Algebra /Trig Apps: Chapter 5 Quadratics Packet In this unit we will: Determine what the parameters a, h, and k do in the vertex form of a quadratic equation Determine the properties (vertex, axis of symmetry, shape) of the quadratic from looking at the coefficients in standard form Determine the roots of a quadratic function by factoring. Determine the roots of a quadratic function by the quadratic formula.
Algebra /Trig Apps: Vertex Form of a Quadratic Equation SWBAT - Identify the coordinates of the vertex of a quadratic equation and the effect of changing a Vertex form of a parabola expresses an equation for the parabola in terms of the vertex. Warm Up: Lab Follow-Up Based on the quadratic modeling activity we did the other day, circle the correct answer for each. 1. If a is positive, the parabola is {happy, sad}.. If a is negative, the parabola is {happy, sad}. 3. If you increase the absolute value of a (make the value of a farther from 0) the parabola gets {skinnier, fatter}. 4. If you decrease the absolute value of a (make the value of a closer to 0), the parabola gets {skinnier, fatter}. 5. Increasing the value of h makes the parabola move to the {right, left}. 6. Decreasing the value of h makes the parabola move to the {right, left}. 7. Increasing the value of k makes the parabola move to the {up, down}. 8. Decreasing the value of k makes the parabola move to the {up, down}. For each of the following, tell what you have to do to get the THICK parabola to model the THIN parabola. By this I mean that you are changing the values of the THICK parabola. 1 Increase/decrease/same 6 value of a Increase/decrease/same 4 value of a 4 The sign of a : Change/remain the same The sign of a : Change/remain the same Increase/decrease/same value of h Increase/decrease/same value of h 5 Increase/decrease/same value of k Increase/decrease/same value of k 3 Increase/decrease/same value of a 6 4 The sign of a : Change/remain the same Increase/decrease/same value of h 6 4 Increase/decrease/same value of a 4 The sign of a : Change/remain the same Increase/decrease/same value of h Increase/decrease/same value of k Increase/decrease/same value of k
The reason why we call this form vertex form is because it indicates where the vertex is. Example: a= h= k= The vertex is therefore ( ). For each of the following, give the coordinates of the vertex. 1. Vertex:. Vertex: 3. Vertex: 4. Vertex: For each of the following, write the equation (in vertex form) of the parabola given the vertex and the value of a. 5. Vertex : (, 5) and a = Equation: 6. Vertex : (-3, 4) and a = 1 Equation: 7. Vertex : (-8, 5) and a = -1 Equation: 8. Vertex : (6, 0) and a = -5 Equation: 3
Match each of the following graphs with their equations. Don t use the graphing calculator for this!! Be prepared to share your thinking for the match you selected. 17 18 19 0 1 a) b) c) d) e) f) g) h) i) j) k) l) m) n) o) p) QUIZ TOMORROW 4
AT Apps: Properties of a Quadratic Function in Standard Form SWBAT Convert Vertex Form Equations to Standard Form Warm Up 1. Answer each of the following questions.. Your answer is written in vertex form y = a(x h) + k 3. Compare your answers with your partner. 4. Be prepared to share your reasoning a) Write the quadratic equation for the parabola with vertex (-5, 3) and a = 3. b) Determine whether the parabola opens upward or downward c) Is the parabola wider or skinnier compared to a parabola with a =? I. Standard Form of a Quadratic Equation: is always the coefficient of the (quadratic) term. is always the coefficient of the (linear) term. is always the coefficient of the no x (constant) term. Determine a, b, and c of each of the following: Re-arrange to be in standard form 1. y = 4x + x 5. y = x - 6x + 9 3. y = x 4 4. y = x - 7x 4 5. y = x 5x + 6 6. y= x - 4x a b c II. Converting an Equation from Vertex Form to Standard Form In order to convert an equation from vertex form to standard form, FOIL the squared term first, distribute the value of the coefficient afterwards if there is one, and then combine any like terms. Example 1: Example : 5
