6.1 Solving Quadratic Equations by Graphing Algebra 2

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1 10.1 Solving Quadratic Equations b Graphing Algebra Goal 1: Write functions in quadratic form Goal : Graph quadratic functions Goal 3: Solve quadratic equations b graphing. Quadratic Function: Eample 1: Write f ( ) 3 in quadratic form. Identif the quadratic term, the linear term, and constant. The graph of a quadratic function is called a. All have an that occurs at the or of the. The is the. Eample : An arrow is shot upward with an initial velocit of feet per second. The height of the arrow h(t), in terms of the time since the arrow was released t, is h( t) t 1t. How long after the arrow is released does it reach its maimum height? What is that height?

2 10 When a quadratic function is set equal to zero, the result is a. A is an equation that can be written in the form. A contains onl variable with. The, or, of a quadratic equation are values of the variable that satisf the equation. Eample 3: Solve 8 0 b graphing. Name the verte and ais of smmetr Eample : Solve m m 1 0 b graphing. Name the verte and ais of smmetr

3 . Solving Quadratic Equations b Factoring Algebra Goal 1: Solve problems b using the guess-and-check strateg. Goal : Solve quadratic equations b factoring. Eample 1 Application: On page 31, read the Application: Tennis paragraph. How long does it take for the ball to hit the ground? h t s0 v0t 1t h t 8t 1t h t 1t v t s 0 0 Zero Product Rule: Eample Application: An arrow is shot upward with an initial velocit of 50 feet per second. The height of the arrow h(t), in terms of the time since the arrow was released t, is h( t) 50t 1t. How long after the arrow is released does it reach its maimum height? What is that height? When does it hit the ground? Eample 3: Solve b factoring. Eample : Solve m 3m 8 0 b factoring.

4 Eample 5: Solve 10 5 b factoring. Eample : Solve 3a 5 ab factoring. Eample 7: Solve h 11 hb factoring. Eample 8: Solve 3 5h 1 hb factoring. Eample 9: Solve a 3 ab factoring.

5 .3 Solving Quadratic Equations b Completing the Square Algebra Goal 1: Solve quadratic equations b completing the square Eample 1: Find the value of c that makes a perfect square. A. c B. 0 c C. 15 c Leading Coefficient of 1 Eample : Solve b completing the square. A. 0 B. a 8a 9 0 C. a 11a 0

6 Leading Coefficient Not Equal to 1 and Imaginar Zeros. Eample 3: Solve b completing the square. A B C D E. g 7g 5 F. 15 0

7 . The Quadratic Formula and the Discriminant Algebra The Golden Gate Bridge Goal 1: Solve quadratic equations b using the quadratic formula. Goal : Use discriminants to determine the nature of the roots of quadratic equations. The Quadratic Formula: The solutions of a quadratic equation of the form given b the following formula. 0, where a 0, are a b c b b ac a Eample 1: Solve each equation b using the quadratic formula. A. 10 B. 30 C. 3 0 D. a 7a 1 0 E. 3 F. 5m 7m 3

8 Eample : A car traveling at meters per second (m/s) and accelerating at the driver brings the car to a complete stop. The equation m / s. After traveling m, t t, where t is the time it takes to stop, can be used to represent this situation. How long did it take the driver to stop the car? The Discriminant: The discriminant is the epression b b b ac ac from the quadratic formula. a The discriminant determines the nature of the roots of a quadratic equation. The table below summarizes all the possibilities. Eample 1 b b 3 b b Value of Discriminant a b ac Perfect Square? ac 0 Yes ac 0 No ac 0 X ac 0 X Nature of Related Nature of Root(s) Graph Eample 3: Find the value of the discriminant for each quadratic equation. Then describe the nature of the roots. A B. 5 0 C D. 9 8

9 . Analzing Graphs of Quadratic Functions Algebra Goal 1: To analze graphs of quadratic functions. Goal : To analze quadratic functions to determine their transformations from the parent graph. Use our graphing calculator to complete the following activit. 1. Hit Window and enter the window below. Xmin = - 9. Xma = 9. Xscl = 1 Ymin =. Yma =. Yscl = 1 Xres = Enter into the Y= portion of our calculator. This graph is called the parent graph for all quadratic equations. All quadratic functions are based on this single function and its graph. Compare ever graph from here on to this graph. Leave this graph on our screen at all times throughout the activit. Section One: Graph each of the following and compare it to the parent graph above. Describe how the new graph changed from the parent graph Graph:. How did the graph change from the parent graph? How do ou think the graph of 1will change (move) from the parent graph? 7. Graph it in our calculator to determine if our conjecture was true or false? True or False Circle One This is called a. Answer this during class discussion.

10 Section Two: Graph each of the following and compare it to the parent graph above. Describe how the new graph changed from the parent graph Graph:. How did the graph change from the parent graph? How do ou think the graph of 3 will change (move) from the parent graph? 1. Graph it in our calculator to determine if our conjecture was true or false? True or False Circle One This is called a. Answer this during class discussion. Section Three: Graph each of the following and compare it to the parent graph above. Describe how the new graph changed from the parent graph Graph:. How did the graph change from the parent graph? How do ou think the graph of will change from the parent graph? This is called a. Answer this during class discussion How do ou think the graph of will change from the parent graph? 3 This is called a. Answer this during class discussion.

11 . Analzing Graphs of Quadratic Functions - Da Algebra Goal: To write a quadratic equation given three points. Eample 1: Write an equation for the parabola. 5 Eample : Write an equation for the parabola. 5

12 Eample 3: Write an equation of the parabola that passes through the points at 9, 3,,3,,7. Eample : Write an equation of the parabola that passes through the points at 0, 3, 1,,,15.

13 .7 Graphing and Solving Quadratic Inequalities Algebra Goal 1: Graph quadratic inequalities. Goal : Solve quadratic inequalities in one variable. > or > shade the parabola. < or < shade the parabola. < and > is a boundar. < and > is a boundar. Eample 1: Graph Eample : Graph. 5 5 Eample 3: Solve

14 Eample : Solve Eample 5: Solve

15 .8 Standard Deviation Algebra Goal 1: Calculate the standard deviation for a set of data. Eample 1: A stud about the effects of different fertilizers on tree growth is being conducted. The trees planted last ear are now 59 cm, 59 cm, 5 cm, 5 cm, 0 cm, 58 cm, 50 cm, and cm tall. Calculate the mean height of the trees and find the standard deviation of the data. Mean X : Standard Deviation X : Eample : The average prices received b U.S. farmers for a dozen eggs during the ears 198 through 199 are as follows: 7, 57, 1, 55, 53, 9, 71, 8, and 58. Find the mean and standard deviation. Mean X : Standard Deviation X :

16 .9 Normal Distribution Algebra Goal 1: Solve problems involving normall distributed data. Normal Distribution: Normal Distribution Curve

17 Eample 1: Determine whether the following data appear to be positivel skewed, negativel skewed, or normall distributed Eample : Determine whether the following data appear to be positivel skewed, negativel skewed, or normall distributed