Building Polynomial Functions

Transcription

1 Building Polynomial Functions NAME 1. What is the equation of the linear function shown to the right? 2. How did you find it? 3. The slope y-intercept form of a linear function is y = mx + b. If you ve written the equation in another form, rewrite your equation in slope y-intercept form. 4. Now, factor out the slope, and rewrite the function as y = m( x + b m ). 5. Choose a second linear function and write it in slope y-intercept form. 6. Graph the function on the axis above, and be sure to label it. 7. Rewrite your second function with the slope factored out (just like you did in Question 4). 8. For each function, what does b m represent on the graph? If you let c, then the form y = m( x c) = b m could be called the slope x-intercept form of a linear equation, where c is the x-intercept. The factor theorem states that if c is a root (x-intercept) of a polynomial function, then (x c) must be a factor of that polynomial function. Note that ( x c) is a factor of the expression. The only other factor is the slope m. 9. From their slope y-intercept form, multiply the two functions together. Resources for Teaching Math 2009 National Council of Teachers of Mathematics

2 10. Graph the resulting function on the same axis as the two lines on the previous page. 11. What kind of function did you get? 12. What relationship do you see between the graph from Question 10 and the lines? and the x-intercepts? and the y-intercepts? 13. Identify the left-most x-intercept on the graph. With a straight-edge, cover everything to the right of that point. What connections do you see relating the signs of the y-values? 14. Identify the right-most intercept on the graph. With a straight-edge, cover up everything to the left of that point. What connections do you see relating to the signs of the y-values? Complete the following sentences. 15. When both lines are above the x-axis, the y-values are and the parabola. 16. When both lines are below the x-axis, the y-values are and the parabola. 17. When one line is above the x-axis and the other is below the x-axis, the parabola. y-value OF L 1 y-value OF L PARABOLA IS ABOVE/BELOW THE x-axis Resources for Teaching Math 2009 National Council of Teachers of Mathematics

3 18. Based on the patterns you saw on the previous page, draw a sketch of the quadratic function that would be obtained from the linear expressions of these lines. 19. Write the equation for each line. 20. To check your sketch in Question 18, multiply the expressions together, and graph the resulting function on the grid above. How accurate was your sketch? Resources for Teaching Math 2009 National Council of Teachers of Mathematics

4 Principles to Actions Clear Mathematical Goals for Student Learning Coherent Activities and Problems Aligned With Mathematical Goals Assess and Advance Student Understanding Allow Productive Struggle Facilitate Discourse To Foster Conceptual Understanding and Procedural Fluency Use Mathematical Representations to Support Learning Use Evidence of Student Thinking to Modify and Improve Instruction

5 What would your learning objective be that would allow you to use this activity? What previous knowledge do students need in order to complete this task? What questions would you ask students during the activity to help them make progress? What would you have students do after completing the task? Strength in Numbers: Collaborative Learning in Secondary Mathematics, Ilana Seidel Horn, NCTM, 2012 Strength In Numbers: Selecting And Setting Up A Task Mathematical Goals Prior Knowledge, Knowledge Needed, What Questions Ways To Solve (Student Eays)? What Misconceptions? What Errors? Expectations: Resources/Tools, Classroom Structure, Recording/Reporting Access to ALL, Ensuring Understanding Strength In Numbers: Supporting Students Exploration Questions for: Getting Started/Making Progress Focus Thinking on Key Mathematical Ideas Access Student Understanding Mathematical Ideas Advance Understanding of Mathematical Ideas Encourage All Students to Share Thinking With Others or to Assess Their Understanding of Their Peers Ideas How will you ensure students remain engaged? What assistance will you give or what questions will you ask frustrated groups? What will you do if a group finished immediately? How will you extend the task? What will you do if a student/group focuses on nonmathematical aspects of the task?

6 Strength In Numbers: Sharing and Discussing the Task How will you orchestrate classroom discussion? What solution paths will be shared? What order? Why? How will this help with the goals of the lesson? What specific questions will you ask so that students will: make sense of the mathematical ideas? expand on, debate and question the solutions being shared? What specific questions will you ask so that students will: make connections among the different strategies that are presented? look for patterns? begin to form generalizations? How will you ensure all students have the opportunity to share their reasoning and thinking?

