Quadratic Equations ALGEBRA 1. Name: A Learning Cycle Approach MODULE 7

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1 Name: ALGEBRA 1 A Learning Cycle Approach MODULE 7 Quadratic Equations The Mathematics Vision Project Scott Hendrickson, Joleigh Honey, Barbara Kuehl, Travis Lemon, Janet Sutorius 016 Mathematics Vision Project Prior work done in partnership with the Utah State Office of Education 01 Licensed under the Creative Commons Attribution-NonCommercial ShareAlike 4.0 Unported license

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3 7.1 CW: Rational Eponents Refer to your Book of Eponent Rules. Write each eponential epression in radical form n t 5 4 Write each radical epression in eponential form a n y n 1. p q n Simplify each of the radical epressions. Use rational eponents if desired. 81 y a b a n

4 7.1 HW: Rational Eponents Fill in the table so each epression is written in radical form and with rational eponents. Radical Form Eponential Form y a b

5 7. CW: Eperimenting with Eponents [This task was adopted from the Illustrative Mathematics Project: Content on this site is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike.0 Unported License] Travis and Miriam are studying bacterial growth. They were surprised to find that the population of the bacteria doubled every hour. 1. Complete the following table and plot the data on the graph provided on page 5. Hours into the study Bacteria population (in thousands) 4. Write an equation for P, the population of the bacteria, as a function of time, t, and verify that it produces correct populations for t = 1,,, and 4 hours. Travis and Miriam want to create a table with more entries; specifically, they want to fill in the population at each half hour. Unfortunately, they forgot to make these measurements so they decide to estimate the values. Travis makes the following claim: If the population doubles in 1 hour, then half that growth occurs in the first half-hour and the other half in the second half-hour. So for eample, we can find the population at t = 1 by finding the average of the populations at t = 0 and t = 1.. Fill in the parts of the table below that you've already computed, and then decide how you might use Travis strategy to fill in the missing data. Also plot Travis data on the graph. Hours Bacteria (in thousands) Comment on Travis idea. How does it compare to the table generated in problem 1? For what kind of function would this reasoning work?

6 Miriam suggests they should fill in the data in the table in the following way: To make the estimates, I noticed that the population increases by the same factor each hour, and I think that this property should hold over each half-hour interval as well. 5. Fill in the parts of the table below that you've already computed in problem 1, and then decide how you might use Miriam s new strategy to fill in the missing data. As in the table in problem 1, each entry should be multiplied by some constant factor in order to produce consistent results (hint: look at your ISN). a. What page are we looking at in your ISN? b. Use this constant factor to complete the table. Plot Miriam s data on the graph. Hours Bacteria (in thousands) Now Miriam wants to estimate the population every 0 minutes instead of every 0 minutes. a. What multiplier would she use for every third of an hour to consistent with the population doubling every hour? b. Use this multiplier to complete the following table. Hours Bacteria (in thousands) 4 8 c. Give a detailed description of how you would estimate the population, P, at 5 t hours. 4

7 Hours vs Bacteria Population (in thousands) 5

8 7. HW: Eperimenting with Eponents READY Topic: Additive and multiplicative patterns The sequences below eemplify either an additive (arithmetic) or a multiplicative (geometric) pattern. Identify the type of sequence, fill in the missing values on the table and write an eplicit equation. 1. Term Value Equation: Type of sequence:. Term Value Equation: Type of sequence:. Term Value Equation: Type of sequence: 4. Term Value Equation: Type of sequence: 6

9 5. Term Value Equation: Type of sequence: Use the graph of the function to find the desired values of the function. 6. Find the value of f() 7. Find where f() = 4 8. Find the value of f(6) 9. Find where f() = What do you notice about the way that inputs and outputs for this function relate? 11. What is the eplicit equation for this function? 7

10 SET Topic: Fill in the missing values of the table based on the growth that is described. 1. The growth in the table is by a factor of four each whole year. Years Bacteria (in thousands) 8 1. The growth in the table is tripled at each whole year. Years Bacteria (in thousands) The growth in the table is tripled at each whole year. Years Bacteria (in thousands) 6 8

