Chapter 9 Notes

1. Chapter 9 Notes Honors Pre-Calculus

2. 9.1 Intro to Conics A Circle is the set of all points, in a plane, such that each point is equidistant from a given point, the center. EQUATION: A Parabola is the set of all points, in a plane, that are equidistant from a given point, the focus, and a given line, the directrix. EQUATIONS:

3. 9.1 cont’d. Opens up/down Opens left/right Vertex : (h,k) Axis of Symmetry : x = h y = k Focus: (h, y+p) (h+p, k) Directrix: y = k – p x = h – p p is the distance from the vertex to the focus or vertex to the directrix. (p is directional) PICTURES

4. 9.1 Examples Find the Standard Form Equation for each Parabola: 1. V (2, 3) Focus (2, 5) 2. V (-1, 4) D: x = 1 Find Everything: Equation: Vertex: Focus: Directrix:

5. 9.2 Ellipses An Ellipse is the set of all points, in a plane, such that the sum of each points’ distances from two fixed points, the foci, is constant. Equations: Center : (h, k) Horizontal -Major Axis- Vertical Vertical -Minor Axis- Horizontal C is the distance from the Center to the Foci. Length of Major Axis – 2a Length of Minor Axis – 2b

6. 9.3 Hyperbolas A Hyperbola is the set of all points, in a plane, such that the difference of each points’ two distances from two fixed points, the foci, is constant. Equations: Center: (h,k) Right/Left OPENSUp/Down a is the distance from Center to vertices. “box” method for graphing, asymptotes. 2 “parabolas” Slopes of asymptotes:

7. 9.5 Parametric Equations If f and g are continuous functions of t on an interval, I. The set of ordered pairs (f(t), g(t)) is a plane curve, C. Parametric Equations t is the parameter Equations: Plot points in order of increasing values of t, you trace the curve in a specific direction. Orientation of the curve. Examples: Eliminating the Parameter 1.Solve for t in one eq. 2.Substitute into 2nd eq. 3.Rectangular Equation

8. 9.5 cont’d. Finding Parametric Equations: Answers:

9. 9.6/9.7 Polar Coordinates& Polar Graphs Directed distance Directed angle r Plot points and give three equivalent points. Polar Axis Coordinate Conversions Rect. To Polar r y x Polar to Rect, P to R R to P Convert:

10. 9.6/9.7 cont’d. GRAPHS: Rose Curves Circle Line n is odd, n petals n is even, 2n petals a is the length of petals Circles Equation Conversion Need to know when sin and cos are (1 or -1). For sin, # petals = # of values. For cos, # petals need 1 more value.