Algebra Project
Name _________________________________ I.D. Number _______________________ Project 2 Evaluation 32 Precalculus: Analytic Geometry & Algebra (MTHH 043 060) Be sure to include ALL pages of this project (including the directions and the assignment) when you send the project to your teacher for grading. Don’t forget to put your name and I.D. number at the top of this page! This project will count for 10% of your overall grade for this course. Be sure to read all the instructions and assemble all the necessary materials before you begin. You will need to print this document and complete it on paper. Feel free to attach extra pages if you need them. To earn full credit, you must justify your solutions by showing your work. When you have completed this project, you may submit it electronically through the online course management system by scanning the pages into either .pdf (Portable Document Format), or .doc (Microsoft Word document) format. If you scan your project as images, embed them in a Word document in .gif image format. Using .gif images that are smaller than 8 x 10 inches, or 600 x 800 pixels, will help ensure that the project is small enough to upload. Remember that a file that is larger than 5,000 K will NOT go through the online system. Make sure your pages are legible before you upload them. Remember that showing your work in arriving at solutions is required for all problems. All graphs must be done on the grids provided or on graph paper. Computer-generated graphs will not be acceptable. Methods learned in this course should be used when solving the problems on this project. Part A: Procedural Skill & Fluency Determine which of the following are linear equations. If an equation is not linear, indicate where it contradicts the definition. 1. 4x = y – 5 2. 4y – 2 = x² Project 2 MTHH 043 3 Find the x and y intercepts of the equation, 3x + 2y = 6, and sketch its graph. Describe and sketch the regions defined by the following inequalities. 4. y ≥ −½x 5. −2x + 2y < 5 Project 2 MTHH 043 6. Sketch the graph of f ( x ) = –x2 – 2x + 3 by locating the x and y intercepts and the extreme point of the parabola. Use the graph to determine the range. List the vertex and tell whether it is a maximum or a minimum. Find the solution sets of the inequalities by graphing the corresponding quadratic functions. 7. − x2 − 2x + 3 0 8. (x+2)2 – 4 0 Project 2 MTHH 043 9. Project 2 Sketch the graph of y -x 2 – 2 . MTHH 043 Name the conic section described by each of the following equations. Include the final equation in standard conics form and your work in arriving at it. Sketch the graphs – for full credit. 10. x2 – y + 2x + 3 = 0 Final Equation: Conic Name: Vertex: Focus: Directrix: 11. 3×2 + 5y2 – 12x + 30y + 42 = 0 Final Equation: Conic Name: Center: Length of Major Axis: Length of Minor Axis: Project 2 MTHH 043 12. 4y2 – 9×2 – 8y + 54x – 113 = 0 Final Equation: Conic Name: Vertices: Center: Asymptotes: 13. x2 + y2 + 4x + 2y – 20 = 0 Final Equation: Conic Name: Center: Radius: Project 2 MTHH 043 Write the standard form equation for each conic section. Include your work. 14. The parabola with focus (5, 4) and vertex (3, 4). 15. The ellipse with center (5, −3) , vertices (5, 1) and (5, −7), and with the length of the minor axis equal to 4. 16. The hyperbola with center (−2, 3), vertices (1, 3) and (−5, 3) , and foci (3, 3) and (−7, 3). 17. The circle with center (−3, 2) that passes through the point (1, 2). Part B: Applications 1. The cash flow per share for Timberland Co. was $0.18 in 2005 and $3.65 in 2010. Write a linear equation in slope-intercept form that gives the cash flow per share in terms of the year. Let 𝑡 = 0 represent 2005. 2. A hockey puck leaves the blade of a hockey stick, bounces off a wall, and travels in a new direction, as shown. Your friend claims the path of the puck forms a right angle. Is your friend correct? Explain. Project 2 MTHH 043 3. The metabolic rate of an ectothermic organism increases with increasing temperature within a certain range. The oxygen consumption 𝐶 of a beetle at certain temperatures 𝑥 can be approximated by 𝐶 = 0.45𝑥 2 − 1.65𝑥 + 50.75, where 𝑥 is the temperature in degrees Celsius. If the oxygen consumption is 150 microliters per gram per hour, what is the air temperature? 4. After being shot into the air, the height of a rocket is ℎ(𝑡) = 80𝑡 − 16𝑡 2, where ℎ is the height in feet above the ground after 𝑡 seconds. a) After how many seconds will the rocket be at a height of 96 feet? b) What is the maximum height the rocket will reach? 5. Competitors in the 10-meter platform diving competition jump upward and outward before diving into the pool below. The height ℎ of a diver in meters above the pool after 𝑡 seconds can be approximated by the equation ℎ(𝑡) = −4.9𝑡 2 + 3𝑡 + 10. When will the diver hit the water? 6. If an elliptical whispering gallery is 80 feet long and 25 feet wide, how far from the center of room should someone stand on the major axis of the ellipse to experience the whispering effect? Round to two decimal places. 7. The orbits of planets around the sun are approximately elliptical with the sun as a focus. The aphelion is a planet’s greatest distance from the sun and the perihelion is its shortest. The length of the major axis is the sum of the aphelion and the perihelion. Earth’s aphelion is 94.51 million miles and its perihelion is 91.40 million miles. Write an equation for Earth’s orbit. Project 2 MTHH 043 8. A bedroom is 3 feet longer than it is wide and has an area of 154 square feet. Find the dimensions of the room. 9. The formula 𝑑 = 0.05𝑠 2 + 1.1𝑠 estimates the minimum stopping distance 𝑑 in feet for a car travelling 𝑠 miles per hour. If a car stops in 200 feet, what is the approximate speed the car was travelling when the driver applied the brakes? 10. The cooling tower for a power plant has sides in the shape of a hyperbola. The tower stands 167.082 meters tall. The diameter at the top is 60 meters. At their closest, the sides of the tower are 40 meters apart. Find an equation that models the sides of the cooling tower. Part C: Conics Graphic Organizer Create a graphic organizer that illustrates the 4 conics we learned about in lesson 9, their graphs, and their equations. Your graphic organizer should include each of the conics, a picture with parts labeled, an actual real world picture of the conic-for example a bridge, the equation, other important features (i.e. foci), and a worked through example. You can create 1 graphic organizer for each conic OR you can create 1 graphic organizer that includes all 4 of the conics. Organize all the information neatly, in a way that makes sense to you. Project 2 MTHH 043 Part D: Logo Equations 1. Find a logo that contains one pair of either parallel or perpendicular lines. Put the actual logo image in a coordinate plane and bold the pair of lines you are using. Find 2 points on each line that you will use and bold those as well. Write 2 equations for each lines: slopeintercept form and point-slope form. Use slope to show that the lines are either parallel or perpendicular. 2. Find a second logo that contains either a circle, ellipses, or hyperbola. Put the actual logo image in a coordinate plane and bold the conic you will be using. Find the key features of the conic (i.e. center, foci, directrix) and write an equation. 3. Find a third logo that contains either a circle, ellipses, or hyperbola. It can not be the same conic shape you used for your second logo. Put the actual logo image in a coordinate plane and bold the conic you will be using. Find the key features of the conic (i.e. center, foci, directrix) and write an equation. Declaration I know that plagiarism is wrong. Plagiarism is to use another’s work and pretend that it is one’s own. I declare that this is my own work. Signature _________________________________ Student I.D. number ________________ Project Submission This project can be submitted electronically. Access the online version of your course and follow the directions provided for Project 2. (Go to Project 2 in Unit 3 on the navigation panel.) For mail submission directions, refer to the enrollment information included with your print materials. Project 2 MTHH 043
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