11 7. CW: Factor Fiin At first, Optima s Quilts only made square blocks for quilters and Optima spent her time making perfect squares. Customer service representatives were trained to ask for the length of the side of the block,, that was being ordered, and they would let the customer know the area of the block to be quilted using the formula: A Optima found that many customers that came into the store were making designs that required a combination of squares and rectangles. So, Optima s Quilts has decided to produce several new lines of rectangular quilt blocks. Each new line is described in terms of how the rectangular block has been modified from the original square block. For eample, one line of quilt blocks consists of starting with a square block and etending one side length by 5 inches and the other side length by inches to form a new rectangular block. The design department knows that the area of this new block can be represented by the epression: A 5 but they do not feel that this epression gives the customer a real sense of how much bigger this new block is (e.g., how much more area it has) when compared to the original square blocks. 1. Can you find a different epression to represent the area of this new rectangular block? You will need to convince your customers that your formula is correct using a diagram (hint: use your Algebra tiles to find the new formula and create the diagram) 9

12 Here are some additional new lines of blocks that Optima s Quilts has introduced. Find two different algebraic epressions to represent each rectangle, and illustrate with a diagram why your representations are correct.. The original square block was etended inches on one side and 4 inches on the other. Diagram: Two algebraic epressions. The original square block was etended 4 inches on only one side. Diagram: Two algebraic epressions 4. The original square block was etended 5 inches on each side. Diagram: Two algebraic epressions 10

13 5. The original square block was etended inches on one side and 6 inches on the other Diagram: Two algebraic epressions Customers start ordering custom-made block designs by requesting how much additional area they want beyond the original area of. Once an order is taken for a certain type of block, customer service needs to have specific instructions on how to make the new design for the manufacturing team. The instructions need to eplain how to etend the sides of a square block to create the new line of rectangular blocks. The customer service department has placed the following orders on your desk. For each, describe how to make the new blocks by etending the sides of a square block with an initial side length of. Your instructions should include diagrams, written descriptions and algebraic descriptions of the area of the rectangles in using epressions representing the lengths of the sides Diagram: Written description: Algebraic descriptions 11

14 Diagram: Written description: Algebraic descriptions Diagram: Written description: Algebraic descriptions 1

15 Diagram: Written description: Algebraic descriptions Some of the orders are written in an even more simplified algebraic code. Figure out what these entries mean by finding the sides of the rectangles that have this area. Use the sides of the rectangle to write equivalent epressions for the area

16 17. What relationships or patterns do you notice when you find the sides of the rectangles for a given area of this type? 18. A customer called and asked for a rectangle with area given by: 7 9. The customer service representative said that the shop couldn t make that rectangle. Do you agree or disagree? How can you tell if a rectangle can be constructed from a given area? 14

17 7. HW: Factor Fiin READY Topic: Creating binomial quadratics Multiply (use the distributive property, write the quadratics in standard form) The answers to problems 1- are quadratics that are represented in standard form. Which coefficient is equal to zero (hint: standard form is: a b c ) Factor the following epressions (Write the epression as the product of two linear factors) Multiply the two linear factors together The answers to problems 9-1 are quadratics that are represented in standard form. Which coefficient is equal to zero? 15

18 SET Topic: Factoring trinomials Find the sides of the rectangles that have each area described below. Write your answers in factored form [e.g. rectangle with one side increased by one and the other side increased by five would be: ] Look back at each factored epression in problems Eplain how it is possible that the coefficient (b) of the middle term can be different numbers in problems when the outside coefficients (a) and (c) are the same. 16

19 GO Topic: Taking the square root of perfect squares Only some of the epressions inside the radical sign are perfect squares. Identify which ones are perfect squares and take the square root. Leave the ones that are not perfect squares under the radical sign. Do not attempt to simplify them (hint: Check your answers by squaring them. You should be able to get what you started with, if you are right.) 7. 17yz a b

20 7.4 CW: The Factor Now that Optima s Quilts is accepting orders for rectangular blocks, their business in growing by leaps and bounds. Many customers want rectangular blocks that are bigger than the standard square block on one side. Sometimes they want one side of the block to be the standard length,, with the other side of the block inches bigger. 1. Draw and label this block. Write two different epressions for the area of the block. Sometimes they want blocks with one side that is the standard length,, and one side that is inches less than the standard size.. Draw and label this block. Write two different epressions for the area of the block. Use your diagram and verify algebraically that the two epressions are equivalent. There are many other size blocks requested, with the side lengths all based on the standard length,. Draw and label each of the following blocks. Use your diagrams to write two equivalent epressions for the area. Verify algebraically that the epressions are equal.. One side is 1 less than the standard size and the other side is more than the standard size. 18

21 4. One side is less than the standard size and the other side is more than the standard size. 5. One side is more than the standard size and the other side is less than the standard size. 6. One side is more than the standard size and the other side is 4 less than the standard size. 7. One side is 4 more than the standard size and the other side is less than the standard size. 8. An epression that has terms in the form: a b c is called a trinomial. Look back at the trinomials you wrote in questions -7. How can you tell if the middle term (b ) is going to be positive or negative? 19

22 9. One customer had an unusual request. She wanted a block that is etended inches on one side and decreased by inches on the other. One of the employees thinks that this rectangle will have the same area as the original square since one side was decreased by the same amount as the other side was increased. What do you think? Use a diagram to find two epressions for the area of this block. 10. The result of the unusual request made the employee curious. Is there a pattern or a way to predict the two epressions for area when one side is increased and the other side is decreased by the same number? Try modeling these two problems, look at your answer to #8, and see if you can find a pattern in the result. a. 1 1 b What pattern did you notice? What is the result of a a? 1. Some customers want both sides of the block reduced. Draw the diagram for the following blocks and find a trinomial epression for the area of each block. Use algebra to verify the trinomial epression that you found from the diagram. a. b

23 1. Look back over all the equivalent epressions that you have written so far and eplain how to tell if the third term in the trinomial epression a b c will be positive or negative. 14. Optima s quilt shop has received a number of orders that are given as rectangular areas using a trinomial epression. Find the equivalent epression that shows the lengths of the two sides of the rectangles. a b. 18 c. 18 d e. 5 4 f. 4 g Write an eplanation of how to factor a trinomial in the form: a b c 1

24 7.4 HW: The Factor READY Topic: Eploring the density of the number line. Find three numbers that are between the two given numbers & 4. & & & & & & & 14 SET Topic: Factoring quadratics The area of a rectangle is given in the form of a trinomial epression. Find the equivalent epression that shows the lengths of the two sides of the rectangle

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26 GO Topic: Graphing parabolas Graph each parabola. Include the verte and at least accurate points on each side of the ais of symmetry. Then describe the transformation in words f g 1.. h b 1 4 4

27 7.5 Warm Up Put the product of columns a and c in column ac. Find two numbers that multiply to the number in column ac and add to the number in b. The first problem has been done for you. a b c ac 1 st number nd number

28 7.5 CW: The Wow Factor Optima s Quilts sometimes gets orders for blocks that are multiples of a given block. For instance, Optima got an order for a block that was eactly twice as big as the rectangular block that has a side that is 1 longer than the basic size,, and one side that is longer than the basic size. 1. Draw and label this block. Write two equivalent epressions for the area of the block.. Optima has a lot of new orders. Use diagrams to help you find equivalent epressions for each of the following a b c. + 4 d e. 7 6

29 . Because she is a great business manager, Optima offers her customers lots of options. One option is to have rectangles that have side lengths that are more than one. For instance, Optima made this cool block. Write two equivalent epressions for this block. Use the distributive property to verify that your answer is correct. 4. Here we have some partial orders. We have one of the epressions for the area of the block and we know the length of one of the sides. Use a diagram to find the length of the other side and write a second epression for the area of the block. Verify your two epressions for the area of the block are equivalent using algebra. a. Area: 7 Side: (+) Equivalent epression for area: b. Area: 5 8 Equivalent epression for area: Side: (+1) c. Area: 7 Equivalent epression for area: Side: (+1) 7

30 5. There s one more twist on the kind of blocks that Optima makes. These are the trickiest of all because they have more than one in the length of both sides of the rectangle! Here s an eample. Write two equivalent epressions for this block. Use the distributive property to verify that your answer is correct. 6. All right, let s try the tricky ones. They may take a little messing around to get the factored epression to match the given epression. Make sure you check your answers to be sure that you ve got them right. Factor each of the following: a b c d

