- 1. Designed and developed by Isaac Benning and Nathaniel Ahaja under the supervision of D.D. Agyei – University of Cape Coast, Ghana. Teacher support materials Topic Quadratics in Polynomial Form : y = ax 2 + bx + k School level SHS 2 Curriculum area Elective Mathematics Class time 80 min ( approximately 2 periods) Teachers’ Guide In this lesson: Quadratics in Vertex Form: y = ax 2 + bx + k, you are provided with three different support materials: Teachers’ guide, the lesson material and student worksheet including student assignment. The activities in the lesson material are planned hand-in- hand with the student worksheet. The teachers’ guide provides an overview of the lesson and step-by-step support to set up the lesson. Before conducting this lesson, be sure to read through this guide thoroughly, and familiarize yourself with the activities in the lesson plan. Introduction When students are introduced to solving quadratic equations in SHS 1, they often don't identify connections between different quadratic function types. Each function has its own mysterious parameters. The behaviour of each graph is learned in isolation. Through the following investigations, students will compare the graph of quadratic equation in the polynomial form, alter their parameters and observe the changes in the numerical data and in the graphical representation. They then are asked to analyse the results, and to form conclusions about quadratic functions in polynomial form. Objectives: The student will: • determine how changes in the parameters of a quadratic equation in the polynomial form affects its graph. • determine how to use the polynomial form of a polynomial quadratic equation to find the location of the vertex on a graph. • apply the vertex of a quadratic function in a realistic setting. Prerequisite Knowledge Students are able to: - determine a linear equation from a given set of data. - evaluate and simplify an expression with variables. - locate the vertex of a parabola on a graph. - substitute values into a given relation to find an unknown variable. Resources • This lesson assumes that your classroom has only one computer, from which you can teach. The presence of a projector is an advantage. For classrooms with enough computers for all your students either working individually or in small groups, this lesson can still be adapted. • Spreadsheet such as Excel. • Pencil, paper, and calculator.
- 2. • Copies of the worksheet for each student or small group of students. Setting up the data and plot windows This guide gives you opportunity and support to utilise ICT for the whole class teaching in generating and analysing quadratic patterns. As the instructor, your core task in the lesson execution is to set up the lesson environment and facilitate activities. The following instructions will give you step-by-step directions in preparing quadratic graphs for demonstration in Excel. You may want to bookmark the activity pages for your students. If you like, make copies of the worksheet for each student. Instructions 1. Before you conduct this lesson in a spreadsheet, it is important that you know some basic use of the spreadsheet. (i.e. entering data, writing formulas or functions, copying formulas and formatting etc.) 2. Any spreadsheet will allow you to input numerical data and then view a plot of these data. Most spreadsheets can display the table and the graph onscreen at the same time. This allows you to experiment with changing values in the table and observing the results in the plot. 3. In Microsoft Excel 5.0 the table and plot can be set up in separate windows as shown. • If the program supports displaying two windows side by side, set the spreadsheet up as follows: a. With 1 window opened, open a second window. Then select the command which will tile with a vertical split. b. Type data in the left window. c. Highlight the cells that contain the data and use the Chart command to create an X-Y Scatter plot in the second window. Make sure to choose to use the data from column 1 as X-data. • If your spreadsheet does not display two windows, you may be able to paste the plot into the spreadsheet. Alternatively, you can toggle back and forth between the plot and table displays. Quadratic Functions - Working with y = ax2+bx+k 4. Input your x-values you would use in plotting your quadratic graph beginning from cell X1 downwards. (You could choose any other column eg. D or E, etc. Be sure you make enough columns for all the variables you may need) 5. Make up an equation in the form y =a*(X1)^2+b*(X1)+k, and enter the formula in cell Y1 (or in the first cell of the next column you chose). Then use the Fill Down command. Note the values of a, b, and k are stored in W5, W6 and W7.
