T HE ClN F DUNN-ill] Learning Objective: The study of parabolic curves through the design of water arcs. Overall Project Plan: 'Qbo 0 You have been hired to design the water arc of a coin fountain. You will 0 Students can work alone or with one partner. 0 Due date is posted in Blackboard. 6 design your fountain, show your fountain on a graph, and represent the a '9 equation in different algebraic forms. PART ONE E: ' ' :7 1. Design your fountain by selecting where the water will launch, where it will land, and the height of the water arc. Follow specications below for your launch point, landing point, and the height of your water arc. Launch and Landing Points: You will use the x-y axis to design your fountain. The circular pool of the fountain has a diameter of 20 feet. Let the surface of the pool correspond to the x-axis with the left side of the pool at the origin (0,0). Choose your launch and landing points (aka where the water comes out of the fountain and where the water goes back into the fountain) to meet the following conditions: , the pool at the origin (0,0). Choose your launch and landing points (aka where the water comes out of the fountain and where the water goes back into the fountain) to meet the following conditions: Your launch point and landing points must be in the water (for example you can't launchr'land from the edge of the pool. In other words, your launch point must be greater than 0 and your landing point must be less than 20.) Hint: To make your algebra manageable for this project, choose integer values for your launch and landing point that create an integer for your input value of the parabola's vertex. For example, ifl use a launch point of 2 feet and a landing point of 18 feet, then the input value of the vertex would be half-way between 2 ft. and 18 ft. Hence the input value of the vertex would be 10 feet (10 is an integer). But, if I choose a launch point of 2 feet and a landing point of 7 feet, then I will get an input value of my vertex of 4.5 feet (4.5 is not an integer). If your input value of your vertex is a non-integer, it will magnify the intensity of the algebra for this project. Height of Fountain: Choose the height of your fountain (aka the height ofthe water arc) to meet the following conditions: The height of the water are must be greater than 6 feet, but less than 30 feet. The height of the water are should be non-integer. An example of a height that would meet the specifications is 28.2 feet. (Students should use different height for their project, this is just an example of non-integer value that meet the conditions of the specications.) Submit for Approval: Students must submit their launch point, landing point, and height to Mrs. Vetter for 2. Graph your fountain using the following guidelines: a Place the side view of your fountain in a first quadrant graph. Have the surface of the pool correspond to the x-axis with the left side of the pool at the origin. - Label the coordinates of the rootsfzeros, vertex, and axis of symmetry. - Graph must be hand drawn on graph paperr'graph grid. 3. Find the factored form of the equation of your fountain y = a(x x1)(x X?) where (x1, 0) & (x2, 0) are the roots of the parabola (i.e. your launch and landing points). Using algebra, solve for the a-value (all algebraic work must be shown). Do not round any values throughout this algebraic process. If you round here, this will cause the remainder of your project to be incorrect. Your a-value must be in simplied fraction form (meaning no decimals in the fraction). For example, 28.2 suppose you are getting when solving for your a-value. To simplify this, you must multiply the 64 282 numerator and denominator hv 1n which oivee and then reduce the fraction to lowest terms which 3. Find the factored form of the equation of your fountain y = a(x x1)(x X?) where (x1, 0) & (x2, 0) are the roots of the parabola (i.e. your launch and landing points). Using algebra, solve for the a-value (all algebraic work must be shown). Do not round any values throughout this algebraic process. If you round here, this will cause the remainder of your project to be incorrect. Your a-value must be in simplied fraction form (meaning no decimals in the fraction). For example, 28.2 suppose you are getting when solving for your a-value. To simplify this, you must multiply the 64 282 numerator and denominator by 10, which gives and then reduce the fraction to lowest terms which 640 141 28.2 282 would be . Do not use or for your a-value. 