Practice: Convert each of the following equations in vertex form to the equivalent equation in standard form y = ax + bx + c by multiplying and simplifying. 7. 8. 9. 10. SUMMARY To convert from Vertex Form, y = a(x h) + k to Standard Form y = ax + bx + c y = (x 3) 4 = [(x - 3)(x 3)] 4 Rewrite the term to be squared as the product of two binomials = (x 6x + 9) 4 FOIL the squared term first = x 1x + 18 4 Distribute the value of the coefficient = x 1x + 14 Combine like terms 6
AT Apps: Axis of Symmetry, Vertex, and Opening of a Quadratic Function SWBAT Determine the Axis of Symmetry, Vertex and Opening of a Parabola given equation in standard form. Warm Up LAB ACTIVITY EXPLORE THE AXIS OF SYMMETRY Every graph of a quadratic function is a parabola that is symmetric about a vertical line through its vertex called the axis of symmetry. There is a relationship between a and b in the quadratic function and the equation of the axis of symmetry. 1. Complete the table.. Compare the axis of symmetry with in your chart. What can you multiply by to get the number in the equation of the axis of symmetry? (Hint: Write and solve an equation to find the value.) Check your answer for each function. 3. Use your answer from Problem to complete the equation of the axis of symmetry of a quadratic function. x = If a>0 (is positive) then the vertex has a minimum point (is happy. ) If a<0 (is negative) then the vertex has a maximum point (is sad. ) Axis of symmetry:, 7
The Vertex has same x-coordinate as the axis of symmetry. Find the y-coordinate obtained by substituting x into the original equation. Practice: Determine the axis of symmetry and vertex of each of the following. 1.. 3. 4. 5. 6. 7. 8. 9. 10. 11. 1. Summary 8
Algebra/Trig: Factoring a Quadratic Expression SWBAT Factor a Quadratic Expression Warm Up Use the distributive property to find each product. Multiply each polynomial 1. x(3 x + 1). 4xy (3x + 6y - 7) There are a number of different ways to factor a quadratic expression. Greatest Common Factor (GCF) The Greatest Common Factor (GCF) is the largest numeric value and variable power that can be divided out of a polynomial. Always start by factoring out any GCF! Example 1: Factor 8x 4 1x 3 16x You can treat the coefficients separately from each variable. First, look for the largest value that is a factor of 8, 1, and 16. 4 is the largest value. Then, for each variable, find the greatest power of that variable that can be divided out of that variable. The greatest power of x that can be divided out of x 4, x 3, and x is x. Think of what will remain when you divide each term by the GCF. Writing the problem this way may help you see what the remaining factor is. Recall that when you divide powers of a variable, you subtract the exponents. Write the GCF on the left, and the remaining factor in parentheses on the right. Be Careful of THIS!! Example : 16x 4 1x 3 4x Example 1: Factor each of these by determining the GCF. 1. 1 ab 30 ac p prt. 3. 1x y 18xy 3 4 4. 6b 36b 3 5. 6. 9
Example : Factor by Grouping Factor each polynomial filling in the blanks. 1.. 10
3) 4) SUMMARY 11
Algebra Trig/ APPS Practice/Homework 15) 16) m 3 + 4m + 6m + 1 17) 18) 19) 0) 6a 3 9a 1 + 8a 1
