph or a perfect circle, but that is what it should be. And now, as we mentioned, we use a solid line because it is a nonstick inequality. And we also said that the origin is part of our shaded region. So we shade the inside of our circle and the inside of our circle includes that origin and will serve as the area that will function as an answer to our inequality. So just like that we are able to graph our inequality and although it didn't come out as a perfect circle because we are doing it handwritten, it should be a perfect circle within an, with a center at zero comma negative one and a radius of three. So I hope that was helpful. And until next time.

Graph each inequality. x^2 + (y + 3)^2 ≤ 16

Key Concepts

Inequalities

Graphing Circles

Coordinate Plane

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