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Intermediate Algebra Unit 7: Applications of Quadratic Equations
Intermediate Algebra
Unit 7: Applications of Quadratic Equations
Intermediate Algebra Unit 7: Applications of Quadratic Equations
Objectives: page
GUIDE SI-IEET: the quadratic formula & methods to solve quadratic equations 2
solving quadratic equations using the Quadratic Formula 3-4
coin fountain project 5-8
quadratic functions practice 9-12
curve fitting with quadratic functions 13-15
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Intermediate Algebra LMJ: Applications of Quadratic Equations
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T H E QUADRATIC FORMULA
Given: ax^ + bx + c = 0
The coefficients (number values) a, b, and c can be used to find the roots of a quadratic equation using:
The Quadratic Formula: _ b ± b 4aC X. —
2a Tvpes of Solutions: . Graphic Representation: Possible Examples:
• 2 real, rational roots 2 x-intercepts x = 72, x = -3
• 1 real, rational root (2 equal roots) 1 x-intercept x = 4
• 2 real, irrational roots , 2 x-intercepts x = 1± V3
METHODS TO S O L V E QUADRATIC EQUATIONS
SOLUTIONS -> ZEROS = X-INTERCEPTS = FACTORS = ROOTS
1. Graphing: zeros can be found where the quadratic equation crosses the x-axis {x-intercepts)
2. Factoring: use the steps for factoring to break down a quadratic expression into factors
3. Quadratic Formula: use the formula to find the roots of a quadratic equation
Solve usin^ the o^uadratic formula, then cross out the letter pair next to your answer. When rounding so^uare roots or final solutions, p^j^T^J^g^^tiej^eat^^ For each letter pair that you DON'T cross out, write the uppercase letter in the box with the lowercase letter.
W2 + 7x + 6 = 0 a 5b^- lib - 1 2 ^ = 0 ?
a - 7m + 2 = 0 i| 2 h ^ - 5 h - 1 1 = 0
1 S X ^ + 2 X = 8-^3x%a^).-8=o g + 8 = 1 5 n e
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44, - 1 . 2 4
.83, - 2 . 4 3
7 4 a ^ + 9 a + 1 = 0 1 Sfc^ = 2 k + 1 8 - - ^ s i c ^ 7 ^ - . 8 - o
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0 . - - i . l 3 2-t'l3fcH
i 8t2 + 6t = 35>->9tVfet--a...̂ {n)py2 + 7 ^ 2y-^23^z^.v7^ o
11 2 q 2 = 14 - q ^ 2 , i * ^ ' , ^ . o 1 s O.SAT^ - 3A: - 9 . 4 = 0
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EXThA: Can You Stop In Time?
/ When a driver needs to stop a car, the approximate stopping distance d (in feet) is given by the formula:
f ^ ^ ^ B d = 0.05u^ + 2.2u, where u is the speed of the car (in — -j^^H miles per hour). Suppose a car travels 200 feet before
stopping (d = 200). How fast was the car traveling?
Intermediate Algebra
(work space for previous page) UniU: Applications of Quadratic Equations
Intermediate Algebra Unit 7: Applications of Quadratic Equations
Coin Fountain Project:
Warm Up:
Recall how to multiply two binomials using the distributive property (FOIL). Practice with these two problems and share your answers with your partner.
0 ( x + 2 ) ( x - 5 ) 0 ( x - 4 ) ( x - 3 )
© Factoring a trinomial is reversing the process you used in questions 1 & 2. . j ; — '5k«if k Factor: x ^ - S x + 6
(J) Now, simplify the following by multiplying: - .4 (x^ - 4x + 8 )
Coin Fountain Project Objectives:
You have been hired to design the water arc of a coin fountain. The pool of the fountain is 20 feet wide, and the water arc is to be greater than 6 feet tall, but less than or equal to 50 feet tall.
launc h landin
50 ft J] point
water-
You will need to determine the location of the launch point and landing points of the arc. You will need to determine the maximum height of the arc. You will need to write an equation that will describe the water arc in terms of height of the arc in relation to the horizontal distance along the pool.
You will submit two examples of this situation in a complete written presentation. (See the directions sheet for details.)
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Coin Fountain Project Example:
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Intermediate Algebra Unit 7: Applications of Quadratic Equations
Coin Fountain Project Directions:
Read the directions carefully! It's a good idea to worl< in PENCIi For your written presentation, submit two different examples of launch and landing points with all of the questions/algebra completed using the separate answer sheet. YOUR EXAMPLES CANNOT BE THE SAME AS THOSE COMPLETED IN CLASS.
(1) On the answer sheet, begin drawing your fountain in the first quadrant graph. Have the surface of the pool correspond to the x-axis with the left side of the pool at the origin. On the graph, show the coordinates that you have chosen for the launch and landing points (these points are called the roots of the parabola.)
(2) On the answer sheet in the corresponding place, fill in the formula y = a(x - x^X^ ~ ^2) where (x^, O) and (x2 , O) are the roots of the parabola and x is tlie point exactly halfway
between the two roots. Choose a height (y) that meets the requirements of the problem and solve for "a", OR, pick an "a" value that will result in the correct height. (Try to pick numbers where the answers will work out without decimals if possible, but .5 is ok.)
(3) On the answer sheet in the corresponding place, rewrite your equation in the form y = ax^ + bx + c, by using FOIL and the distributive property. (Watch your signs - use the warm-up examples to help you.)
c Type your equation into the y = menu and adjust the window to graph your parabola only in the first quadrant. Use the table or the trace tool to verify your launch and landing points, making sure you created the correct equation.
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sz o JO CO
You can further verify your answer using the table feature OR find the maximum:
a) Use 2"'' T R A C E ^ C A L C MENU
•I b) When it asks for the left bound, move the cursor somewhere just to the left of the maximum point and press ENTER.
c) Repeat for the right bound - move the cursor somewhere just to the right of the maximum point and press ENTER.
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d) When it prompts you to guess, just press ENTER.
e) The maximum point should be given
Complete the graph:
• Include the maximum point and complete the parabola.
• Graph at least four other points (two on each side of the max) for a total of seven points, and connect to form the parabola.
• Clearly label all seven coordinates: for the launch and landing points, the maximum point and all other points on the parabola.
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Intermediate Algebra UnjiZ: Applications of Quadratic Equations
On the answer sheet in the corresponding place, answer the following questions 4 - 5:
(4) After one foot of horizontal distance after your launch point, how high will your fountain be? , Answer this question for both examples provided. Use the table or the trace feature in the calculator to answer this question. Write your answer in a complete sentence.
(5) At what horizontal distances will the fountain be 5 feet high? Use the quadratic formula to answer this question, showing all algebraic work and rounding answers to the nearest tenth.
(6) On the answer sheet in the corresponding place, answer the following questions only once after you've completed your second fountain:
a) What do the x-axis and y-axis represent?
b) What do the roots represent?
c) Which fountain had the more pleasing appearance and why do you think so?
Grading Scale
^ Graphs correctly and neatly labeled (scale, points on the parabola). [6 pts. each] •/ Question #2: correct roots used and a value chosen. [5 pts. each]
Question #3: quadratic equation written, showaii stepsfortuiicredit. [3 pts. each] ^ Question #4: question answered (no work required) [2 pts. each] ^ Question #5: question answered using quadratic formula, show aii steps for fuii credit. [6 pts. each] ^ Question #6: questions (a) - (c) answered in complete sentences. [6 pts.]
Total- 50 points
Neatness countsl Work carefully, show all steps and follow directions to earn maximum points.
Coin Fountain Project - EXTRA CREDIT
This optional piece is only to be completed once you've finished the rest of the project!
(1) Pick one of your examples from the main assignment (or use a new one).
(2) Determine the value of "a" that will produce a 50 ft high arc for your launch and landing point. [4] (3) Create a second arc that intersects the first one and write an equation for the new arc. [5] (4) Complete a new graph for this example and find the point of intersection algebraically,
and clearly label the axes and the points. [6]
Extra Credit - 15 points
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Intermediate Algebra Unit 7: Applications of Quadratic Equations
Quadratic Functions Practice:
Situation: You have been hired to design the water arc of a coin fountain. The pool of the fountain is 30 feet wide. Your fountain must begin 5 feet from the edge of the pool and be the same distance away from the opposite edge of the pool.
Use the information above to answer the following questions: ^(^z.s^ o }
(1) Set up the equation using the formula y = a( x - xi)( x - Xa). Choose a value for "a" that will make the fountain less than 160 feet high, but more than 100 feet high. Find the height of the fountain for the "a" value you picked.
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(2) Write your equation in y = ax^ + bx + c form.
(3) state how high this fountain will be at 12 feet of horizontal distance from the edge of the pool.
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Intermediate Algebra Unit 7: Applications of Quadratic Equations
(4) At what horizontal distance will this fountain be approximately 25 feet high? Showing all . work, use the quadratic formula to solve for the answers. Round to the nearest tenth.
O ^ 'i^x''•t"iSx. -Mi.S X = - b + Vb^ - 4 a c
2a
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(5) Graph your fountain, using a total of 7 points (as done on the project). Be sure to include the roots and the maximum point as 3 of your 7 points. Clearly label your axes for full credit.
1 S C A L E : X-axis : one box = 1
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y-axis : one box = 10
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Intermediate Algebra Unit 7: Applications of Quadratic Equations
Situation: You have been hired to design the water arc of a coin fountain. The pool of the fountain is 40 feet wide. Your fountain must begin 13 feet from the edge of the pool and be the same distance away from the opposite edge of the pool.
(1) Set up the equation using the formula y = a( x - X i ) { x - X2). Choose an //ifeger value for "a" that will make the fountain less than 100 feet high, but more than 50 feet high. Find the height of the fountain for the "a" value you picked.
Use the information above to answer the following questions:
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(2) Write your equation in y = ax^ + bx + c form.
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(3) State how high this fountain will be at 15 feet of horizontal distance from the edge of the pool.
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Intermediate Algebra Unit 7: Applications of Quadratic Equations ^--50
(4) At what horizontal distance will this fountain be approximately'30 feet high? Showing all work, use the quadratic formula to solve for the answers. Round to the nearest hundredth
O- - Z x ^ ' ^ S O y - 7 32- x = - b ± V b ' - 4 a c
2a
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(5) Graph your fountain, using a total of 7 points (as done on the project). Be sure to include the roots and the maximum point as 3 of your 7 points. Clearly label your axes for full credit.
S C A L E : X-axis : one box = 1 y-axis : one box = 10
Intermediate Algebra Unit 7: Applications of Quadratic Equations
Curve Fitting with Quadratic Functions:
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Write a quadratic function that fits each set of data points:
1. (1 ,2 ) , (3 ,34 ) , (-2, 14) 2. (1,9), (2, 9), (3, 5)
3. (0,25), (3, 1), (6,49) 4. (0, 1), (3, 10), (5, 26) b - ' ^ e-<2 I ^ n u m b e r
Jill participated in a study to determine the total time required to bring her car to a stop after she becomes aware of danger. Jill's reaction time includes the time between recognizing danger and applying the brakes plus the time for stopping after applying the brakes. The data table below gives the stopping distance d (in feet) of Jill's car traveling at speed s (in miles per hour) from the time that she notices danger.
V V -> La.
s (mi/h) 14 40
50 202
64 314
5. Write a quadratic function to model the given data.
6. Use your model to predict the stopping distance when the speed is 30 mi/h.—=?«g :^3Q
7. Use your model to find the speed when the stopping distance is 138.8 ft. — ^ <!• - / 3 ^ . S
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Intermediate Algebra Unit 7: Applications of Quadratic Equations
Imagine an experiment conducted on IVIars in winicli an object is propelled vertically from a height ho above the surface of the planet and \N\ih an initial velocity Vq. The height is measured at three different points in time, and the data are recorded in the given table.
8. Write a quadratic function that fits the data.
t (sec) h(ft) 1 19 2 21 3 11
9. Algebraically find the approximate maximum height reached by the object, rounded to the nearest hundredth. » ^ C ^ W ( i ^ ' i ^ 4 - ^ o ( ' ^ > 5 = ^
~ J : S . , " i f _
10. Algebraically find how long will it take for the object to reach its maximum height, rounded to the nearest hundredth.
•h.-=^ % - t f f w e - {.(fi'l 5-Cc
11. Use the features in the calculator to find how long after the object is propelled from its initial height will it take for the object to return to the surface of Mars.
12. Use your function and calculator features to predict the height of the object when t is 2.5.
-". W ^ U t - n - ^ f t
13. Use your function and calculator features to predict the time(s) when the object is at a height of 16 feet.
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Intermediate Algebra Unit 7: Applications of Quadratic Equations
A company keeps track of items produced and profit over several months. Three data points are shown in the given table.
14. Use the data in the table to create a quadratic function that describes the profit as a function of the number of items produced.
^. ^ 2 -Number of items Profit
produced 5 $51 10 $56 15 $11
15. Showing all algebra, use your function to predict the maximum profit possible.
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16. Showing all algebra, use your function to predict the level of production that will maximize profit.
Write a quadratic function that fits each set of given data points:
17. (1,-1) , (2, 5), (3, 13) 18. (1 , 13), (4, 7), (5,-3)
19. (2, 18), (6, 10), (8,-6) 20. (-2,-7), (1,8) , (2, 21)
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2 1 . (0 ,4) , (1,5), (3, 25) 22. (-3, 7), (-1,-5), (6, 16)
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Intermediate Algebra Unit 7: Applications of Quadratic Equations
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