(b) Equation for a parabola: c) Direction of Opening (1) If a is positive, it opens (2) If a is negative, it opens

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1 Section.1 Graphing Quadratics Objectives: 1. Graph Quadratic Functions. Find the ais of symmetry and coordinates of the verte of a parabola.. Model data using a quadratic function. y = 5 I. Think and Discuss A. Quadratic Functions 1. Form a) Quadratic term b) Linear Term c) Constant Term. Graph of a quadratic a) Symmetrical (1) Def () Ais of Symmetry (a) Def b) Verte (1) () () (b) Equation for a parabola: c) Direction of Opening (1) If a is positive, it opens () If a is negative, it opens B. Which points are on the graph of the function y = 5? 1. (1, -7). (, 0). (0, -5). (-, 11) C. What do you think the greatest eponent is for a quadratic function? for a linear function? D. Try This 1. Tell whether each function is linear or quadratic? a). y = ( )( ) b). y = ( + ) c). y = ( + 5 ) d). y = ( 5). Find the ais of symmetry, verte, and determine if the verte is a maimum or minimum point for the following: y = Use you calculator to verify your answers to problem. Homework: p all, 59- odds Precalculus Chapter Page 1

2 Section. Solving Quadratic Equations by Graphing Objectives: 1. Solve quadratics by graphing I. Solving Quadratic Functions Graphically A. Solution (Root or Zero) 1. Algebraic Definition. Graphical definition (What would happen if the graph didn't cross the -ais or just touched it?) B. Solve the given graph: y = 5. C. Solve the following using your graphing calculator. 1. y = +. y = Homework: p all, 19, 7 Precalculus Chapter Page

3 Objective: 1. Solve polynomials Section. Solving polynomials by Factoring I. Work Together A. Solve for each variable in the following equations. = 0 y = 0 5y = 0 Why were these easy to solve? B. Solve for each variable in the following equations. ab = 1 5y = 75 Can I use the same method I did with the problems in section A above? Eplain C. Wrap Up II. Think and Discuss A. The Process Precalculus Chapter Page

4 B. Try the following = 0 = 0 ( )( ) = 0 ( )( ) = 0 ( ) = 0 or ( ) = 0 ( ) = 0 or ( ) = 0 = or = = or = + = 0 1 = 0 ( )( ) = 0 ( )( ) = 0 ( ) = 0 or ( ) = 0 ( ) = 0 or ( ) = 0 = or = = or = + = 0 1 = 0 ( )( ) = 0 ( )( ) = 0 = 0 or ( ) = 0 or ( ) = 0 = 0 or ( ) = 0 or ( ) = 0 = or = or = = or = or = C. What is wrong with the following problems? Find the error; eplain the error in this person's thought process; and correct the problem. 5 = 15 + = 5 5 ( 9)( + 5) = 15 + = ( 9) = 15 or ( + 5) = 15 + = 5 = or = + 5 = 0 ( 9)( + ) = 0 ( 9) = 0 or ( + ) = 0 = 9 or = Homework: p. 0, 15-1 odds Precalculus Chapter Page

5 Section.B Modeling Real World Data Modeling Data A. The table shows the average temperature in Gatlinburg, TN, for each month. Plot the points on a graph. Would it be useful to represent this data with a linear model? Eplain. Month Temp Feb() 5 Apr() 7 Jun() Aug() Oct() 71 Nov(11) 5 B. Finding Equations to model Quadratic Functions 1. Find an equation to model the data mentioned, using your graphing calculator. (Hint: It is done the same way you did the linear functions, ecept for one thing.). Use the equation you just found to predict the average temperature in September.. How close was it to the actual temp of 1? Homework: p all Precalculus Chapter Page 5