320 64 640 Also, do not use 0.440625 (result of dividing numerator by denominator) for your a-value. Students are only allowed to use the simpiified fraction form of the a-value. (all algebraic work must be shown) Hint: To solve for the a-value, since you already know your vertex from your fountain design, then you can use the vertex to solve for the a-value. We did this back in module 2! Another Hint: If you have found the correct a-value, then if you put your equation into Desmos you will see a parabola that has your exact launch point, landing point, and height. If you are not seeing the equation match up to the launchflandingfheight, then you have made a mistake in your algebraic calculations when nding the a-value. Convert your equation from #3 to transformation (vertex) form, y= a(x - h)2 + k. (there is not much algebraic work to show here, but you should explain your reasoning) Hint: If you put this equation into Desmos, this parabola should show the exact launchflandingfheight which will verify that you have the correct equation. Convert your equation from #4 to standard form y = ax2 + bx + c. All work should be in simplified fraction form (aka no decimals and no rounding). (all algebraic work must be shown) Hint: Use Desmos to check your equation. If it is not producing the exact launchflandingfheight, then you have made a mistake in your algebraic calculations. Using the standard form of your equation (from #5), use the quadratic formula to nd your roots (show all work). Note, if you have completed the previous steps correctly, then you should get the same roots that you started out with when you designed your fountain. Here you are just algebraically demonstrating that the quadratic formula will get you back to the same values you started with when you designed your launch point and landing point. If you are not getting back to the same values that you started out with at the beginning of the project, then there is an algebraic mistake in one of steps. (all algebraic work must be shown) PART TWO In Part 1, you used three forms of the equation; you used the factored form, the transformation form, and the standard form. Explain the pro's and con's of each form of the equation. For example, which form is it easiest to identify the vertex? the roots? the y-intercept? Or sketch a graph? Use complete sentences and correct mathematical language. This part should consist of 3 to 5 sentences. CONCLUSION The conclusion must address each of the following What you enjoyed about the project What parts of the project were challenging for you What you would do differently in completing the project A brief sentence (or two) describing how parabolas are part of our everyday lives This part should consist of 5 8 sentences. CONCLUSION The conclusion must address each of the following What you enjoyed about the project What parts of the project were challenging for you What you would do differently in completing the project A brief sentence (or two) describing how parabolas are part of our everyday lives This part should consist of 5 8 sentences. Project Report Except for your graph (which is hand drawn), your project report must be typed and using appropriate mathematical formatting and notation (equation editor or something similar). All math must be typed! You are NOT going to submit handwritten math, any scanned version of your handwritten math, or pictures of handwritten math. The only part of your project that will not be typed will be the graph. You can take a picture (or scan) of your graph and insert this into the project report. For typing the math, you should NOT type your math as regular text which would look something like this: y = -141!320(x-2)(x-8) y = -141f320(x"2 10x +16) etc. NOT going to submit handwritten math, any scanned version of your handwritten math, or pictures of handwritten math. The only part of your project that will not be typed will be the graph. You can take a picture (or scan) of your graph and insert this into the project report. For typing the math, you should NOT type your math as regular text which would look something like this: y = -141f320(x-2)(x-8) y = -141!320(x"2 10x +16) etc. You SHOULD use equation editor or something similar so that your math is in proper format. Note all students have access to Microsoft word and equation editor as a student at Wake Tech. If you don't know how to use equation editor, please seek out some help. How to Use Eguation Editor in Microsoft Word (opens in new window 141 =__ _2 _ y 32006 )(x 8) 141 2 = x 10x+16 y 320( ) etc. Your report should be in this order 0 a title page with your name(s), date, and title of your fountain You SHOULD use equation editor or something similar so that your math is in proper format. Note all students have access to Microsoft word and equation editor as a student at Wake Tech. If you don't know how to use equation editor, please seek out some help. How to Use Equation Editor in Microsoft Word (opens in new window) 141 (x - 2) (x-8) 320 141 V= (x -10x+16) 320 etc. Your report should be in this order . a title page with your name(s), date, and title of your fountain . all of Part 1 with algebraic work shown to support your answers. Graph is embedded image within the paper (i.e. not attached separately) all of Part 2 with complete sentences . the conclusion