Pre-Calculus Honors

Name: ______

Period # ______

Unit 10 Lesson 2 Do Now

Find an equation in standard form of a parabola that satisfies the given conditions. Focus (3, 4) and directrix x = 5. Sketch a graph of the parabola labeling 3 coordinate points.

Name: ______

Period # ______

Unit 10 Lesson 2 Do Now

Find an equation in standard form of a parabola that satisfies the given conditions. Focus (3, 4) and directrix x = 5. Sketch a graph of the parabola labeling 3 coordinate points.

Pre-Calculus Honors

Book Reference 8.1

Unit 10 Lesson 2: Conic Sections and Ellipses

1. Do Now: Read through and mark up the following text and table.

An ellipse is the set of all points in the plane whose distances from two fixed points in the plane have a constant sum. The fixed points are the foci of the ellipse.

The line through the foci is called the focal axis. The chord lying on the focal axis is the major axis of the ellipse. The chord through the center perpendicular to the focal axis is the minor axis of the ellipse. The length of the major axis is 2a and the length of the minor axis is 2b.

The center of the ellipse is midway between the foci.

The points where the ellipse intersects the major and minor axis are the vertices of the ellipse.

Ellipse with center (h, k)
Standard Equation / /
Focal axis / y = k / x = h
Foci / /
Vertices / /
Semi major axis / a / a
Semi minor axis / b / b
Pythagorean Relation / /

2. Group Practice: Label the vertices, center, semi major and minor axis, and foci on the following diagram.

/ Center: ______
Vertices: ______
Semi-major axis: ______
Semi-minor axis: ______
Foci: ______
Focal axis: ______
Algebraic Model: ______

3. Guided Practice: Representing an Ellipse Verbally, Numerically, Algebraically, and Graphically.

Verbal / Graphical
Find an equation in standard form for the ellipse that satisfies the given conditions. The foci are (-2, 1) and (-2, 5); the major axis is 8. / Directions: Label the vertices and center of the ellipse. Draw and label your major and minor axis.

Analytical Properties of the Ellipse / Algebraic
Answer the following questions about the ellipse.
1. Is the focal axis horizontal or vertical? How can you determine the focal axis using the verbal?
2. What is the center of the ellipse? How can you determine this using the verbal?
3. What is the major and minor axis of the ellipse? How can you determine this using the verbal? / Directions: Write a function of an ellipse, in standard form described in the verbal.
______

(Long Block)

Verbal / Graphical
Find an equation in standard form for the parabola that satisfies the given conditions
Focus (-5, 3); Directrix y = 9 / Directions: Label two other points on the parabola. Use your focal width, focus, and axis. Draw your directrix and focus on your graph.

Analytical Properties of Parabola / Algebraic
Answer the following questions about the parabola.
1. Which way does the parabola open?
______
2. What is the vertex of the parabola?
______
3. What is the axis of the parabola?
______
4. What is the focal length and focal width of the parabola? Label what the focal length and focal width means graphically on the graph above.
______
______/ Directions: Write a function in standard form that represents the parabola in the verbal.
______

Unit 10 Lesson 2 Problem Set

The Ellipse

1. Write the standard form equation of an ellipse that has foci (±2,0) and a major axis length of 10.

2. Write the standard form equation of an ellipse that has major axis endpoints (1, -4) and (1, 8) and a minor axis length of 8.

3. Write the standard form equation of an ellipse where the foci are (1, -4) and (5, -4) and the major axis endpoints are (0, -4) and (6, -4).

4. Find the center, vertices, and foci of the following ellipse:

5. Find the standard form of the equation of an ellipse with a = 2c and vertices of (-4, 4) and (4, 4).

6. A semielliptical arch over a tunnel for a road through a mountain has a major axis of 100 feet and a height at the center of 40 feet.

(a.) Draw a rectangular coordinate system on a sketch of the tunnel with the center of the road entering the tunnel at the origin. Identify the coordinates of the known points.

(b.) Find an equation of a semi elliptical arch over the tunnel.

(c.) Determine the height of the arch 5 feet away from the edge of the tunnel.

7(a) Solve the system of equations and graphically by finding the intersection point on the coordinate plane below.

7(b) Check you solution by solving the system algebraically using your method of choice.

Answer Key

1. Write the standard form equation of an ellipse that has foci (±2,0) and a major axis length of 10.

2. Write the standard form equation of an ellipse that has major axis endpoints (1, -4) and (1, 8) and a minor axis length of 8.

3. Write the standard form equation of an ellipse where the foci are (1, -4) and (5, -4) and the major axis endpoints are (0, -4) and (6, -4).

4. Find the center, vertices, and foci of the following ellipse:

Center: (7, -3)

Vertices: (-1, -3) (15, -3)

Foci: