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Module 1 – Quadratic Functions
Classroom Task: 1.1 Something to Talk About – A Develop Understanding Task An introduction to quadratic functions, designed to elicit representations and surface a new type of pattern and change (F.BF.1, A.SSE.1, A.CED.2) Ready, Set, Go Homework: Quadratic Functions 1.1 Classroom Task: 1.2 I Rule – A Solidify Understanding Task Solidification of quadratic functions begins as quadratic patters are examined in multiple representations and contrasted with linear relationships (F.BF.1, A.SSE.1, A.CED.2) Ready, Set, Go Homework: Quadratic Functions 1.2 Classroom Task: 1.3 Scott’s Macho March – A Solidify Understanding Task Focus specifically on the nature of change between values in a quadratic being linear. (F-‐BF, F-‐LE) Ready, Set, Go Homework: Quadratic Functions 1.3 Classroom Task: 1.4 Rabbit Run – A Solidify Understanding Task Focus on maximum/minimum point as well as domain and range for quadratics (F.BF.1, A.SSE.1, A.CED.2) Ready, Set, Go Homework: Quadratic Functions 1.4 Classroom Task: 1.5 Look out Below – A Solidify Understanding Task Examining quadratic functions on various sized intervals to determine average rates of change (F.BF.1, A.SSE.1, A.CED.2) Ready, Set, Go Homework: Quadratic Functions 1.5 Classroom Task: 1.6 Tortoise and Hare – A Solidify Understanding Task Comparing quadratic and exponential functions to clarify and distinguish between type of growth in each as well as how that growth appears in each of their representations (F.BF.1, A.SSE.1, A.CED.2, F.LE.3) Ready, Set, Go Homework: Quadratic Functions 1.6 Classroom Task: 1.7 How Does it Grow – A Practice Understanding Task Incorporating quadratics with the understandings of linear and exponential functions (F.LE.1, F.LE.2, F.LE.3) Ready, Set, Go Homework: Quadratic Functions 1.7
Set Topic: Recognizing linear, exponential, and quadratic equations. In each set of 3 functions, one will be linear and one will be exponential. One of the three will be a new category of function. List the characteristics in each table that helped you to identify the linear and the exponential functions. What are some characteristics of the new function? Find an explicit and recursive equation for each. 8. Linear, exponential, or a new kind of function. A. x f x( ) 7 128 8 256 9 512 10 1024 11 2048 Characteristics? Explicit equation: Recursive equation:
B. x f x( ) 7 49 8 64 9 81 10 100 11 121 Characteristics? Explicit equation: Recursive equation:
C. x f x( ) 7 13 8 15 9 17 10 19 11 21 Characteristics? Explicit equation: Recursive equation:
F. x f x( ) -‐2 9 -‐1 6 0 5 1 6 2 9 Characteristics? Explicit equation: Recursive equation:
9. Graph the functions from the tables in #8. Add any additional characteristics you notice from the graph. Place your axes so that you can show all 5 points. Identify your scale. Write your explicit equation above the graph. A. equation:
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18. George is loading freight into an elevator. He notices that the weight limit for the elevator is 1000 lbs. He knows that he weighs 210 lbs. He has loaded 15 boxes into the elevator. Each box weighs 50 lbs. Identify the rate of change for this situation.
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Ready, Set, Go!
Ready Topic: Adding and multiplying binomials Add 1. 2𝑥 + 7 + 5𝑥 + 3 2. 6𝑥 − 1 + 𝑥 − 10 3. 8𝑥 + 3 + 3𝑥 − 4 4. −5𝑥 + 2 + 7𝑥 − 13 5. 12𝑥 + 3 + −4𝑥 + 3 Multiply. (Use the distributive property.) 6. 2𝑥 + 7 5𝑥 + 3 7. 6𝑥 − 1 𝑥 − 10 8. 8𝑥 + 3 3𝑥 − 4 9. −5𝑥 + 2 7𝑥 − 13 10. 12𝑥 + 3 −4𝑥 + 3 11. 𝑥 + 5 𝑥 − 5 12. Compare your answers in problems 1 – 5 to your answers in problems 6 – 10 respectively. In problems 1 – 5 you were asked to add the binomials and in problems 6 – 10 you were asked to multiply them. Look for a pattern in the answers. How are they different? 13. The answer to #11 is a different “shape” than the answers in 6 – 10, even though you were multiplying. Explain how it is different from the other products. Try to explain why it is different. Think of 2 more examples of multiplication of two binomials that would do the same thing as #11. 14. Try adding the two binomials in #11. 𝑥 + 5 + 𝑥 − 5 =________________ Is this answer a different “shape” than the answers in problems 1 – 5 ? Explain.
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21. Compare the perimeter to the area in each of problems (15 – 20). What do the perimeters and areas have in common? In what way are the numbers and units in the perimeters and areas different?
