Back To CourseMath 103: Precalculus
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Tyler has tutored math at two universities and has a master's degree in engineering.
When we first learn how to make graphs in algebra, people usually start off by learning how to graph points. Next, we learn lines.
Then, we learn how to change exponents in the equation to get curves.
If you look carefully at the images above, you'll see that each time, the y variable didn't change. It just sat on the left side of the equation, all by itself. Notice below how curves begin to form, but they only curve up and down.
Since we made curves by changing the x variable's exponent, what would happen if we changed the y variable, too? If we raise the y variable to the second power, we get curvature in a different direction. In special instances where both x and y are squared, we get the equation of a circle.
Circles are very special shapes, and their graphs are just as special. Where other graphs have many properties, circles are made up of two things: a center and radius. If we know the center and radius, we can draw a circle on the graph and write its formula.
There is a special equation we use for circles. The formula is (x-h)^2 + (y-k)^2 = r^2. A circle is very basic shape but has a complicated formula. To find out if an equation is that of a circle, there are four important things to remember. The first is that the x and y terms are squared. The second is all terms in the equation are positive. The third is the center point of the circle is (h,k). Finally, r represents the radius of the circle.
h and k are the coordinates of the center, but they won't always be given in order. An easy way to remember is by looking at each section of the equation. In the first section, we have x and h in the same set of parentheses. Since they are together, you can remember that h is the x value of the center point. In the second section, we have y and k in the same set. Since they are together, we know that k is the y value of the center point. By knowing this formula, you can quickly see the coordinates for the center point of a circle.
It's pretty easy to remember that r stands for radius. The tricky part is that you must remember r is squared in the equation. Sometimes, it will just be written as a whole number. In that case, to find the radius you must take the square root. Remember, all three sections of the equation should be squared, even if the radius is shown as a whole number.
With any equation, it's important we know how to draw it on a graph. You may have drawn circles since you were a child, but circles on a graph have to be located in just the right spot. The center point is the most important point, so it's important to mark it first. There's a catch - it isn't actually part of the graph. Mark the center point lightly, because it is just a reference point. For the equation (x-2)^2 + (y-1)^2 = 5^2, where is the center? h=2 and k=1. These are coordinates of the center point (2,1). Remember, if h and k are confusing, the number with x is the x coordinate. The number in parentheses with y is the y coordinate of the center.
The other important part of this graph is the radius. In this case, the radius r = 5. To draw this graph, we will start at the center point and use the radius to mark points up, down, left, and right. In this case, we start at the point (2,1) and move up 5 units. Mark that point. Go back to the center and move down 5 units. Do the same by starting at the center and going left and right 5 units. Use these 4 points, seen below, as a guide as you draw your circle.
When you see a circle on the graph, the two key things to know are still the center and the radius. In the circle above, let's find the center point. Remember, the center point isn't actually on the graph, so we will mark it lightly and erase it when we are finished. The center of the circle will be halfway between the top and bottom of the graph. It will also be halfway between the left and right sides. We find the center by calculating the midpoint between each set of points. The midpoint is the point half the distance between two points. To find the midpoint, we take the x values, add them, and divide by 2. We do the same with the y values. This gives us the midpoint.
In this case, we want to take the midpoint of the top and bottom point of the circle. The top point of the circle is (-2,4). The bottom point is at (-2,-2). We find the midpoint by adding the x values and dividing by 2. -2+-2=-4, -4/2 = -2. Now do the same for the y values. 4+-2=2, 2/2 = 1. The midpoint is (-2,1) Just to make sure this is a perfect circle, we need to find the midpoint between the left and right points. Finding the midpoint of the x values gives us -5 + 1 = -4, -4/2 = -2. The midpoint of the y values gives us 1 + 1 = 2; 2/2 = 1. The midpoint for left and right is the same. The center of the circle is (-2,1), where h=-2 and k=1.
To find the radius of this circle, we can start at the center and count how many units it takes for us to get to any point on the circle. If we start at the center and go up, we move three units to get to the top of the circle. The radius of this circle is r=3.
We take the center and the radius and put them into our circle formula. The formula is (x-h)^2 + (y-k)^2 = r^2. Substitute in h=-2, k=1, and r=3. This gives us (x- (-2) )^2 + (y-1)^2 = 3^2. Simplify the two, and square the three, and we get our equation (x+2)^2 + (y-1)^2 = 9. Notice that the x and y are squared, and they are positive. This is the equation of the circle.
Even though x and y are both squared, a circle is nothing to be afraid of. All we need to know are the center and the radius. If we know these two things, we can graph a circle or write its formula. The center point gives us h and k, and the radius gives us r. We can either substitute this into the equation or put it right on the graph. Shapes can help us learn, even in math.
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Back To CourseMath 103: Precalculus
11 chapters | 81 lessons