7 Investigating Graphs, Day 1 Perhaps you have enjoyed riding on a Ferris wheel at an amusement park. Carlos and his friends are celebrating the end of the school year at a local amusement park. Carlos has always been afraid of heights, and now his friends have talked him into taking a ride on the Ferris wheel. As Carlos waits nervously in line, he has been able to gather some information about the wheel. By asking the ride operator, he found out that this wheel has a radius of 25 feet with a center 30 feet above the ground. With this information, Carlos is trying to figure out how high he will be at different positions on the wheel. 1. How high above the ground will Carlos be when he is at the very top of the wheel? 2. How high will he be when he is at the very bottom of the wheel? 3. How high will he be when he is at the positions farthest to the left or the right on the wheel? Because the wheel has ten spokes that are evenly spaced, Carlos wonders if he can determine the height of the positions at the ends of each of the spokes as shown in the diagram. Carlos has just finished studying right triangle trigonometry, and wonders if that knowledge can help him. 4. Find the height of each of the points labeled A through J on the Ferris wheel diagram below. Represent your work on the diagram so it is clear to others how you have calculated the height at each point. 1 based on Integrated Math, The Mathematics Vision Project, 2013

8 Carlos has also been carefully timing the rotation of the wheel and has observed the following additional fact: The Ferris wheel makes one complete rotation counterclockwise every 20 seconds. 5. Using this new information, how many degrees does the Ferris wheel rotate per second (angular speed)? 6. How high will Carlos be 2 seconds after passing position A on the diagram? 7. Calculate the height of a rider at each of the following times t, where t represents the number of seconds since the rider passed position A on the diagram. Keep track of any patterns you notice in the ways you calculate the height. As you calculate each height, record the time and height on the diagram as the coordinates (time, height). Elapsed time, t, since passing position A 1 sec Calculations Height of the rider 2 sec 2.5 sec 3 sec 4 sec 5 sec 8. Examine your calculations for finding the height of the rider during the first 5 seconds after passing position A. During this time, the angle of rotation of the rider is somewhere between 0 and 90. Write a general formula for finding the height of the rider during this time interval. 2 based on Integrated Math, The Mathematics Vision Project, 2013

9 9. Calculate the height of a rider at each of the following times t, where t represents the number of seconds since the rider passed position A on the diagram. Keep track of any patterns you notice in the ways you calculate the height. As you calculate each height, record the time and height on the diagram as the coordinates (time, height). 6 sec 8 sec 10 sec 12 sec 15 sec 18 sec 19 sec 20 sec 23 sec 28 sec 35 sec 38 sec 40 sec 10. How might you find the height of the rider in other quadrants of the Ferris wheel, when the angle of rotation is greater than 90? 3 based on Integrated Math, The Mathematics Vision Project, 2013

10 Investigating Graphs, Day 2 Recall the following facts for the Ferris wheel in the previous tasks: The Ferris wheel has a radius of 25 feet. The center of the Ferris wheel is 30 feet above the ground. The Ferris wheel makes one complete rotation counterclockwise every 20 seconds. 1. Based on the data you calculated, as well as any additional insights you might have about riding on Ferris wheels, sketch a graph of the height of a rider on this Ferris wheel as a function of the elapsed time since the rider passed the position farthest to the right on the Ferris wheel. We can consider this position as the rider s starting position at time t = 0. Be sure to include the starting position. 2. Write the equation of the graph you sketched in question 1. 4 based on Integrated Math, The Mathematics Vision Project, 2013

11 3. We began this task by considering the graph of the height of a rider on a Ferris wheel with a radius of 25 feet and center 30 feet off the ground, which makes one revolution counterclockwise every 20 seconds. How would your graph change if: a. the radius of the wheel was larger? or smaller? b. the height of the center of the wheel was greater? or smaller? c. the wheel rotates faster? or slower? 4. Of course, Ferris wheels do not all have this same radius, center height, or time of rotation. Describe a different Ferris wheel by changing at least one of the facts listed above. Description of my Ferris wheel: 5. Sketch a graph of the height of a rider on your Ferris wheel from question 4 as a function of the elapsed time since the rider passed the position farthest to the right on the Ferris wheel. 6. Write the equation of the graph you sketched in question 5. 5 based on Integrated Math, The Mathematics Vision Project, 2013

12 7. How does the equation of the rider s height change if: a. the radius of the wheel is larger? or smaller? b. the height of the center of the wheel is greater? or smaller? c. the wheel rotates faster? or slower? 8. Write the equation of the height of a rider on each of the following Ferris wheels t seconds after the rider passes the farthest right position. a. The radius of the wheel is 30 feet, the center of the wheel is 45 feet above the ground, and the angular speed of the wheel is 15 degrees per second counterclockwise. b. The radius of the wheel is 50 feet, the center of the wheel is at ground level (you spend half of your time below ground), and the wheel makes one revolution clockwise every 15 seconds. 6 based on Integrated Math, The Mathematics Vision Project, 2013