- 3. 6. Set the cursor over cell Y1 to note the formula. You should see: = a*(X1)^2+b*(X1)+k. (The ^ symbol is used for exponents in a spreadsheet and the * symbol must be used for multiplication.) 7. Try altering the values of a, b, and k. It is difficult to see how the parabola changes because the scale automatically adjusts. Therefore, set a = 1, b = 0 and k = 0 and proceed to execution of the lesson in the lesson plan. (It is important to set up this before the lesson begins.) Lesson Plan LESSON 1: Quadratics in Polynomial Form : y = ax 2 + bx + k (Double Lesson) Lesson plan and timing Activity Approximate time (in minutes) Introduction 10 Execution of the lesson 60 Conclusion 05 Ending the lesson 05 Total 80 Introduction (10 minutes) The graph of the function y = ax 2 + bx + k has several properties. In this activity, you will examine how the shape of the parabola changes as the values of a, b, and k are modified. You will also determine how this equation will help you find x- and y-coordinates of the vertex on the graph as well as the axis of symmetry. You will be introduced to a realistic application of this function illustrating how these properties are useful. Begin the lesson with a real-world example of a golf activity on a park. In this scenario ask students to describe the path of the ball when it is hit at one end of a park to the other. Try to lead students to identify that the movement of the ball is parabolic as shown in the diagram below and help them to link this to a quadratic curve. 40m/s 300 In the second scenario, ask students to predict the path of a ball thrown vertically up and help them to realise that real world phenomena can be modelled with mathematical functions. Prepare students for the following activities (activities: 1.0 – 3.0) by organizing them in small groups (2-3 students per group). Assign specific roles to members in the group e.g. presenter, recorder and chairperson. Execution of the lesson (60 min) Activity 1 : The shape of the parabola Prepare the graph y = ax 2 + bx + k by setting a =1, b= 0 and k=0 before beginning the lesson on an overhead project. By organizing students in small groups (2-3 students per group), guide them to observe and describe the changes as various parameters of the quadratic equation are altered. 1.1 Varying the value of a (set b=0 and k=0) • Direct each group to observe and describe the changes in the graph as we increase the value of a systematically on the spreadsheet (eg. Set a = 2, 4, 8, 10, 20, 40). Get students to record the observations in the students’ worksheet in their groups (it is necessary that the teacher present the different graphs on the same sheet to help bring out the concept). • Set the values of a in the reverse order (40, 20, 10, 8, 4, 2, 1, 0) and get students to record their
- 4. observations. • Get students to observe how the graph changes when a changes from positive to negative numbers. Set the value of a to be zero and continue decreasing the value of a to negative numbers. • Guide groups to compare their observation notes and note down their differences. • Ask representatives of few groups to report the results to the whole class. • Discuss group results with students. (Verify results by graphical representations on the spreadsheet if necessary). Some discussion points could be: i. Increasing the value of a makes the parabola steeper (narrower). ii. Decreasing the value of a flattens (widens) out the parabola. iii. When the value of a = 0 we have a line on the x-axis (connection between quadratic and linear functions). iv. As we continue to decrease the value of a through negative values, the parabola opens downwards and gets steeper (narrower). 1.2 Varying the value of k (set a = 1 and b = 0) Repeat the process for the activity by varying the value of k and guide students to observe and record changes on graph. Some discussion points could be: i. Increasing the value of k moves the graph up (vertically) without altering the shape. ii. Decreasing the value of k moves the graph down (vertically) without altering the shape. 1.3 Varying the value of b (set a = 1 and k = 0) Set a = 1 and k = 0 then vary the value of b as follows: • To increase (2,4,6,8,10 and 20) • To zero • To decrease (-2,-4,-6,-8 and -10). Have students to observe and record the values of the x-coordinate to help predict the effect on the parabola and the path the vertex follow as b varies. Some discussion points could be: Varying the value of b affects the position of the graph as follows: • If b>0, the vertex is located on the left of the y-axis. • If b<0, the vertex is located on the right of the y-axis. • If b=0, the vertex is located at the y-axis. • Increasing or decreasing the value of b decreases or increases the path of the vertex vertically respectively. • The vertices of the curve follow a parabolic path as b is increasing or decreasing. • The parabola formed by the path of the vertices opens in a direction opposite to that of the graph whose b is being altered. 2.0 The vertex of the parabola in terms of ‘a’, ‘b’, and ‘k’ and the axis of symmetry. • Vary the values for a and b eg., (a,b) = (1,4), (1,6), (2.5,5) ,(1,-4), (1,-10) (-2,2), (2,8) and (3,9) keeping k =0 to obtain different graphs on the spreadsheet. • Have students to determine the x-coordinate of every vertex by generating a data table comprising of these values. • Have students predict the x-coordinates of the vertex in terms of a and b. They should arrive at the answer x = -b . Have students to realize that to get the y coordinate you 2a substitute the x coordinate in the quadratic equation. • Have them predict the line of symmetry x = – b. 2a 3.0 A real world phenomenon
- 5. Allow students to apply their knowledge in a real life application by solving the problem on the worksheet. Call a representative each from some selected groups to present their findings for classroom discussion. Conclusion (5min) A quadratic function in the form y = ax 2 + bx + k becomes wider as |a| decreases and narrower as |a| increases. The parabola opens up when a > 0 and opens down when a < 0. The leading coefficient a is the only coefficient that changes the shape of the graph. The position of the vertex is determined by varying the value of b. If b>0, the vertex is located on the left of the y-axis. If b<0, the vertex is located on the right of the y-axis. If b=0, the vertex is located on the y-axis. The x-coordinate of the vertex is given by –b/2a and the y-coordinate can be found by substituting the value of x in the quadratic b equation. The axis of symmetry x =- . Increasing or decreasing the value of k moves the graph up 2a (vertically) or down (vertically) without altering the shape. Ending the Lesson (5 min) Give assignments out to students as indicated on the assignment sheet.
