Document Sample

7.2 Solving Quadratic Equations Algebra 2 Mrs. Spitz Spring 2007 Objectives Solve quadratic equations by graphing, Find the equation of the axis of symmetry and find the coordinates of the vertex of the graph of a quadratic function solve quadratic equations by factoring Assignment pp. 319-320 #6-45 odd Definition of a Quadratic Function A quadratic function is a function that can be described by an equation of the form y = ax2 + bx + c, where a ≠ 0. Generalities The path of an object when it is thrown or dropped is called the TRAJECTORY of the object. A bouncing object like the tennis ball in the photo will have a trajectory in a general shape called a parabola. Generalities Equations such as y = 6x – 0.5x2 and y = x2 – 4x +1 describe a type of function known as a quadratic function. Graphs of quadratic functions have common characteristics. For instance, they all have the general shape of a parabola. Generalities The table and graph can be used to illustrate other common characteristics of quadratic functions. Notice the matching values in the y-column of the table. 6 y = x2 – 4x + 1 x x2 – 4x + 1 y -1 (-1)2 – 4(1) + 1 6 4 0 (0)2 – 4(0) + 1 1 x=2 1 (1)2 – 4(1) + 1 -2 2 2 (2)2 – 4(2) + 1 -3 3 (3)2 – 4(3) + 1 -2 5 4 (4)2 – 4(4) + 1 1 -2 5 (5)2 – 4(5) + 1 6 (2, -3) -4 Generalities Notice in the y-column of the table, -3 does not have a matching value. Also notice that -3 is the y-coordinate of the lowest point of the graph. The point (2, -3) is the lowest point, or minimum point, of the graph of y = x2 – 4x + 1. 6 y = x2 – 4x + 1 x x2 – 4x + 1 y -1 (-1)2 – 4(1) + 1 6 4 0 (0)2 – 4(0) + 1 1 x=2 1 (1)2 – 4(1) + 1 -2 2 2 (2)2 – 4(2) + 1 -3 3 (3)2 – 4(3) + 1 -2 5 4 (4)2 – 4(4) + 1 1 -2 5 (5)2 – 4(5) + 1 6 (2, -3) -4 Maximum/minimum points For the graph of y = 6x – 0.5x2, the point (6, 18) is the highest point, or maximum point. The maximum point or minimum point of a parabola is also called the vertex of the parabola. The graph of a quadratic function will have a minimum point or a maximum, BUT NOT BOTH!!! Axis of Symmetry The vertical line containing the vertex of the parabola is also called the axis of symmetry for the graph. Thus, the equation of the axis of symmetry for the graph of y = x2 – 4x + 1 is x = 2 In general, the equation of the axis of symmetry for the graph of a quadratic function can be found by using the rule following. Equation of the Axis of Symmetry The equation of the axis of symmetry for the b graph of x y = ax2 + bx + c, where a ≠ 0, is 2a Ex. 1: Find the equation of the axis of symmetry and the coordinates of the vertex of the graph of y = x2 – x – 6. Then use the information to draw the graph. First, find the axis of NOTE: for symmetry. y = x2 – x – 6 b x a = 1 b = -1 c = -6 2a 1 x ( ) 2 1 1 x 2 Ex. 1: Find the equation of the axis of symmetry and the coordinates of the vertex of the graph of y = x2 – x – 6. Then use the information to draw the graph. 1 2 1 Next, find the vertex. y ( ) 6 Since the equation of 2 2 the axis of symmetry is x 1 1 = ½ , the x-coordinate of the vertex must be ½ . 6 You can find the y- 4 2 coordinate by 1 2 24 substituting ½ for x in y = x2 – x – 6 . 4 4 4 25 The point ( ½, -25/4) is the vertex of the graph. 4 This point is a minimum. Generalities The table and graph can be used to illustrate other common characteristics of quadratic functions. Notice the matching values in the y-column of the table. 2 y = x2 – x – 6 x x2 – x – 6 y -2 (-2)2 – (-2) – 6 0 5 -1 (-1)2 – (-1) – 6 -4 -2 x=½ 0 (0)2 – (0) – 6 -6 1 (1)2 – (1) – 6 -6 -4 2 (2)2 – (2) – 6 -4 3 (3)2 – (3) - 6 0 -6 This point is a minimum! -8 ½, -25/4) Solving Quadratic Equations Graphically SOLVING QUADRATIC EQUATIONS USING GRAPHS The solution of a quadratic equation in one variable x can be solved or checked graphically with the following steps: STEP 1 Write the equation in the form ax 2 + bx + c = 0. STEP 2 Write the related function y = ax 2 + bx + c. STEP 3 Sketch the graph of the function y = ax 2 + bx + c. The solutions, or roots, of ax 2 + bx + c = 0 are the x-intercepts. Checking a Solution Using a Graph 1 2 Solve x = 8 algebraically. Check your solution graphically. 2 SOLUTION 1 2 Write original equation. x = 8 2 x 2 = 16 Multiply each side by 2. x= 4 Find the square root of each side. CHECK Check these solutions using a graph. Checking a Solution Using a Graph CHECK Check these solutions using a graph. 1 Write the equation in the form ax 2 + bx + c = 0 1 2 x =8 Rewrite original equation. 2 1 2 x –8=0 Subtract 8 from both sides. 2 2 Write the related function y = ax2 + bx + c. y = 1 x2 – 8 2 Checking a Solution Using a Graph CHECK Check these solutions using a graph. 2 Write the related function – 4, 0 4, 0 y = ax2 + bx + c. y = 1 x2 – 8 2 1 3 Sketch graph of y = x2 – 8. 2 The x-intercepts are 4, which agrees with the algebraic solution. Solving an Equation Graphically Solve x 2 – x = 2 graphically. Check your solution algebraically. SOLUTION 1 Write the equation in the form ax 2 + bx + c = 0 x2 – x = 2 Write original equation. x2 – x – 2 = 0 Subtract 2 from each side. (x-2)(x+1)=0 Factor and set equal to zero. x–2=0 x+1=0 x=2 Solve. These are your x-intercepts. x = -1 2 Write the related function y = ax2 + bx + c. y = x2 – x – 2 Solving an Equation Graphically 2 Write the related function y = ax2 + bx + c. y = x2 – x – 2 – 1, 0 2, 0 3 Sketch the graph of the function y = x2 – x – 2 From the graph, the x-intercepts appear to be x = –1 and x = 2. Solving an Equation Graphically From the graph, the x-intercepts – 1, 0 2, 0 appear to be x = –1 and x = 2. CHECK You can check this by substitution. Check x = –1: Check x = 2: x2 – x = 2 x2 – x = 2 ? ? (–1) 2 – (–1) = 2 22 – 2 = 2 1+1=2 4–2=2

DOCUMENT INFO

Shared By:

Categories:

Tags:

Stats:

views: | 5 |

posted: | 2/7/2012 |

language: | |

pages: | 21 |

OTHER DOCS BY hx6s897

How are you planning on using Docstoc?
BUSINESS
PERSONAL

By registering with docstoc.com you agree to our
privacy policy and
terms of service, and to receive content and offer notifications.

Docstoc is the premier online destination to start and grow small businesses. It hosts the best quality and widest selection of professional documents (over 20 million) and resources including expert videos, articles and productivity tools to make every small business better.

Search or Browse for any specific document or resource you need for your business. Or explore our curated resources for Starting a Business, Growing a Business or for Professional Development.

Feel free to Contact Us with any questions you might have.