13 Adapting Tasks Jeopardy (Reversibility) 1. Find an equation for a parabola whose graph satisfies the given condition. a) Has a vertex at (-2, -5) b) Has a y-intercept at (0, -6) c) Has x-intercepts at (3,0)and (5,0) d) Has x-intercepts at the origin and ( 4,0) 2. Solution Sets Given below are the graphs of two lines, y= 0.5x+5 and y= 1.25x+8, and several regions and points are shown. Note that C is the region that appears completely white in the graph. For each region and each point, write a system of equations or inequalities, using the given two lines, that has the region or point as its solution set and explain the choice of,, or = in each case. (You may assume that the line is part of each region.) Adapted from Illustrative Mathematics Goal: find and interpret solutions to systems of equations and inequalities. Generalization a b a b log a b log a log b These equations are common misconceptions. Answer these questions about the two equations above: a. Find values for a and b for which this expression is false. b. Find values for a and b for which this expression is true. c. Find another pair of values for each case. d. Find all values for a and b that make the expression true. Goal: understand the meaning of equality and solutions to an equation

14 4. Consider y a x b c, where a, b, and c are constants and a 0. Kyle claims that this equation will always have two roots. Sandy claims that this equation will always have zero roots. a. Create an equation that supports Kyle s claim. Explain your choice in equation. b. Create an equation that supports Sandy s claim. Explain your choice in equation. c. Create an equation that shows both Kyle and Sandy are incorrect. Explain your choice in equation. d. Given conditions on the relationships between a, b, and c for each scenario to be true. Flexibility 5. Find the solution to 4 3x 4 8 using at least three approaches. Goal: Identify different approaches to solve an exponential equation in one variable. 6. Which of the following could be an expression for the function whose graph is shown? Explain. a) (x+12) 2 +4 b) (x - 2) 2-1 c) (x+18) 2-40 d) (x -10) 2-15 e) -4(x+2)(x+3) f) (x+4)(x - 6) g) (x -12)(-x+18) h) (20 - x)(30 - x) Goal: Connect algebraic and graphical representations of a quadratic function. 7. The x-intercepts of y f x are -1, 3, and 6. Find the x-intercepts of (a) y f 2x (b) y 2 f x (c) y f x 2 Goal: Connect properties of functions and transformations.

15 The Beautiful Connection Between Polynomials and Probability Brian Shay, Mathematics Teacher, San Dieguito Union High School Warm Up: Find these products. Feel free to use different representations or procedures! A. (x + 1) 2 B. (x + x 2 ) 2 C. (x + x 2 + x 3 ) 2 D. (x + x 2 ) 3 Part 1 We will be flipping a coin for this set of problems. Heads gets you one point while Tails gets you two points! 1. If you flip a coin twice. What options are there? What s the likelihood for each option? 2. What if we now flip a coin three times and answer the same questions. 3. Do the same process with four flips. What patterns do you notice? Why do you think these patterns are happening? 4. What do you wonder will happen with five flips? How about n flips? Part 2 Let s now move up to a spinner. This spinner has three regions, each equal in size. The regions are labeled 1, 2, and 3, and you earn that number of points for landing in that particular region. 1. Spin twice and sum the results. What options are there? What s the likelihood for each option? 2. Now do the same process with three spins and answer the same questions. 3. If we spun four times, what is the most likely sum we would get? What might be an efficient way of tracking and calculating these probabilities?

16 Part 3 A. Let s turn our attention to the standard six-sided die. 1. Create a polynomial representing the outcomes possible when rolling a six-sided die once. 2. We all know that rolling a sum of 7 is the most likely result when rolling two dice. Verify that using a polynomial. B. Instead of summing the values on the face, score as follows: two points for rolling a multiple of three and 1 point for not rolling a multiple of three. 3. What is the most likely result if we play this new game rolling twice and track the sum? Part 4 Let s now move back to a spinner. This spinner still has three equally sized regions, yet they are now labeled with the values 3, 4, and Spin twice and sum the values. What options are there? What s the likelihood for each option? 2. What would be the most likely sum if we spun three times? What is the probability of getting this sum? 3. How would you go about finding these probabilities if we spun four times? Part 5 We go purchase a new spinner, pictured here on the right. If the spinner lands in Region A, you earn 1 point, if it lands in Region B, you earn 11 points, and if it lands in Region C, you earn 5 points. 1. We spin twice and sum the points we earn. Find the polynomial to represent this game. What is the most likely sum? 2. What is the most likely sum if the spinner is spun three times?