Concept Category 14 Conics in the Rectangular Coordinate

Concept Category #14 Conics in the Rectangular Coordinate System 6 A I can derive the equations of circles, parabolas, ellipses and hyperbolas given the general equation in the second degree. I can find the key characteristics for each conic (i. e. , focus, directrix, vertices, co-vertices, foci, asymptotes, minor axis, major axes). Foundational Skill: I can recognize and graph conic functions, including ellipses and hyperbolas.

Brainteaser: A radio tower services a 10 mile radius. You stop your car 5 miles east and 9 miles north of the tower. Will you be able to receive radio waves from the tower?

Task Find the equation of the circle centered at the origin(in terms of x and y) that passes through the point (7, 24).

I. Classifying Conics & Analyzing Graphs A. Definition Conic Sections: The curves formed when a plane intersects the surface of a right cylindrical cone.

B. Visual

A. Definition Conics can be classified by computing the discriminant B 2 – 4 AC, of equations of the form Ax 2 + Bxy + Cy 2 + Dx + Ey + F = 0 B = 0 for the conics we will be working with

Example Identify each of the following conics:

Example Identify each of the following conics:

A. Definition • Parabola is the set of all points in a plane whose distance from a fixed point is equal to its distance from a fixed line. Directrix Focus

B. Visual

Recall Parabolas? Standard form for the equation of a parabola: Vertex: (h, k) Axis of symmetry: x = h focus: (h, k + p) directrix: y = – p+k where p = 1/4 a Vertex: (h, k) Axis of symmetry: y = k focus: (h + p, k) directrix: x = – p+h where p = 1/4 a

C. Process Sketch the graph of y 2 + 2 y + 8 x + 17 = 0. Specify its vertex, focus, directrix and axis of symmetry. Determine the type of conic by using the discriminant

Standard form for the equation of a circle: center: (h, k) radius: r

Example What is the equation for the circle shown below?

Graph the following function.

Goal Problems: Determine choices for practice Recall & Reproduction Graph the following functions: 1. 2. 3. State the equation of a circle in general form which has a center at (5, -3) and a radius of 9. Routine Graph the following functions: 1. 2. 3. Write the center-radius equation of a circle that is tangent to the x-axis, with a center located at (4, -6).

Ellipses Standard Equation: OR

For Both Types: • Center (h, k) • a is the length of the semi-major axis (2 a is the length of the major axis) • b is the length of the semi-minor axis (2 b is the length of the minor axis) • c : vertices: (h + a, k) and (h – a, k) foci: (h + c, k) and (h – c, k) vertices: (h, k + a) and (h, k – a) foci: (h, k + c) and (h, k – c)

Hyperbolas: For Both: center: (h, k) OR vertices: (h – a, k) and (h + a, k) foci: (h – c, k) and (h + c, k) asymptotes: a a 0 vertices: (h , k – a) and (h, k + a) foci: (h, k – c) and (h, k + c) asymptotes:

Example: Graph the following function:

Example: Graph the following function:

Example: Graph the following function: 9 x 2 -72 x-16 y 2 -32 y-16=0

Determine the type of conic and justify your reasoning. Then sketch the graph

Goal Problems: Determine choices for practice Recall & Reproduction Routine Sketch the graph of -4 x 2+9 y 2 -48 x-72 y+144 = 0

Non-Routine 1. 2. Write an equation for the ellipse with vertices (4, 0) and (– 2, 0) and foci (3, 0) and (– 1, 0).

Hyperbolas: For Both: center: (h, k) OR vertices: (h – a, k) and (h + a, k) foci: (h – c, k) and (h + c, k) asymptotes: a a 0 vertices: (h , k – a) and (h, k + a) foci: (h, k – c) and (h, k + c) asymptotes:

Example: Graph the following function:

Example: Graph the following function:

Goal Problems: Determine choices for practice Recall & Reproduction Routine

Non-Routine

Active Practice: CCSS Recall & Reproduction Routine Non-Routine G-GPE-3. Read p 160 - 165, § 3. 3 p 165 G-GPE- #1 -40 3. 1 (CA) Read p 167 -172, § 3. 4 p 172 #3 -30, 33 -42 Read p 173 -179 § 3. 5 p 180 #9 -28 G-GPE-3. § 11. 1 p 646 #1 -8 G-GPE 3. 1 (CA) § 11. 2 p 659 #1 -12 § 11. 1 p 646 #9 -28 § 11. 2 p 660 #36 -40 § 11. 2 p 659 #13 -35 Modeling: § 11. 1 p 646 #29 -32 § 11. 2 p 660 #41, 42 Read p 173 -179 § 3. 5 p 179 #1 -4