31 7.5 HW: The Wow Factor READY Topic: Comparing arithmetic and geometric sequences The first and fifth terms of each sequence are given. Fill in the missing numbers. 1. Arithmetic 1875 Geometric Arithmetic 1875 Geometric Arithmetic 1875 Geometric

32 SET Topic: Writing an area model as a quadratic epression Write two equivalent epressions for the area of each block. Let be the side length of each of the large squares Problems 4-6 all contain the same number of squares measuring a. What is different about the images? and 1 b. How does this difference affect the quadratic epression that represents them? c. Describe how the arrangement of the squares and rectangles affects the factored form 0

33 Topic: Factoring quadratic epressions when a 1 Factor the following quadratic epressions

34 GO Topic: Find the equation of the line of symmetry of a parabola Given the -intercepts of a parabola, write the equation of the line of symmetry 4. -intercepts: (-, 0) and (, 0) 5. -intercepts: (-4, 0) and (16, 0) 6. -intercepts: (-, 0) and (5, 0) 7. -intercepts: (-14, 0) and (-, 0) 8. -intercepts: (17, 0) and (, 0) 9. -intercepts: (-.75, 0) and (.5, 0)

35 7.6 CW: Lining Up Quadratics Graph each function and find the verte, the y-intercept and the - intercepts. Be sure to properly write the intercepts as points 1. y 1 Line of symmetry: Verte: -intercepts: y-intercept:. f 6 Line of symmetry: Verte: -intercepts: y-intercept:

36 . g 4 Line of symmetry: Verte: -intercepts: y-intercept: 4. Based on these eamples, how can you use a quadratic function in factored form to: a. Find the line of symmetry of the parabola? b. Find the verte of the parabola? c. Find the -intercepts of the parabola? d. Find they-intercept of the parabola? e. Find the vertical stretch 4

37 5. Graph the two functions, f() and g(), on the same graph Linear function 1: f 1 Linear function : g 1 6. On the same graph as #5, graph the function P that is the product of the two linear functions. What shape is created (hint: make a table) 7. Describe the relationship between -intercepts of the linear functions and the -intercepts of the function P. Why does this relationship eist? 8. Describe the relationship between the y-intercepts of the linear functions and the y-intercept of the function P. Why does this relationship eist? 5

38 9. Given the parabola to the right, sketch two lines that could represent its linear factors 10. Write an equation for each of these two lines. 11. How did you use the and y intercepts of the parabola to select the two lines? 1. Are these the only two lines that could represent the linear factors of the parabola? If so, eplain why. If not, describe the other possible lines. 1. Use your two lines to write the equation of the parabola. Is this the only possible equation of the parabola? 6

39 7.6 HW: Lining up Quadratics READY Topic: Multiplying Binomials Multiply the following binomials Describe the relationship between the middle coefficient (b) and the outside coefficients (a and c) 7

40 SET Topic: Factored Form of a Quadratic Function Given the factored form of a quadratic function, identify the verte, intercepts, and vertical stretch of the parabola. 11. y y 6 1. y 5 7 a. Verte: a. Verte: a. Verte: b. -int: b. -int: b. -int: c. y-int: c. y-int: c. y-int: d. stretch: d. stretch: d. stretch: y 7 7 a. Verte: y 8 4 a. Verte: y 5 9 a. Verte: b. -int: b. -int: b. -int: c. y-int: c. y-int: c. y-int: d. stretch: d. stretch: d. stretch: y a. Verte: 18. y 5 5 a. Verte: 19. y a. Verte: b. -int: c. y-int: d. stretch: b. -int: c. y-int: d. stretch: b. -int: c. y-int: d. stretch: 8

41 GO Topic: Verte Form of a Quadratic Equation Given the verte form of a quadratic function, identify the verte, intercepts, and vertical stretch of the parabola. y 4 0. y 6 1. y 1 8. a. Verte: a. Verte: a. Verte: b. -int: b. -int: b. -int: c. y-int: c. y-int: c. y-int: d. stretch: d. stretch: d. stretch: y y y a. Verte: a. Verte: a. Verte: b. -int: b. -int: b. -int: c. y-int: c. y-int: c. y-int: d. stretch: d. stretch: d. stretch: 6. Did you notice that the parabolas in problems 11, 1, & 1 are the same as the ones in problems, 4, & 5 respectively? If you didn t, go back and compare the answers in problems 11, 1, & 1 and problems, 4, & 5. Prove that a b c