6 Section. Completing the Square Goals: 1. To solve quadratic equations by completing the square. Work Together FOIL These 1. ( ). ( + 5). ( 7) Making perfect squares 1. + = ( ). + + = ( + ). 0 + = ( ) Solve the following problems without setting equations equal to zero. 1. =. ( ) = 1. (1 + ) = = = Wrap Up Think and Discuss Completing the square is a method to solve quadratic equations when they do not factor. Method 1. Get constant on one side of the equation.. Factor out the coefficient in front of the.. Make the variable side into a perfect square (reminder what ever you add to one side must be done to the other).. Square root both sides and simplify. Eamples = = = = 0 Homework: p., odds, 5-1 odds, -7 odds, 55- all Precalculus Chapter Page

7 Section.5A The Quadratic Formula and Discriminant Goals:. To solve quadratic equations using the quadratic formula.. To use the discriminant to determine the nature of the roots of the quadratic equation. Work Together Solve using the Complete the Square Method a + b + c = 0 Can you write a formula to solve for in all quadratics? Wrap Up Think and Discuss II. The Quadratic Formula is a method to solve quadratic equations when they do not factor. A. Formula: If a b c + + = 0, then ( ) ± ( ) ( )( ) ( ) ( ) = where 0 B. Eamples = = = = 0 III. The Discriminant (Determines Nature of the Roots) A. Formula: If a + b + c = 0, then. B. Translation 1. If D > 0, then. If D = 0, then. If D < 0, then C. Eamples Determine the nature of the roots 1. = = 5 Homework: p. 17 1, 15-7 odds, 9-9 odds,, 7-0 all Precalculus Chapter Page 7

8 Section.5B Sum and Product of Roots Goals:. To find the sum and product of the roots of a quadratic equation. 5. To find all possible integral roots of a quadratic equation.. To find a quadratic equation to fit a given condition. Work Together Solve the following quadratics = = = 0 Find the sum and product of each problem s roots. 1. Sum =. Sum =. Sum = Product = Product = Product = Can you make a conclusion? (Do you see a pattern with the original quadratic and the sum and products?) Wrap Up Think and Discuss Sum and Product Theorem: Formula: If the roots of a + b + c = 0 are r 1 and r, then Eamples Find the quadratic given its roots. 1. Roots are and -. Roots are 5 and 1 *. One root is 5 + i *. One root is 1+ *Note: a + bi ( a + b c ) is a root iff a bi ( a b c ). Homework: worksheet Precalculus Chapter Page

9 Section. Analyzing Graphs of Quadratic Functions Goals: 1. To graph quadratic equations of the form y = a( h) + k and identify the verte, the ais of symmetry, and the direction of the opening.. To determine the equation of the parabola from given information about the graph. I. Terms A. Verte B. Ais of Symmetry C. Parent Graph: y = II. Dynamics of y = a( h) + k A. What does h do? 1. Graph: y = ( ). Graph: y = ( + ) B. What does k do? 1. Graph: y = +. Graph: y = C. What does a do? 1. Graph: y =. Graph: y = Re-graph these two equations but us a negative coefficient D. Wrap Up y = a( h) + k 1. h. k Verte: Ais of Symmetry: Precalculus Chapter Page 9

10 . a E. Eamples: 1. Name the verte, ais of symmetry and direction of opening. a) y = ( + 11) + b) y = ( ) +. Put the following quadratics into a) y = + + b) y = a( h) + k y = c) y = Graph the following: a) y = ( + ) 1 b) y = = ( 1) + c) y ( ) III. Find the equation of the parabola A. Eamples: (,1) (,-) (,-) - -. Parabola passes through the verte (5, ) and the point (, ). Homework: p. 5 1,, (15-)/, 9, 0, 7, 51, 5, 55- all, 7 Precalculus Chapter Page

11 Section.7 Graphing and Solving Inequalities Goals: 1. To graph quadratic inequalities.. To solve quadratic inequalities in one variable. I. Graphing Quadratic Inequalities A. Same linear functions. B. Eamples 1. y < ( ) 1 y II. Solving quadratic inequalities A. Graphically: 1. 0 > < B. Algebraically 1. 0 > < 0 Homework: p. 1,, 15, 19,5, -9 all, 1-1 odds,, 51-5 all, 59- all Precalculus Chapter Page 11