Chapter 10 Quadratic Relations

In this chapter you should … Learn to write and graph the equation of a circle. Learn to find the center and radius of a circle and use it to graph the circle .

Conics Definition: A conic section is the intersection of a plane and a cone. By changing the angle and location of intersection, we can produce a circle, ellipse, parabola or hyperbola; or in the special case when the plane touches the vertex: a point, line or 2 intersecting lines. Circle Ellipse Parabola Hyperbola

Graphing Conics x2 + y2 = 25 9x2 + 16y2 = 144 x2 - y2 = 9

What am I ?? x2 - 2y2 = 4 6x2 + 6y2 = 600 4x2 + 25y2 = 100

Parabolas

Parabolas Vertex form: y = a(x – h)2 + k where the vertex is (h, k). If a is negative, the parabola opens down. If a is positive, the parabola opens up. If is greater than 1, the parabola is tall & skinny. If is less than 1, the parabola is short & fat. If is equal to 1, the parabola is average. Notice for the vertex that the sign of “h” is opposite its sign in the parentheses.

Examples for parabolas y = -2 (x – 5)2 + 3 y = ¼ (x + 3)2 - 9 vertex: ( , ) opens: shape: vertex: ( , )

More examples for parabolas y = -x2 – 10 y = (x+5)2 vertex: ( , ) opens: shape: vertex: ( , )

Graphing Parabolas Remember that a vertical line, the “axis of symmetry”, goes through the vertex. Points on each side of this line are reflections: they will have the same y-coordinate. To graph a parabola, first graph the vertex and lightly sketch the axis of symmetry. Consider whether the parabola opens up or down. Choose an x just to the right or left of the vertex. Find the y-value and graph the point, then graph its reflection across the axis of symmetry. Repeat with another point.

Example of graphing Graph y = 2(x + 3)2 – 5 vertex: ( -3, -5) opens: upward axis of symmetry: x = -3 complete: (-2, _____) and reflection (___, ___) complete: (-1, _____) and reflection (___, ___)

Graphing Practice graph y = ½(x – 2)2 – 4 graph y = -2(x – 5)2

Writing in Vertex Form An equation of a parabola in quadratic form: y = ax2 + bx + c How will you graph it if it’s not in vertex form? Find the vertex the way you did in Chapter 5! The x-coordinate of the vertex is on the axis of symmetry: Find the value of (x, y) at the vertex. This is now (h, k). Get the value of “a” from the x2 term. Now write the equation in the form y = a( x – h)2 + k.

Practice with vertex form Write y = 2x2 + 20x + 53 in vertex form. Find the vertex, axis of symmetry, direction of opening; graph. vertex: x = = -5. y = 2(-5)2 + 20(-5) + 53 = 3. vertex: (-5, 3); axis of symmetry: x = -5. a = 2 (coefficient of x2); parabola opens upward. vertex form of equation: y = 2(x + 5)2 + 3

Practice vertex form with these y = 2x2 – 12x + 9 y = 1/2x2 + 2x – 8 y = -x2 + 2x + 1 y = -6x2 – 12x - 1 Answers y = 2(x – 3)2 – 9 y = ½(x + 2)2 – 10 y = -(x – 1)2 + 2 y = -6( x + 1)2 + 5

Graphing Now, can you graph those four problems? Remember the steps: axis of symmetry (h) vertex (h, k) direction of opening (a) a couple of points & their reflections

Solving Quadratic Systems Method 1: Solve the system algebraically: Subtract the like terms to eliminate the x2 terms Solve for y. Substitute your answer to get the value of x. You might get more than one y; if so, you also get more than one x. Your answer would have two (x, y) solutions.

Method 2: Solve the system graphically Re-write each equation in terms of y = .... Remember the there are two square roots of y2: and . Graph each of the equations and find their points of intersection, using the graphing calculator.

10-3 Circles What you’ll learn … Learn to write and graph the equation of a circle. Learn to find the center and radius of a circle and use it to graph the circle 2.09 Use the equations of parabolas and circles to model and solve problems. a) Solve using tables, graphs, and algebraic properties. b) Interpret the constants and coefficients in the context of the problem.

A circle is the set of all points in a plane that are distance r from a given point, called the center. The distance r is called the radius. If r is the radius of a circle with a center at the origin, then the equation of the circle can be written in the form x2  +  y2  =   R2

Memorize this!!!  The Standard Form of the Equation of a Circle                                         (h, k) is the center r is the radius (x, y) is any point on the circle                                                                                                

Example 1 Writing an Equation of a Circle Write an equation of a circle with center (-4,3) and radius 4. Write an equation of a circle with center (5,-2) and radius 8.

Example 2a Using Translations to Write an Equation Write an equation for the translation of x2 + y2 =9 four units left and three units up. Then graph the translation.

Example 2a Using Translations to Write an Equation Write an equation for the translation of x2 + y2 =1 nine units right and two units down. Then graph the translation.

Example 4 Finding the Center and Radius Find the center and radius of the circle with equation (x – 16)2 + (y +9)2 = 144. Find the center and radius of the circle with equation (x + 8)2 + (y +3)2 = 121.

Example 4b Finding the Center and Radius Find the center and radius of the circle with equation x2 + y2 – 8x - 4y + 19 = 0. Find the center and radius of the circle with equation x2 + y2 + 6y - 27 = 0.

Example 4c Finding the Center and Radius Find the center and radius of the circle with equation x2 – 10x + y2 + 4y - 7 = 0. Find the center and radius of the circle with equation x2 + 2x + y2 - 10y - 38 = 0.

Example 5 Graphing a Circle using Center and Radius Graph (x + 1)2 + (y – 3)2 = 25 Graph (x - 4)2 + (y + 2)2 = 49

In this chapter you should have … Learned to write and graph the equation of a circle. Learned to find the center and radius of a circle and use it to graph the circle .