42 7.6C CW: Writing Quadratic Functions from Graphs Use your notes to write a quadratic function (in verte, intercept, or standard form) for each graph. 1. What do you know? verte: y intercept: intercepts: Which function form(s) is the easiest to write from the information you can find on the graph? verte form intercept form Function: standard form. What do you know? verte: y intercept: intercepts: Which function form(s) is the easiest to write from the information you can find on the graph? verte form intercept form standard form Function: 40

43 . What do you know? verte: y intercept: intercepts: Which function form(s) is the easiest to write from the information you can find on the graph? verte form intercept form standard form Function: 4. What do you know? verte: y intercept: intercepts: Which function form(s) is the easiest to write from the information you can find on the graph? verte form intercept form standard form Function: 41

44 7.7 CW: I ve Got a Fill-in For each problem below, you are given a piece of information that tells you a lot. Use what you know about that information to fill in the rest. 1. You get this: y 1 Fill in this: Factored form of the equation -intercepts and y-intercept. You get this: y 6 Fill in this: Verte form of the equation verte and y-intercept 4

45 . You get this: Fill in this: Factored form of the equation Standard form of the equation 4. You get this: Fill in this: Verte form of the equation Standard form of the equation 4

46 5. You get this: y 6 16 Fill in this: Either form of the equation other than standard form Verte -intercepts and y-intercept 6. You get this: y 1 1 Fill in this: Either form of the equation other than standard form Verte y-intercept 7. You get this: y Fill in this: Either form of the equation other than standard form Verte -intercepts and y-intercept 44

47 7.7 HW: I ve Got a Fill-in READY Topic: Identifying and interpreting parts of an equation A golf-pro practices his swing by driving golf balls off the edge of a cliff into a lake. The height of the ball above the lake (measured in meters) as a function of time (measured in seconds and represented by the variable t) from the instant of impact with the golf club is: t 4.9t The epressions below are equivalent: standard form: 4.9t 19.6t 58.8 factored form: 4.9 t 6 t 4.9 t 78.4 verte form: 1. Which epression is the most useful for finding how many seconds it takes for the ball to hit the water? Why?. Which epression is the most useful for finding the maimum height of the ball? Justify your answer.. If you wanted to know the height of the ball at eactly.5 seconds, which epression would help the most to find the answer? Why? 4. If you wanted to know the height of the cliff above the lake, which epression would you use? Why? 45

48 SET Topic: Finding multiple representations of a quadratic One form of a quadratic function is given. Fill-in the missing forms. 5. a. Standard Form b. Verte Form c. Factored Form 5 y d. Table (include the verte and points on either side of the verte). Show the first and second differences y e. Graph 6. a. Standard Form b. Verte Form y 1 c. Factored Form d. Table (include the verte and points on either side of the verte). Show the first and second differences y e. Graph 46

49 7. a. Standard Form y 10 5 b. Verte Form c. Factored Form d. Table (include the verte and points on either side of the verte). Show the first and second differences y e. Graph 8. a. Standard Form b. Verte Form c. Factored Form d. Table (include the verte and points on either side of the verte). Show the first and second differences y e. Graph 47

50 9. a. Standard Form b. Verte Form c. Factored Form d. Table (include the verte and points on either side of the verte). Show the first and second differences y e. Graph GO Topic: Factoring quadratics Verify each factorization by multiplying the linear factors Factor the following quadratic epressions, if possible (some will not factor)

51 17. m 16m s c 11c. Which quadratic epression above could represent the area of a square? Eplain. 4. Would any of the epressions above NOT be the side-lengths for a rectangle? Eplain 49

