In this lesson, we're going to focus on graphing a circle. This is the standard equation of a circle with the center at the origin. x squared plus y squared is equal to r squared. So let's say if we have this equation, x squared plus y squared is equal to 4. What are the coordinates of the center, and what is the value of the radius? The center is just going to be the origin, 0, 0. And r squared is equal to 4. Therefore, the radius is the square root of 4. It's 2. So relative to the origin, you need to travel 2 units to the right, 2 units to the left, 2 units up, and 2 units down.
And then plot each of those points. And then simply connect those points. by means of a circle.
So that's how you can graph a circle. But now sometimes the origin, or rather the center may not be the origin. Let's say it might be translated h units left or right and k units up or down.
So in this equation the center is h comma k. So let's say if we have the formula x minus 2 squared plus y plus 3 squared. Let's say that's equal to 16. What are the coordinates of the center and what is the radius? The center is going to be 2, negative 3. You simply need to change the sign.
If you set x minus 2 equal to 0, and if you find the value of x, x will equal... 2. The radius is simply the square root of 16, so it's 4. Now let's go ahead and make a graph. So the first thing that we want to do is we want to plot the center, which is 2, negative 3. So it's over here.
And then we want to use the radius to find the other points. So we want to travel 4 points, or 4 units to the right. That will take us to the point 6, negative 3. And then 4 units to the left. left relative to the center that will take us to the point negative 2 negative 3 then we need to go up 4 units negative 3 plus 4 is 1 so that will take us to the point 2 comma 1 and then down 4 units negative 3 minus 4 is negative 7 so that will take us to the point 2 negative 7 and then simply plot the four points that are on the edge of the circle consider this equation x squared plus y squared plus 8x minus 6y plus 21. This is an equation of a circle in non-standard form. So given this equation, how can we graph the circle?
What we need to do is we need to put it in standard form. So first, let's put the x variables next to each other. And then let's leave a space, and let's put the y variables.
next to each other. Now the 21, I'm going to move it to the right side. So it's going to be negative 21 on the right side, since it's positive 21 on the left side.
And we need to complete the square. Half of a is 4. So we're going to add 4 squared to both sides. On the left side, I'm going to write it as 4 squared. But on the right side, I'm going to write it as 16, since 4 times 4 is 16. Half of 6 is 3, so I'm going to add 3 squared, which is 9. Don't worry about the negative sign.
When you square something, it's going to be positive. So now let's factor x squared plus 8x plus 16. It's a perfect square trinomial. Now you can literally see everything you need.
It's going to be x. And then whatever this sign is, plus this number before you square it, 4 squared. Two numbers that multiply to 16 but add to 8 are 4 and 4. So it's x plus 4 times x plus 4. And it works out that way. Now to factor the next perfect square trinomial, it's going to be this variable, which is y.
Whatever this sign is, minus. and then this number before you square so y minus 3 squared On the right, 16 plus 9 is 25. Negative 21 plus 25 is 4. Now that we have the equation in standard form, we need to find the center and the radius. The center is going to be negative 4, positive 3. All you have to do is change the sign.
And the radius is square root of 4, which is 2. Keep in mind, this is r squared. So let's just rewrite this information. And now let's go ahead and let's graph it. Most of the equation, or the circle, will be on the right side. Actually, the left side.
Since it's negative 4, positive 3. So let's plot the center. Negative 4, 3 is right there in that position. And then we need to go up 2 units from the center.
So that will take us to the point negative 4, 5. And then down 2 units. And then 2 to the right. And 2 to the left. And so let's go ahead and graph it. My circles are not the best, but that's pretty much what it is.
So now you know how to graph a circle if you're given the equation in non-standard form. Let's say if you're given the center of the circle, let's say it's 2, negative 5, and you're also given the radius of the circle, which is 3. What is the equation of the circle in standard form? So first, we need to realize that h is equal to 2. It's the x-coordinate of the center.
And k is negative 5. And here's the standard equation. It's x minus h squared plus y minus... k squared which is equal to r squared so H is 2 k is negative 5 so it's y minus negative 5 and R is 3 so this is going to be x minus 2 squared and then y minus negative 5 is the same as y plus 5 and 3 squared is 9. So that's how you can write the equation of the circle in standard form.
If you want to write it in non-standard form, you can simply FOIL x minus 2 squared and FOIL y plus 5 squared. Combine like terms and adjust the equation however you want to. But that's it.
That's all you need to do to write the equation in standard form. Let's try another example. Let's say the center is and the radius of the circle is 6. Go ahead and write the equation in standard form. So we can see that H is negative.
3, k is positive 7. So let's start with this equation, x minus h squared plus y minus k squared is equal to r squared. So h is negative 3. k is positive 7 and r is 6. So this is going to be x plus 3 squared plus y minus 7 squared, which is equal to 36. And so that's the equation of a circle in standard form.