Algebra/Trig Apps: Difference of Perfect Squares Factoring SWBAT: Factor the Difference of Perfect Squares (DOPS) Warm Up 1. Factor Using the GCF. Multiply (x + 3)(x 3) Recognize a difference of two squares: the coefficients of variable terms are perfect squares, powers on variable terms are even, and there is a MINUS sign between them. Example: Factor 100x 6 9y This is a Difference of Perfect squares. 100 and 9 are perfect squares, and the powers of x and y are even. Determine what each term is a perfect square of. 100 is 10, 9 is 3, x 6 is (x 3 ) and y is the square of y. Careful! Don t forget that 1 is a perfect square! Differences of perfect squares can be factored by the pattern a b = (a + b)(a b). In this case, the a is 100x 6 so the a=10x 3 and b is 9y so b=3y. Examples: 1. x 9. a 4b 3. x 16 4. 5. 4x y 6. 4a 9 7. 49a 81b 4 8. 1 9 m n 9 16 4 13
Practice: Factor each of the following. Summary 14
Algebra Trig/Apps Exit Ticket 15
Name Date Algebra/Trig Apps: Factoring Quadratic Trinomials of the form SWBAT: Factor quadratic trinomials of the form x + bx + c. Warm Up 1. Factor by Grouping.. Factor using GCF. Mini-Lesson Do you recognize the pattern??? Quadratic trinomials are the results of FOILing two binomials. You Try!!! Complete the Diamond Multiply (x + )(x + 5) = = Notice the constant term in the trinomial; it is the product of the constants in the binomials. You can use this fact to factor a trinomial into its binomial factors. (Find two factors of c that add up to b) 16
ax + bx + c Example 1: First Sign is Positive and Last Sign is Positive Factor: x + 6x + 8 Factor: x + 5x Answer: ( ) ( ) Answer: ( ) ( ) Practice 1: Factor. 1. x + 5x + 6. x + 8x + 1 3. x + 6x + 5 Answer: ( ) ( ) Answer: ( ) ( ) Answer: ( ) ( ) 4. x + 6x + 9 5. x + 10x + 1 6. x + 11x 17
Example : First Sign is Negative and Last Sign is Positive Factor: x - 10x + 4 Factor: x - 7x Answer: ( ) ( ) Answer: ( ) ( ) Practice : Factor. 7. x - 8x + 15 8. x - 6x + 8 9. x - 7x + 10 Answer: ( ) ( ) Answer: ( ) ( ) Answer: ( ) ( ) 10. x - 5x + 6 11. x - 13x + 40 1. x - 6x Example 3: First Sign is Positive or Negative and Last Sign is Negative Factor: x + x - 0 Answer: ( ) ( ) 18
Practice 3: Factor 13. x + x 15 14. x + 3x 10 15. x + 6x - 40 Answer: ( ) ( ) Answer: ( ) ( ) Answer: ( ) ( ) 16. x - x 3 17. x - x 15 18. x - x - 48 Challenge Problem: 1) )Factor: x 4 + 18x + 81 19
Summary: Example: Factor: x 5x - 50 0
ALGEBRA TRIG/APPS Homework 1
Algebra/Trig Apps: Factoring Quadratic Trinomials of the form ax + bx + c where a>1 SWBAT: Factor Quadratic Trinomials of the form ax + bx + c where a> 1 Warm Up The area of the rectangle below is represented by the polynomial x + 8x + 7. Find the binomials that could represent the lengths and width of the rectangle. A = x + 8x + 7 The a value is called the leading coefficient. It is always the coefficient of the x term. Because of the leading coefficient, we can t use the previous method to do factor. There are a number of tricks that we can use to figure out how to factor, but we will use the method called Splitting the Middle. Don t worry too much about WHY it works, just learn to use it first, and hopefully someday (soon) you will see why it works. Factor by Grouping Splitting the Middle Example 1: Factor 6x + 19x + 10
Example : Factor. Example 3: Factor. Practice: 1. x 4 8x 3. 3x 8x 3 3. 6x 7x 4. 1x 8x 15 5. 4x 8x 3 6. 3x 8x 3 7. 6x 7x 8. 1x 8x 15 3
9. 4x 8x 3 10. 3x 8x 3 11. 6x 7x 1. 1x 8x 15 13. 4x 8x 3 14. 3x 8x 3 15. 6x 7x 16. 1x 8x 15 Summary: 4
ALGEBRA TRIG/APPS Homework Factor each of the following using the Splitting the Middle Method. 5
ALGEBRA TRIG/APPS SWBAT: Factor a Trinomial Completely Warm Up Factor each. 1.. 3. 4. 5. Factor by Grouping. 6
Factoring Trinomials Completely In the previous lesson, we saw how to factor a trinomial of the form bx c by employing the diamond method. In each of those cases, the coefficient of the quadratic ( ) term was always one, and thus not written. It is also possible to factor trinomials of the form a bx c where the coefficient a is a number other than 1 by combining two factoring methods into the same problem. y = 5x - 1 7
