Section 5.0A Factoring Part 1

Transcription

1 Section 5.0A Factoring Part 1 I. Work Together A. Multiply the following binomials into trinomials. (Write the final result in descending order, i.e., a + b + c ). ( 7)( + 5) ( + 7)( + ) ( + 7)(3 + 5) (3 )( + 7) B. Where did the first term ( a ) of the trinomial come from in each of the problems above? (Use the words like product, sum, First terms, Outer terms, Inner terms, and Last terms) C. Where did the last term (c) of the trinomial come from in each of the problems above? (Use the words like product, sum, First terms, Outer terms, Inner terms, and Last terms) D. Where did the middle term (b) of the trinomial come from in each of the problems above? (Use the words like product, sum, First terms, Outer terms, Inner terms, and Last terms) E. Wrap Up Algebra Page 1

2 II. Think and Discuss A. Factoring Reversing the FOILing process. 1. Factor the following trinomials (i.e., break them apart to be two binomials again) = ( )( ) = ( )( ) + = 3. 1 ( )( ) Remember what you just discovered and what we discussed on the previous page (i.e., start with the first two questions and then verify with the last question). + =. 3 ( )( ) = ( )( ) B. Factoring can also be the reverse of distributing (i.e., what is in common with all the terms?) 1. Try the following y 10 y = ( ) Remember what you do when you distribute 3 ( y + ),. + y + 3 = ( ) now do the opposite. C. What happens when both are together (reverse distribution and reverse FOILing)? Which do you do first? Always do reverse distribution first, then the reverse FOILing. 1. Try the following 3 1. r + 3r 5 r = ( )= ( )( ) 3. r + 3r + 1 r = ( )= ( )( ) 3. r 1 r ( )= ( )( = ) Homework: p. P Basic: (3 )/3 Algebra Page

3 Section 5.0B Factoring Part Objective: 1. Factoring polynomials with or more terms. I. Work Together A. Multiply the following binomials. 1. ( a)( y + b). ( + 7)( y + ) 3. ( )( + 3). ( + 7)(3 + 5) B. Looking at the binomials you just multiplied, can you figure out a way to factor the following? 1. a + b + ay + by = ( )( ) y + + y = ( )( ) = ( )( ) = ( )( ) C. Wrap Up Algebra Page 3

4 II. Think and Discuss A. Factor the following using the grouping method a + 5 a = ( )+( )= ( )+ ( )=( )( ). rs + st + 3r + 3 t = ( )+( )= ( )+ ( )=( )( ) 3. 5y 10 y + = ( )+( )= ( )+ ( )=( )( ) + = ( )+( )= ( )+ ( )= ( )( ) = ( )+( )= ( )+ ( )=( )( ) B. What happens if none of the methods I use is able to factor the problem? Then the polynomial is prime. Try the following y Homework: p. 7 Basic: 0-5 all; all Etended: 7,7 Algebra Page

5 Section 5.1 Graphing Quadratics Objectives: 1. Graph Quadratic Functions. Find the ais of symmetry and coordinates of the verte of a parabola. 3. Model data using a quadratic function. f ( ) = + 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) Equation for a parabola: b) Verte (1) () (3) c) Direction of Opening/Etrema (1) If a is positive, it opens and has a value. () If a is negative, it opens and has a value. B. Eamples 1. Find the y-intercept, ais of symmetry, -coordinate of the verte, make a table of values that includes the verte, and graph for the following equation: y = Determine if ( ) f = has a maimum or minimum value, find that value, and state the domain and range for the function. 3. A souvenir shop sells about 00 coffee mugs each month for $ each. The shop owner estimates that for each $0.50 increase in the price, he will sell about 10 fewer coffee mugs per month. How much should the owner charge for each mug in order to maimize the monthly income from their sales? What is the maimum monthly income the owner can epect to make from these items?. Use you calculator to verify your answers to problem 3. Homework: Day 1: p. 5 Basic: 1-11 odds, 1; Etended: 1, 5 Day : p. 55 Basic: - evens; Etended: 0, 7 Algebra Page 5

6 Section 5.1B 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(10) 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. 3. How close was it to the actual temp of 1? Homework: p. 5 Basic: 1- all Algebra Page