- 6. Students’ Worksheet Introduction The graph of the function y = ax 2 + bx + k has several properties. In this worksheet, you are provided with activities that will help you examine how the shape of the parabola changes as the values of a, b, and k are modified. You will also determine how the parameters of the equation will help you find the x- and y-coordinates of the vertex. You will be introduced to a realistic application of this function illustrating how these properties are useful. 1.0 The shape of the parabola As the parameters are altered in the quadratic equation, observe the slides and notice the changes in the shape of the graph. Record your observations as in the questions below: 1.1 Varying the value of a Question 1.1a. How does the graph change when a changes from positive to negative? Sketch. i. When a is positive ii. When a is negative Question 1.1b. For what values of a will the vertex of the parabola be a maximum (the highest point on the graph)? Question 1.1c. For what values of a will the vertex be a minimum (the lowest point on the graph)? Question 1.1d. What happens to the graph when a = 0? Why? Question 1.1e How does the graph change as the value of a increases? How does it change if the value of a decreases? Question 1.1f.
- 7. Which of the following parabolas will appear wider: y = −2x2 + x − 5 or y = 4x2 − 2x + 2? (Check your answer by graphical representation on slide.) Question 1.1g. Which of the following parabolas will open downwards: y = 2x2 − x − 5 or y = −4x2 + 2x + 2? (Check your answer by graphical representation on slide.) 1.2 Varying the value of k Question 1.2b What happens to the graph when the value of k is increased or decreased? • When k is increased, the graph moves (upwards / downwards). Underline the correct answer. • When k is decreased, the graph moves (upwards / downwards). Underline the correct answer. 1.3 Varying the value of b Question 1.3 Record the value of x in the x-coordinate of the vertex as we alter the values of b (when a = 1 and k=0) b 6 4 2 0 -2 -4 -6 x Question 1.3a How do changes in the value of b affect the vertex? When b > 0, the vertex lie ………………………………………….. . When b < 0, the vertex lie ………………………………………….. . When b = 0, the vertex lie ………………………………………….. . Question 1.4a How far will the vertex of y = −2x 2 + x − 5 be from the vertex of y = −2x 2 + x − 1? Question 1.4b Determine a function that is wider, opens in the opposite direction, and has vertex lower than y = 0.5x2 + 2x -3. Explain how you determined this result.
- 8. 2.0 The x coordinate of the vertex of the parabola in terms of ‘a’, ‘b’, and ‘k’ and the axis of symmetry when k = 0 Following the changes in the values of a and b, fill in the values of x-coordinate of the respective vertices. a 1 -1 2.5 1 1 2 b 4 6 5 -4 -10 8 x Question 2.1 Determine the value of x in terms of 'a' and 'b' in the table. Question 2.2 Calculate the coordinates of the vertex for the equation y = −x 2 + 4x − 5. Question 2.3 The equation y = 2 x 2 + 8x + 3 has a vertex at (−2, −5). Without graphing this equation, determine whether the vertex will be the minimum or the maximum y-value on the parabola? How? Question 2.4 Which point on the parabola will the axis of symmetry always pass through? Does the formula for finding the x-coordinate of the vertex help you to find the axis of symmetry of a parabola? Question 2.5 If the quadratic equation y = x 2 + bx − 3 has an axis of symmetry at x = 3, what is the value of b? Explain how you know. 3.0 A real world phenomenon The following function describes the vertical position of a falling object as a function of time. 1 2 h = gt + v o t + ho , where 2 h represents the height of the ball
- 9. t represents the time that the ball is in the air g represents the gravitational acceleration on the object (-10 m/s 2 on earth) v o represents the launch velocity (or initial velocity) ho represents the launch height (or initial height) Question 3a Compare this function with y = ax 2 + bx + k by completing the table below. Components of Components of 1 y = ax 2 + bx + k h = gt 2 + v o t + ho 2 y is the same as h x is the same as ________ a is the same as ________ ________ is the same as vo k is the same as ________ Question 3b. What is the leading coefficient of the equation describing the path of the falling object if you are measuring length in meters and time in seconds? Give a numeric value without any letters. Explain how you determined this result. Question 3c. A ball is launched vertically 1.85 meters from the ground at a velocity of 20 meters per second. What is the maximum height of the throw? Explain how you determined this result
- 10. Assignment 1. Which of the following parabolas opens upward and appears narrower than y = −3x 2 + 2x − 1? 2 A. y = 4x − 2x − 1 B. y = −4x 2 + 2x − 1 C. y = x 2 + 4x D. y = −2x 2 + x + 3 2. Which of the following parabolas is 2 units higher than y = 2x 2 − 4x − 1, but has the same shape and opens in the same direction? A. y = 2x 2 − 6x − 1 B. y = 2x 2 − 2x − 1 C. y = 2x 2 − 4x − 3 D. y = 2x 2 − 4x + 1 3. What are the coordinates of the vertex of the parabola y = x 2 + 2x − 1? A. (−2, 5) B. (−1, −2) C. (−2, −1) D. (−1, 2) 4. Which of the following equations has an axis of symmetry with the equation x = 1.5? A. y = 2x 2 − 6x − 1 B. y = 2x 2 − 3x + 1 C. y = x 2 + 6x − 3 D. y = 2x 2 + 3x + 2