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Challenge Problem: Recall that the volume of a rectangular solid (a box) is given by V L W H. If a particular rectangular solid has a volume of 5 15x 10, how would you represent the length, width and height of the solid? Justify your answer. SUMMARY Exit Ticket 9
ALGEBRA TRIG/APPS Factoring Completely Homework 30
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ALGEBRA TRIG/APPS: MORE FACTORING COMPLETELY Warm - Up 1.. Some polynomials cannot be factored into the product of two binomials with integer coefficients, (such as x + 16), and are referred to as prime. Other polynomials contain a multitude of factors. "Factoring completely" means to continue factoring until no further factors can be found. More specifically, it means to continue factoring until all factors other than monomial factors are prime factors. You will have to look at the problems very carefully to be sure that you have found all of the possible factors. To factor completely: 1. Search for a greatest common factor. If you find one, factor it out of the polynomial.. Examine what remains, looking for a trinomial or a binomial which can be factored. 3. Express the answer as the product of all of the factors you have found. 3
ALGEBRA TRIG/APPS Example 1: Factoring Completely FACTOR: 10x - 40 Practice: Factoring Completely 33
Example : Factoring Completely Factor: 8 Factor: Practice: Factoring Completely 4. 10 5. 6. 7. 8. 9. 3. 4x + 4x + 36 10. 4. x 3-8x + 16x 34
Challenge Problem: Summary: EXIT TICKET Factor Completely Homework 35
ALGEBRA TRIG/APPS Homework 36
ALGEBRA TRIG/APPS REVIEW FOR TEST SWBAT: Apply their knowledge on Factoring Factor. Station # 1 Common Monomial Factors (GCF) 1) 9x 1x 5 ) 4x 3 6x + 10x 3) Factor by Grouping. 4) 5) Factor. Station # Difference of Two Squares D.O.T.S 1) x² - 49 ) 36x² - y² 3) 64 - y² 4) 9a² - 11y² 5) a 6 9b 1 6) 5x 4-144y² 37
Station # 3 Factoring Trinomials Diamond 1) x² + 1x + 0 ) x² - 10x + 4 3) x² + 3x 18 4) x² - 7x + 1 5) x² - 6x - 7 6) x² - x 56 Station # 4 Factoring Completely 1) ax² - a ) 4a 36 3) 1x 3y² 4) 9a 4 36b 4 5) 3x + 15x 4 6) x 4 3x 3 40x² 38
Station # 5 Word Problems 1) The area of rectangle is represented by x + 9x + 18. Find the binomials that could represent the lengths and width of the rectangle. ) The Volume of rectangular prism is represented by p 3-1p + 35p. Find the factors that would represent the length, width, and height of the rectangular prism. 39
Review SWBAT: Apply Their Knowledge on Factoring. A) 4t A) y 4 A) 5n 9 B) 3t 6 B) y B) 3n 4 C) t C) y 3 C) 15n D) t 6 D) y D) 3n 9 4. Factor each expression using the GCF. 5. 6. 7. 8. 40
9. 10. 11. A) (x + 6)(x + 1) A) (x - 3)(x - 7) A) (x + 5)(x + 10) B) (x + 5)(x + 1) B) (x - 3)(x + 7) B) (x 5)(x 3) C) (x - 5)(x + 1) C) (x + 10)(x + 11) C) (x + 5)(x + 3) D) (x + )(x + 3) D) (x + 3)(x - 7) D) (x 5)( x + 3) Factor each binomial. 1. 13. 5x 4 14. A) (b - 8)(b - ) A) (5x + )(5x - ) B) (b + 4)(b + 4) B) (15x + )(10x - ) C) (b + 8)(b + ) C) (x + )(5x - ) D) (b - 4)(b + 4) D) (5x + )(5x + ) 15. 16. A) 3x 3 (x - 9) B) 3x 3 (x + 3)(x - 3) C) 3x 3 (x + 3)(x + 3) D) 9x 3 (x - 9) 17. 41
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Name: Date: 1.. A box has a volume given by the trinomial + 3. What are the possible dimensions of the box? Use factoring completely. a. b. c. d. 43
Name: Date: Algebra/Trig Apps: Factoring Completely Warm Up Factor each. 1.. These are multi-step factoring problems. You should attack the problems in the following order: 1. Look for a GCF. Look for a Difference of Two Squares, or Sum or Difference of Two Cubes 3. Try to Factor Trinomials to two binomial factors. (use either trial & error or the Factoring algorithm on back side) Examples: Consider the trinomial 3x + 15x + 18 (a) What is the GCF of each term in the trinomial? (b) Write the trinomial as a product Involving its GCF. (c) How does the trinomial inside of the parentheses now factor? (d) Write 3x + 15x + 18 in its completely factored form. 44
