Circles, Parabolas, Ellipses, and Hyperbolas. Table of Contents. Title Slide. Table of Contents. Equations of the Curves centered at the origin (0,0). Equations for Curves not centered at the origin. Circles. Parabolas. Ellipse. Hyperbolas. Example 1 Circle.
Circles, Parabolas, Ellipses, and Hyperbolas
Table of Contents
Equations of the Curves centered at the origin (0,0)
Equations for Curves not centered at the origin
Example 1 Circle
How to Solve Example 1 Circle
Example 2 Parabola
Solution to Example 2 Parabola
Example 3 Ellipse
How to Solve Example 3 Ellipse
Example 4 Hyperbola
How to Solve Example 4 Hyperbola
Centered at (h, k)
or centered at (h, k)
Circles are a special type of ellipses. There is a center that is the same distance from every point on the diameter. In the equation the center is at (h, k). The distance from the center to any point on the line is called the radius of the circle. From the equation to find the radius you take the square root of r2.
A parabola is a curve that is oriented either up, down, left, or right. The vertex of the parabola is at (h, k). In the equation the h value added or subtracted to x moves the parabola left and right. If you subtract the value of h the parabola moves to the right. If you add the value of h the parabola moves to the left. The parabola can be made skinnier and wider by changing the value of a in the equation. If a is a whole number the parabola will become skinnier; if it’s a fraction the parabola will become wider. When you add or subtract a value of k the parabola moves up or down by the value. Parabolas are symmetrical across the line through the vertex of the parabola.
The center of the ellipse is at (h, k). The radius of ellipses are not a constant distance from the center. To find the distance to the curve from the center you have to find the distance from the center to the curve for the x and y separately, these points are called vertices. The vertices are on the major axis and minor axis. The major axis is the longer axis and the minor axis is the shorter axis through the center of the ellipse. To find the distance from the center in the x direction you take the square root of a2. To find the distance from the center in the y direction you take the square root of b2. You then will have two points on the x direction and two points in the y direction and you use these four points to draw your ellipse. Ellipses are symmetrical across both of there axis's.
Hyperbolas look like two parabolas opening in opposite directions. The equations of the asymptotes are y= k + (b/a)(x-h). The asymptotes help you to graph the hyperbola. The center of the hyperbola is also at (h, k). The vertices of the hyperbola depend on whether the hyperbolas open left and right or up and down. You can determine which way the hyperbola opens by looking to see if the x or y term has a negative sign. In the equation in the upper left corner the y term has the negative and since the y has the negative the hyperbolas open left and right. When opening left and right the vertices are (h+a, k). For the equation in the upper right corner the x value has the negative sign which means that the hyperbola opens up and down. The vertices for an equation that opens up and down are (h, k+b).
4.Next you plot the four points that you just found.
5.Then connect the four points around the center to create your circle.
6. Next we pick at least two more values of x
to find values of y’s for to be able to graph
the parabola accurately. Lets use x = -4 and
So for x = -4, y = 2(-4+2)2+1 = 2(-2)2+1= 9
and for x = 0, y = 2(0+2)2+1= 9.
So we have the points (-4,9) and (0,9).
7. Now we can graph the center of the
parabola (-2,1) and the points that we have
found to lie on the parabola (-1,3), (-3,3),
(-4,9), and (0,9).
8. Next we can connect the pints and continue
the ends of the curve up to create the
6. To graph the hyperbolas we need to graph our asymptotes.
7. Now we need to plot the two vertices.
8. We can now draw in our hyperbola by drawing two parabola looking curves through the vertices that get close to the asymptotes but never touch the asymptotes and so that the parabola shaped curves open left and right.
Click on the correct answer to move to the next question.
What type of object/curve is produced by the equation below curve? Also find the center from the equation.
A. Circle (6,4)
B. Ellipse (6,2)
C. Ellipse (6,4)
D. Hyperbola (6,4)
E. Circle (6,2)
Click on the correct answer to continue.
What is the center of this hyperbola? What are the equations for the asymptotes from this hyperbolic equation?
A. Center (7,9) Asymptotes 9+(36/25)(x-7) and 9-(36/25)(x-7)
B. Center (6,5) Asymptotes 5+(36/25)(x-6) and 5-(36/25)(x-6)
C. Center (5,6) Asymptotes 25+(7/5)(x-36) and 25-(7/5)(x-36)
D. Center (7,9) Asymptotes 9+(6/5)(x-7) and 9-(6/5)(x-7)
Click on the correct answer to move to the next problem.
What type of object/ curve is given by the equation below? What is the center of the equation?
A. Circle Center (3,2)
B. Ellipse Center (2,10)
C. Parabola Center (2,10)
D. Parabola Center (3,2)
Click the correct answer to continue.
What is the center and the radius of the following circle equation?
A. Center (1,10) Radius = 10
B. Center (10,1) Radius = 1
C. Center (0,1) Radius = 10
D. Center (1,0) Radius = 10
Print and complete the following worksheets.
Writing Equations from Graphs
Equations of Curves
Barnett, Raymond A., Michael R. Ziegler, and Karl E. Byleen. PreCalculus graphs and models. Boston: McGraw Hill, 2005.
Calvert, J. B. “Ellipse”. 2005. 3 Dec.2009 http://mysite.du.edu/~jcalvert/math/ ellipse.htm
Dawkins, Paul. “Algebra.” 2009. 3 Dec. 2009 <http://tutorial .math.lamar.edu/classes/alg/hyperbolas.aspx>.
Picture from google images: <http://www.stickergiant.com/Merchant2/imgs/450/ss35_450.jpeg>