52 7.8 CW: Throwing an Interception The -intercept(s) of the graph of a function f() are often very important because they are the solution to the equation f() = 0. In previous tasks, we learned how to find the -intercepts of the function by factoring, which works great for some functions, but not for others. In this task we are going to work on a process to find the -intercepts of any quadratic function that has them. We ll start by thinking about what we already know about a few specific quadratic functions and then use what we know to generalize to all quadratic functions with -intercepts. 1. Consider the graph of the function 1 1 f a. Complete the table and graph the function f() b. What is the equation of the line of symmetry? c. What is the verte of the function? d. What are the -intercepts? e. How far are the -intercepts from the line of symmetry? f. How far above the verte are the -intercepts? g. What is the value of f() at the -intercepts? 50

53 . Consider the graph of the function f 6 4 a. Complete the table and graph the function f() b. What is the equation of the line of symmetry? c. What is the verte of the function? d. What do you estimate the -intercepts to be? e. About how far are the -intercepts from the line of symmetry? f. How far above the verte are the -intercepts? g. What is the value of f() at the -intercepts? 51

54 . Consider the graph of the function f 5 a. Complete the table and graph the function f() b. What is the equation of the line of symmetry? c. What is the verte of the function? d. What do you estimate the -intercepts to be? e. About how far are the -intercepts from the line of symmetry? f. How far above the verte are the -intercepts? g. What is the value of f() at the -intercepts? 5

55 7.8 HW: Throwing an Interception READY Topic: converting units of measure While working with areas it sometimes essential to convert between units of measure, for eample changing from square yards to square feet. Convert the areas below to the desired measure. (Hint: area is two dimensional, for eample 1 yd = 9 ft because ft along each side of a square yard equals 9 square feet.) 1. 7 yd = ft. 5 ft = in. 1 mile = ft Set Topic: Transformations and Parabolas, Symmetry and Parabolas 4. Eamine the differences and similarities between each quadratic function a. Graph each of the quadratic functions. f() = g() = 9 h() = ( + ) 9 b. How do the functions compare to each other? c. What are the -intercepts of g()? d. What are the coordinates of the points corresponding to the -intercepts in g() in each of the other functions? g() (same answers from part c) (, 0) (, 0) f() h() e. How do the coordinates compare to each other? 5

56 5. Eamine the differences and similarities between each quadratic function a. Graph each of the quadratic functions. f() = g() = 4 h() = ( 1) 4 b. How do the functions compare to each other? c. What are the -intercepts of g()? d. What are the coordinates of the points corresponding to the -intercepts in g() in each of the other functions? g() (same answers from part c) (, 0) (, 0) f() h() e. How do the coordinates compare to each other? 54

57 GO Topic: Evaluating functions Use the given functions to find the missing values. (Check your work using a graph.) 6. f() = a. f(0) = b. f() = c. f() = 0 d. f() = 0 7. f() = a. f(0) = b. f() = c. f() = 0 d. f() = 0 8. f() = ( ) a. f(0) = b. f() = c. f() = 0 d. f() = 0 9. f() = ( + 1) + 8 a. f(0) = b. f() = c. f() = 0 d. f() = 0 55

58 7.8B CW: Quadratic Formula 1. Factor each epression. a b c d What patterns did you notice? The standard form of a quadratic equation is y = a + b + c; what is the relationship between b & c in the problems above? If the c term is not a then we can manipulate the equation and create square factors to solve quadratic equations as long as we use the properties of equality. Eample: Equation to solve = 0 Can t make a square factor Move the c term to the other side ( +? )( +? ) = What two numbers (that are also the same) add up to the b term? What is special about this number? What number did we add to the left side of the equation? What do we need to do on the right side of the equation? What is special about this number? ( + )( + ) = ( + )( + ) = Write the factor as a perfect square ( + ) = Take the square root of each side (what do you need to remember?) Solve for (you should have two numbers) 56

59 . Solve = 0 using the same method. 4. Let s try to solve + 7 = 0. It s a little trickier since a is not 1. Equation to solve + 7 = 0 Can t make a square factor Move the c term to the other side Can t make a square factor Divide both sides by a ( +? )( +? ) = What two numbers (that are also the same) add up to the b term? What is special about this number? What number did we add to the left side of the equation? What do we need to do on the right side of the equation? What is special about this number? ( + )( + ) = ( + )( + ) = Write the factor as a perfect square ( + ) = Take the square root of each side (what do you need to remember?) Solve for (you should have two numbers) 57