Go Topic: Greatest Common Factor (GCF) Find the GCF of the given numbers. 22. 15abc2 and 25a3bc 23. 12x5y and 32x6y 24. 17pqr and 51pqr3 25. 7x2 and 21x 26. 6x2, 18x, and -‐12 27. 4x2 and 9x 28. 11x2y2, 33x2y, and 3xy2 29. 16a2b, 24ab, and 16b 30. 49s2t2 and 36s2t2
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1.3 Scott’s Macho March A Solidify Understanding Task
After looking in the mirror and feeling flabby, Scott
decided that he really needs to get in shape. He joined a
gym and added push-ups to his daily exercise routine. He
started keeping track of the number of push-ups he
completed each day in the bar graph below, with day one
showing he completed three push-ups. After four days, Scott was certain he can continue this
pattern of increasing the number of push-ups for at least a few months.
1 2 3 4
1. Model the number of push-ups Scott will complete on any given day. Include both explicit and recursive equations.
Scott’s gym is sponsoring a “Macho March” promotion. The goal of “Macho March” is to raise money for charity by doing push-ups. Scott has decided to participate and has sponsors that will donate money to the charity if he can do a total of at least 500 push-ups, and they will donate an additional $10 for every 100 push-ups he can do beyond that. 2. Estimate the total number of push-ups that Scott will do in a month if he continues to increase
the number of push-ups he does each day in the pattern shown above.
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Ready, Set, Go! Ready
Topic: Fundamental Theorem of Arithmetic
The prime factorization of a number is given. Multiply each number to find the whole number that each factorization represents. 1. 2!×3!×5! 2. 3!×5!×7! 3. 5!×11!×13! The following problems are factorizations of numerical expressions called quadratics. Given the factors, multiply to find the quadratic expression. Add the like terms. Write the x2 term first, the x-‐term second, and the constant term (term without an x) last. 4. 𝑥 + 5 𝑥 − 7 5. 𝑥 + 8 𝑥 + 3 6. 2 𝑥 − 9 𝑥 − 4 7. 3 𝑥 + 1 𝑥 − 4 8. 2 3𝑥 − 5 𝑥 − 1 9. 2 5𝑥 − 7 3𝑥 + 1
Set Use first and second differences to identify the pattern in the tables as linear, quadratic, or neither. Write the recursive equation for the patterns that are linear or quadratic. 9. Pattern?
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Ready, Set, Go!
Ready Topic: applying the slope formula Calculate the slope of the line between the given points. Use your answer to indicate which line is the steepest. 1. A (-‐3, 7) B (-‐5, 17) 2. H (12, -‐37) K (4, -‐3) 3. P (-‐11, -‐24) Q (21, 40) 4. R (55, -‐75) W(-‐15, -‐40)
Set Topic: Investigating perimeters and areas
Adam and his brother are responsible for feeding their horses. In the spring and summer the horses graze in an unfenced pasture. The brothers have erected a portable fence to corral the horses in a grazing area. Each day the horses eat all of the grass inside the fence. Then the boys move it to a new area where the grass is long and green. The porta-‐fence consists of 16 separate pieces of fencing each 10 feet long. The brothers have always arranged the fence in a long rectangle with one length of fence on each end and 7 pieces on each side making the grazing area 700 sq. ft. Adam has learned in his math class that a rectangle can have the same perimeter but different areas. He is beginning to wonder if he can make his daily job easier by rearranging the fence so that the horses have a bigger grazing area. He begins by making a table of values. He lists all of the possible areas of a rectangle with a perimeter of 160 ft., while keeping in mind that he is restricted by the lengths of his fencing units. He realizes that a rectangle that is oriented horizontally in the pasture will cover a different section of grass than one that is oriented vertically like this. So he is considering the two rectangles as different in his table.
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5. Fill in Adam’s table with all of the arrangements for the fence. (The first one is done for you.) Length in “fencing” units
Width in “fencing” units
Length in ft. Width in ft. Perimeter (ft) Area (ft)2
1 unit 7 units 10 ft 70 ft 160 ft 700 ft2
2 units 160 ft
3 units 160 ft
4 units 160 ft
5 units 160 ft
6 units 160 ft
7 units 160 ft
6. Discuss Adam’s findings. Explain how you would rearrange the sections of the porta-‐fence so that Adam will be able to do less work. 7. Make a graph of Adam’s investigation. Let length be the independent variable and area be the dependent variable. Label the scale. 8. What is the shape of your graph? 9. Explain what makes this function be a quadratic.
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14. 15. 16. Examine the graph at the right on the interval (0, 1). Which function do you think is growing faster? 17. Now focus on the interval (2 , 3). Which function is growing faster in this interval?
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Ready, Set, Go!
Ready Topic: Evaluating exponential functions Find the indicated value of the function for each value of x. x = −2,−1,0,1,2,3{ } 1. f x( ) = 3x 2. g x( ) = 5x
3. h x( ) = 10x 4. k x( ) = 12
⎛⎝⎜
⎞⎠⎟x
5. m x( ) = 13
⎛⎝⎜
⎞⎠⎟x
Set The Sears Tower in Chicago is 1730 feet tall. If a penny were let go from the top of the tower, the position above the ground s(t) of the penny at any given time t would be s t( ) = −16t 2 +1730 .
6. Fill in the missing positions in the chart. Then add to get the distance fallen.
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7. How far above the ground is the penny when 7 seconds have passed?