Name: Date: Practice: Factor Completely 45
Name: Date: AlgebraTrig Summary: Factoring Completely Look for GCF Factor it out! (Don't throw it away!) Difference of perfect squares? x - a Difference of perfect cubes? x 3 ± a 3 Quadratic Trinomial x + bx + c Quadratic trinomial ax +bx + c where a 1 4 term polynomial (or more...) (x-a)(x+a) ± (x )(x ) Look for two numbers that multiply to c and add to b Splitting the Middle METHOD FACTOR BY GROUPING Everything, but in Vertex Form Vertex Form of a Quadratic Axis of symmetry: vertex: has the same effect on the appearance of the graph as it does in standard form. 46
Name: Date: Algebra/Trig: Solving Quadratic Equations - Finding the Roots by Factoring SWBAT: Solve Quadratic Equations by Factoring WARM UP The roots or zeros of a quadratic equation are where the parabola the axis. A of a function is an x- value that makes the function ( ) equal zero. A of a function is also the same as the x-intercept. The roots or zeros of a quadratic equation may have two roots, one root or no roots. 47
Name: Date: Solving (Roots) by Factoring The roots (also called zeros or solutions) of a quadratic equation are where the graph of the equation hits the x-axis, or where y=0 In order to determine the roots of a quadratic, set the quadratic to 0, factor, and solve. Example: Determine the roots of 1. Set y = 0, or set the quadratic =0 if there is no y.. Factor the quadratic 3. Make a t chart. Set each factor = 0 4. Solve each linear factor. Practice: Solve each of the following by factoring. 1) x + 11x + 8 = 0 ) p + 7p + 10 = 0 3) n 11n + 30 = 0 4) m + 1m + 35 = 0 5) v + 7v = -1 6) a = -49 14a 7) r = 6r 8) b + b = 35 48
Name: Date: 9) n 4n = 1 10) x + 16 = -10x 11) b = b + 1) x + 16 = -10x 13) x 4 = 0 14) k + 9k = -14 15) x 13x = -7x 16) n + 7n = 0 17) p + 5p = 14 18) r = -r + 8 49
Name: Date: Summary Exit Ticket Which graph represents the function: 50
Name: Date: Algebra Trig/Apps Solving Quadratic Equations by the Quadratic Formula HOW TO SWBAT Use the Quadratic Formula to Solve Quadratic Equations Warm - Up The Quadratic Formula is derived from the process of completing the square, which is beyond the scope of this class. ± Where is the coefficient of the term, is the coefficient of the term, and is the constant term. The Quadratic Formula can be used to solve ANY quadratic, even those that you can factor (rational roots) and those you cannot (irrational.) Example 1: ± ± Please note that you get the same answer if you solved by factoring. The roots are:, { } 51
Name: Date: Example : Please note that you get the same answer as the example in completing the square. The roots are: { ± ± } Example 3: ± ± Please note that this needs to be simplified as per all rules of radicals! ± The roots are: { } DIRECTIONS: One student should solve the equations in column A and one student should solve the equations in column B. When each has finished their side, switch and check each other s work. If you both agree the answers are correct, I will verify your work. A B 5
Name: Date: Solve each of the following by the quadratic formula. Problems 1-10 will have rational solutions, and problems 10-0 will have irrational solutions. 53
Name: Date: SUMMARY The solutions of some quadratic equations are not rational, and cannot be factored. For such equations, the most common method of solution is the quadratic formula. The quadratic formula can be used to solve ANY quadratic equation, even those that can be factored. Be sure you know this formula!!! Example: Note: The equation must be set equal to zero before using the formula. Name Date EXIT TICKET Use the quadratic formula to find all roots of the equation x 3x = 0 54