60 5. Try the same process with a quadratic equation in standard form: a + b + c = 0 6. Find the -intercepts of the following quadratic functions. a. f() = b. f() = c. f() = d. f() =

61 7.8C CW: Quadratic Formula Practice Use the quadratic formula to find the -intercepts. Use the -intercepts to find the line of symmetry and the verte. Write the function in verte form and factored form. f 1. Line of symmetry: -intercepts: Verte: Verte form: Factored form: f Line of symmetry: -intercepts: Verte: Verte form: Factored form: 59

62 Use the quadratic formula to find the solutions = = = = 7 60

63 = = = = 0 61

64 7.9 CW: Curbside Rivalry Carlos and Clarita have a brilliant idea for how they will earn money this summer. Since the community in which they live includes many high schools, a couple of universities, and even some professional sports teams, it seems that everyone has a favorite team they like to root for. In Carlos and Clarita s neighborhood these rivalries take on special meaning, since many of the neighbors support different teams. They have observed that their neighbors often display handmade posters and other items to make their support of their favorite team known. The twins believe they can get people in the neighborhood to buy into their new project: painting team logos on curbs or driveways. For a small fee, Carlos and Clarita will paint the logo of a team on a neighbor s curb, net to their house number. For a larger fee, the twins will paint a mascot on the driveway. Carlos and Clarita have designed stencils to make the painting easier and they have priced the cost of supplies. They have also surveyed neighbors to get a sense of how many people in the community might be interested in purchasing their service. Here is what they have decided, based on their research. A curbside logo will require 48 in of paint A driveway mascot will require 16 ft of paint Surveys show the twins can sell 100 driveway mascots at a cost of $0, and they will sell 10 fewer mascots for each additional $5 they charge 1. If a curbside logo is designed in the shape of a square, what will its dimensions be? A square logo will not fit nicely on a curb, so Carlos and Clarita are eperimenting with different types of rectangles. They are using a software application that allows them to stretch or shrink their logo designs to fit different rectangular dimensions.. Carlos likes the look of the logo when the rectangle in which it fits is 8 inches longer than it is wide. What would the dimensions of the curbside logo need to be to fit in this type of rectangle? As part of your work, write a quadratic equation that represents these requirements. 6

65 . Clarita prefers the look of the logo when the rectangle in which it fits is 1 inches longer than it is wide. What would the dimensions of the curbside logo need to be to fit in this type of rectangle? As part of your work, write a quadratic equation that represents these requirements. Your quadratic equations on the previous two problems probably started out looking like this: ( + n) = 48 where n represents the number of inches the rectangle is longer than it is wide. The epression on the left of the equation could be multiplied out to get and equation of the form + n = 48 If we subtract 48 from both sides of this equation we get + n 48 = 0. In this form, the epression on the left looks more like the quadratic functions you have been working with in previous tasks, y = + n Consider Carlos quadratic equation where n = 8, so = 0. Describe at least two different strategies, other than guess-and-check, you could use to solve the equation. 5. After much disagreement, Carlos and Clarita agree to design the curbside logo to fit in a rectangle that is 6 inches longer than it is wide. What would the dimensions of the curbside logo need to be to fit in this type of rectangle? As part of your work, write and solve a quadratic equation that represents these requirements. 6

66 6. What are the dimensions of a driveway mascot if it is designed to fit in a rectangle that is 6 feet longer than it is wide (see the requirements for a driveway mascot given in the bulleted list above)? As part of your work, write and solve a quadratic equation that represents these requirements. 7. What are the dimensions of a driveway mascot if it is designed to fit in a rectangle that is 10 feet longer than it is wide (see the requirements for a driveway mascot given in the bulleted list above)? As part of your work, write and solve a quadratic equation that represents these requirements. 64

67 Carlos and Clarita are also eamining the results of their neighborhood survey, trying to determine how much they should charge for a driveway mascot. Remember, this is what they have found from the survey: They can sell 100 driveway mascots at a cost of $0, and they will sell 10 fewer mascots for each additional $5 they charge. 8. Write an equation, make a table, and sketch a graph (on the same set of aes) for the price of the driveway mascot for each $5 increment,, in the price. Equation Table number of $5 increments, price of the mascot m() Sketch the Graph 9. Write an equation, make a table, and sketch a graph for the number of driveway mascots the twins can sell for each $5 increment,, in the price of the mascot. (Note: zero $5 increments is a price of $0) Equation Table number of $5 increments, number they can sell, s() Sketch the Graph 65