7 Section 5. 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 by graphing: + + = 0. C. Solve by graphing: 0 = + 3 D. State the consecutive integers between which the roots are located for the given table. E. Find the zeros using your graphing calculator. 1. y = 3 +. y = F. The highest bridge in the United States is the Royal Gorge Bridge in Colorado. The deck of the bridge is 1053 feet above the river below. Suppose a marble is dropped over the railing from a height of 3 feet above the bridge deck. How long will it take the marble to reach the surface of the water, assuming there is no air resistance? Use the formula h( t) = 1t + h, where t is the time in seconds and h0 is the initial height above the water in feet. 0 Homework: Day 1: p. 3 Basic: 1- all, 7-13 odds, 30-3 all; Etended: 50, 5 Day : p. 7 Basic: (3-1)/3 and p.,70 Algebra Page 7

8 Objective: 1. Solve polynomials Section 5.3 Solving Quadratics by Factoring I. Work Together A. Solve for each variable in the following equations. 3 = 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 Algebra Page

9 B. Try the following = = 0 ( )( ) = 0 ( )( ) = 0 ( ) = 0 or ( ) = 0 ( ) = 0 or ( ) = 0 = or = = or = + 3 = 0 1 = 0 ( )( ) = 0 ( )( ) = 0 ( ) = 0 or ( ) = 0 ( ) = 0 or ( ) = 0 = or = = or = 3 + = 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. 3 5 = = ( 9)( + 5) = 15 + = ( 9) = 15 or ( + 5) = = 5 = or = = 0 ( 9)( + ) = 0 ( 9) = 0 or ( + ) = 0 = 9 or = Homework: p. 7 Basic:13-15 all, 3, 0, 79; Etended: 55, 0, 70 Algebra Page 9

10 Section 5.A Square Roots I. General Information A. Definition of Square Roots B. Note: Square roots have more than one root. 1. Eamples 3 has two square roots.. The nonnegative root is called the principal root. a) 3 is asking for the principle root. b) 3 is asking for the opposite of the principle root. c) ± 3 is asking for the both roots. C. Problems Find each root ± 19 II. Radical Epressions A. Properties 1. a b =. a b = B. Eamples on Simplifying ± III. Rationalizing the Denominator A. Protocol B. Eamples Homework: Day 1: p. 99 Basic: (3-30)/3 Day : p. 3 Basic: 1-1 all Algebra Page 10

11 Section 5. The Quadratic Formula and Discriminant Goals: 1. To solve quadratic equations using the quadratic formula.. To use the discriminant to determine the nature of the roots of the quadratic equation. I. 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 1. = = = = II. 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 3. If D < 0, then C. Eamples Determine the nature of the roots = = = = 0 Homework: p. 97 Basic: 1-13 odds, 3; Etended: 0, 3, 1 (If roots are comple roots, do not solve) Algebra Page 11

12 Section 5.B Sum and Product of Roots Goals: 3. To find the sum and product of the roots of a quadratic equation.. To find all possible integral roots of a quadratic equation. 5. 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 = 3. 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 3 and -. Roots are 5 and 1 *3. One root is 1+ 3 *Note: ( a + b c ) is a root iff ( a b c ). Homework: p. 30 Basic: 1-5 all Algebra Page 1

13 Section 5.7 Transformations with 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: 10 y = ( + 3) B. What does k do? 1. Graph: y = +. Graph: 10 y = C. What does a do? 1. Graph: y 10 =. 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: 3. a Algebra Page 13

14 E. Eamples: 1. Name the verte, ais of symmetry and direction of opening. a) y = ( 3) + b) y ( ) = Put the following quadratics into a) y y = a( h) + k = + + b) y = 3 + Graph this one Reminder: How do I find b the verte? = a III. Find the equation of the parabola Eamples: 1. (,1) 3 (,-) (,-) Parabola passes through the verte (1, ) and the point (3, ). Homework: Day 1: p. 30 Basic: 5-7 all,,,, 70, 7 Day : p. 30 Basic: 1- all, 9, 19, 3, 3; Etended: 1, Algebra Page 1

15 Section 5. 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 < ( 3) 1 y + II. Solving quadratic inequalities A. Graphically: > (use calculator) B. Algebraically Homework: Day 1: p. 315 Basic: 1-3 all, 1,, 7; Etended: - all Day : p. 315 Basic: -7 all, 9-1 all, 5; Etended:, 55 Algebra Page 15