8. How far has it fallen when 7 seconds have passed?
9. Has the penny hit the ground at 10 seconds? Justify your answer.
10. The average rate of change of an object is given by the formula r = dt, where r is the rate of
change, d is the distance traveled, and t is the time it took to travel the given distance. We often use some form of this formula when we are trying to calculate how long a trip may take. If our destination is 225 miles away and we can average 75 mph, then we should
arrive in 3 hours. 225 mile75 mph
= 3 hours⎡
⎣⎢
⎤
⎦⎥ In this case you would be rearranging the formula
so that t = dr.
However, if your mother finds out that the trip only took 2 ½ hours, she will be upset. Use the rate formula to explain why.
11. How is the slope formula like the formula for rate? m = y2 − y1x2 − x1
12. Find the average rate of change for the penny on the interval [0, 1] seconds.
13. Find the average rate of change for the penny on the interval [6, 7] seconds.
14. Explain why the penny’s average speed is different from 0 to 1 second than between the 6th
and 7th seconds.
15. What is the average speed of the penny from [0,10] seconds?
16. What is the average speed of the penny from [9,10] seconds?
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17. Find the first differences on the table where you recorded the position of the penny at each second. What do these differences tell you?
18. Take the difference of the first differences. (This would be called the 2nd difference.) Did your answer surprise you? What do you think this means?
Go Topic: Evaluating functions 19. Find f 9( ) given that f x( ) = x2 +10. 20. Find g −3( ) given that g x( ) = x2 + 2x + 4. 21. Find h −11( ) given that h x( ) = 2x2 + 9x − 43. 22. Find r −1( ) given that r x( ) = −5x2 − 3x + 9.
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Ready, Set, Go!
Ready Topic: Recognizing functions Identify which of the following representations are functions. If it is NOT a function state how you would fix it so it was. 1. D = {(4,-‐1) (3, -‐6) (2, -‐1) (1, 2) (0, 4) (2, 5)} 2. The number of calories you have burned
since midnight at any time during the day.
3.
4. x -‐12 -‐8 -‐6 -‐4 f(x) 25 25 25 25
5.
6.
Set Topic: Comparing rates of change in linear, quadratic, and exponential functions
The graph at the right shows a time vs. distance graph of two cars traveling in the same direction along the freeway. 7. Which car has the cruise control on? How do you know? 8. Which car is accelerating? How do you know?
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9. Identify the interval in figure 1 where car A seems to be going going faster than car B. 10. Identify the interval in figure 1 where car B seems to be going faster than car A. 11. What in the graph indicates the speed of the cars? 12. A third car C is now shown in the graph (see figure 2). All 3 cars have the same destination. If the destination is a distance of 12 units from the origin, which car do you predict will arrive first? Justify your answer.
Go Topic: Identifying domain and range from a graph. State the domain and range of each graph. Use interval notation where appropriate. 13. domain ____________ range ____________ 14. domain ____________ range ____________ 15. domain ____________ range ____________ 16. domain ____________ range ____________
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17. domain ____________ range ____________ 18. domain ____________ range ____________ 19. Describe the domain and range in figure 1 and figure 2. domain ____________ range ____________ domain ____________ range ____________
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Ready, Set, Go!
Ready
Topic: transforming lines 1. Graph the following linear equations on the grid.
(See figure 1.) The equation y = x has been graphed for you. For each new equation explain what the number 3 does to the graph of y = x. Pay attention to the y-‐intercept, the x-‐intercept, and the slope. Identify what changes in the graph and what stays the same.
a.) y = x + 3
b.) y = x – 3
c.) y = 3x
2. The graph of y = x is given. (See figure 2.) For each equation predict what you think the number -‐2 will do to the graph. Then graph the equation.
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a.) y = x + (-‐2) Prediction:
b.) y = x – (-‐2) Prediction:
c.) y = -‐2x Prediction:
Set Topic: Distinguishing between linear, exponential, and quadratic functions For each relation given:
a. Identify whether or not the relation is a function. (If it’s not a function, omit b – d.) b. Determine if the function is Linear, Exponential, Quadratic or Neither. c. Describe the type of growth. d. Express the relation in the indicated form.
3. I had 81 freckles on my nose before I began using vanishing cream. After the first week I had 27, the next week 9, then 3 . . .
Make a graph. Label your axes and the scale. Show all 4 points.
a. Function? b. Determine if the function is Linear,
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Go Match the function on the left with the equivalent function on the right. 7. f x( ) = −2x + 5 a. f x( ) = 5 2( )x 8.
b.
9. I put $7000 in a savings account that pays 3% interest compounded annually. I plan to leave it in the bank for 20 years. The amount I will have then.
c. f (1) = 2; f (n +1) = f (n)+ 2n
10. The area of the triangles below.
d.
11. f 0( ) = 5; f n( ) = 2∗ f n −1( ) e. y + x = 0
12. f 0( ) = 5; f n( ) = f n −1( )− 2 f. 𝑦 = 𝑥 − 1 𝑥 + 3
13. x -‐7.75 -‐¼ ½ 11.6 f (x) 7.75 ¼ -‐½ -‐11.6