68 10. Write an equation, make a table, and sketch a graph for the revenue the twins will collect for each $5 increment in the price of the mascot. What is revenue? How do you find this? Equation Table number of $5 increments, revenue, r() Sketch the Graph 11. The twins estimate that the cost of supplies will be $50 and they would like to make $000 in profit from selling driveway mascots. Therefore, they will need to collect $50 in revenue. Find the prices they should charge to collect at least $50 in revenue. 1. What price should the twins charge for each mascot if they want to make the most profit? 66

69 7.9 HW: Curbside Rivalry READY Topic: Finding -intercepts for linear equations 1. Find the -intercept of each equation below. Write your answer as an ordered pair. Consider how the format of the given equation either facilitates or inhibits your work. a. 4y 1 b. y 5 y c. d. y 4 1 e. y 7 6 f. 5 y 10. Which of the linear equation formats above facilitates your work in finding -intercepts? Why?. Using the same equations from question 1, find the y-intercepts. Write your answers as ordered pairs a. 4y 1 b. y 5 y c. d. y 4 1 e. y 7 6 f. 5 y Which of the formats above facilitate finding the y-intercept? Why? 67

70 SET Topic: Solve Quadratic Equations, Connecting Quadratics with Area For each of the given quadratic equations, (a) describe the rectangle the equation fits with. (b) What constraints have been placed on the dimensions of the rectangle? Solve the quadratic equations below

71 GO Topic: Factoring Epressions Write each of the epressions below in factored form B Warm Up Rank the three quadratic equations on ease of solving (e.g. finding the -intercepts). Eplain why you arrived at the rankings you did and find the -intercept of the easiest one. Easy: Medium: Hard:

72 7.10 Warm Up Find the quadratic equation from the given table: f()

73 7.10 CW: Writing Quadratic Functions w/ Matrices Write an eplicit quadratic function for each table of values. 1. f() System of equations 1-1 Matri a= Quadratic Function: check: b= c=. f() System of equations Matri a= Quadratic Function: check: b= c= 71

74 . f() System of equations Matri a= Quadratic Function: check: b= c= 4. f() System of equations Matri a= Quadratic Function: check: b= c= 7

75 7.10 HW: Writing Quadratic Functions w/ Matrices Create a matri for the system of equations that can be used to find the solution. 4y z 0 y 5z y 5z 1. y 4z 1 4 4y z 4 6y 4z 1 We have learned several different methods for solving quadratic equations. Use the most efficient method for each of the equations below. Check your solution(s). Solve for

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77 Homework Checklist Section Pg Problems Checked? 75

78 Module 7 KUDOs: Quadratic Equations Algebra 1 Honors I will be able to Essential Questions: How do I use algebraic epressions to model and solve problems? What are the limits of mathematical modeling? Know: (understand the meaning of these terms, memorize the formulas) o Integer o coefficient o rational number o factor o radical o term o irrational number o quadratic formula o epression o function o o o o roots, zeroes, solutions verte line of symmetry parabola Understand: Big ideas B: beginning of the unit L: right after the lesson T: before the unit test I understand... Not yet Got it Understand that polynomials are similar to integers and can be added, subtracted, multiplied, and divided Quadratic functions have a distinct pattern of growth and can be interpreted as the product of two linear factors Homework problems that apply, tet pages, Do B: beginning of the unit L: right after the lesson T: before the unit test I can do the following... Not yet Got it Factor quadratic epressions Homework problems that apply, tet pages, questions I have Use the properties of eponents to simplify or compare epressions with rational eponents Solve quadratic equations: by inspection (e.g., for = 49), taking square roots, the quadratic formula, [completing the square], and factoring Graph quadratic equations and identify important features (intercepts, maimum, minimum, line of symmetry, verte, stretch, domain, range) Write quadratic functions in standard form, verte form, and intercept form and describe features highlighted by each form Create the standard form of a quadratic equation using a matri. (solve a system of equations with variables) [graphing calculator] 76