Name: Date: Algebra Trig/Apps: SWBAT Solve Quadratic Word Problems WARM - UP Find the zeroes of the quadratic function f(x) = x + 5x 7 Algebra/Trig Apps: Quadratics Word Problem Sampler: Each of the following problems requires you to determine the roots, one of the coordinates of the vertex, or both. Determine by careful reading which the problem requires and find it. Show all work!! Helpful reading hints: Hits the ground, is empty, is zero : roots maximum, minimum, height : vertex Example: A superhero is trying to leap over a tall building. The function gives the superhero s height in feet as a function of time. The building is 61 feet high. a. Will the superhero make it over the building? b. If he makes it over the building, determine how long it takes him to hit the ground. 55
Name: Date: For each question, you need to ask yourself: Is the question asking me to find the root (x-intercept) of the function? Is the question asking me to find the max/min? If so, is the question asking me for the x or y coordinate? 1. A ball is thrown straight up at an initial velocity of 54 feet per second. The height of the ball t seconds after it is thrown is given by the formula. How many seconds after the ball is thrown will it return to the ground? Are you being asked to find the roots (x- intercept? or a max/min? (Vertex coordinates). The height of an object, h(t), is determined by the formula h( t) 16t 56t, where t is time, in seconds. Will the object reach a maximum or a minimum? Explain or show your reasoning. Are you being asked to find the roots (x- intercept? or a max/min? (Vertex coordinates) 3. A model rocket is launched from ground level. Its height, h meters above the ground, is a function of time t seconds after launch and is given by the equation h 49. t 686. t. What would be the maximum height, to the nearest meter, attained by the model? Are you being asked to find the roots (x- intercept? or a max/min? (Vertex coordinates) 56
Name: Date: 4. The height, h, in feet, a ball will reach when thrown in the air is a function of time, t, in seconds, given by the equation h( t) 16t 30t 6. Find, to the nearest tenth, the maximum height, in feet, the ball will reach. Are you being asked to find the roots (x- intercept? or a max/min? (Vertex coordinates) 5. When a current, I, flows through a given electrical circuit, the power, W, of the circuit can be determined by the formula W 10I 1I. What amount of current, I, supplies the maximum power, W? Are you being asked to find the roots (x- intercept? or a max/min? (Vertex coordinates) 6. Barb pulled the plug in her bathtub and it started to drain. The amount of water in the bathtub as it drains is represented by the equation, where L represents the number of liters of water in the bathtub and t represents the amount of time, in minutes, since the plug was pulled. Determine the amount of time it takes for all the water in the bathtub to drain. Are you being asked to find the roots (x- intercept? or a max/min? (Vertex coordinates) 57
Name: Date: 7. The cross-section of a tunnel through a mountain can be modeled by the equation where x all of the measurements are in feet. How wide is the tunnel? Are you being asked to find the roots (x- intercept? or a max/min? (Vertex coordinates) 8. A skating rink manager finds that revenue R based on an hourly fee F for skating is represented by the function. What hourly fee will produce the maximum revenues? Are you being asked to find the roots (x- intercept? or a max/min? (Vertex coordinates) 9. After a heavy snowfall, Joe and Karin made an igloo. The dome of the igloo is in the shape of a parabola and the height of the igloo in inches is given by a) Joe wants to place a support in the middle of the igloo, along the axis of symmetry. How far from the edge of the igloo should he place the support? b) Neither Joe nor Karin can stand up inside the igloo. How tall is the center of the igloo? 58