Welcome Accelerated Algebra 2! U7H3: Worksheet 10.3 #15-22, 24-25, 27-30, 33 Complete on graph paper Updates: U7Q1 will be March 23 rd U7T will be April 3 rd
Agenda (1) Warm-Up! (2) Review U7H1 + U7H2 (3) Review Circles (4) (5) Closure
Warm-Up! (1) Write the equation of a circle when the center is at (4, -2) and the radius is 12. (2) Write the equation of a line that is tangent to x 2 + y 2 = 225 at (9, -12).
Review U7H1 U7H1: Worksheet 10.1 #11-13, 32-35, 54-55 10.2 #3-7odd, 10-17
Conic Sections I will leave one page blank for supplemental notes. Today s Lesson #: ---- Topic: LO#3: Write and graph the equation of an ellipse.
Demonstration of how a circle becomes an ellipse. (Geogebra)
Ellipse o If you pulled the center of a circle apart into two points, it would stretch the circle into an ellipse. These two points are called the focal points or foci (represented by the letter c). - The distance from one foci to a point on the ellipse to the second foci will be constant throughout the entire ellipse. Why are the foci important? (Demonstration using cardboard, nails, string, and pencil)
Ellipse Continued o Ellipse has two axes a major and minor axis. Major axis passes through both foci and is the longer axis. The endpoints of the major axis are the vertices of the ellipse. Minor axis is the shorter axis. The endpoints on the minor axis are the co-vertices of the ellipse. The major and minor axis are perpendicular and intersect at the center of the ellipse.
a is the distance from the center to the vertex. b is the distance from the center to the co-vertex. c is the distance from the center to the focus.
Whiteboards Identify the major axis; is it horizontal or vertical? What are the vertices? Identify the minor axis; is it horizontal or vertical? What are the co-vertices.
Whiteboards Identify the major axis; is it horizontal or vertical? What are the vertices? Identify the minor axis; is it horizontal or vertical? What are the co-vertices.
The a, b, and c are all related in one formula. We are going to derive this formula!
Notice how the fraction with a will tell you whether the ellipse is vertical (y) or horizontal (x).
Example #1 (a) Write an equation in standard form for the ellipse with center (0, 0), vertex at (6, 0), and co-vertex at (0, 4). Then graph. (b) Write an equation in standard form for the ellipse with center (0, 0), co-vertex at (5, 0), and vertex at (0, -9). Then graph.
Whiteboards Write an equation in standard form for each ellipse with center (0, 0).
Whiteboards Write an equation in standard form for each ellipse with center (0, 0).
Example #2 (a) Graph the ellipse: (b) Graph the ellipse:
Whiteboards!
Example #3 An ellipse has its center at the origin. A vertex is at (-13, 0) and a co-vertex is at (0, 5). What are the coordinates of the focus? Write the equation of the ellipse.
Example #4 (a) Write an equation in standard for the ellipse with the center at the origin, vertex at (5, 0), and focus at (-2, 0). What is the length of the minor axis? (b) Write an equation in standard for the ellipse with the center at the origin, co-vertex at (-3, 0), and focus at (0, 1). What is the length of the major axis?
Example #5: Find the center, vertices, co-vertices, foci, domain, and range of the ellipse. (a) (b)
Whiteboards! 1. Write an equation in standard form for each ellipse with center (5, 3). (a) Vertex (5, 7); Co-Vertex (3, 3) (b) Vertex (9, 3); Focus (7, 3)
Whiteboards! 2. A city park in the form of an ellipse with equation x 2 y 2 50 + 20 = 1 measured in meters, is being renovated. The new park will have a length and width double that of the original park. (a) Write the equation of the new park. (b) Find the dimensions of the new park.
Exit Ticket Describe what all the variables in the following ellipse equation represent in your own words.
Exit Ticket 1) Write an equation of the ellipse with center (-3, 1). The ellipse has a vertex at (-3, 5) and a co-vertex at (-1, 1). 2) Identify the vertices and co-vertices from the given equation.