- 1. Designed and developed by David Annan and Letecia Anima under the supervision of D.D. Agyei- University of Cape Coast, Ghana Teacher support materials Topic Quadratics in Vertex Form : y = a(x-p) 2 + q 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 = a(x-p 2 + q, 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 sketching quadratic functions in SHS 1, they often don't identify connections between effects the changes in the parameters have on the graph of the functions. Each function has its own mysterious parameters. In this lesson students will compare the graph of a quadratic to its equation in vertex form, vary the terms of the equation and explore how the graph changes in response. They then are asked to analyse the results, and to form conclusions about quadratic functions in the vertex form. Objectives: The student will: • determine how changes in the parameters of a quadratic function in the vertex form affects its graph • determine how to use the vertex form of a quadratic function to find the coordinates of the vertex on a graph. • determine the y-intercept of a parabola. • determine the minimum and maximum values of a parabola. Prerequisite Knowledge Students are able to: - find the solutions of a quadratic equation in the polynomial form. - locate the vertex of a parabola on a graph. - complete the squares of a quadratic function in the form y = ax2 + bx + k. Resources
- 2. • 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. • Copies of teacher support materials (including teachers’ guide) for teachers (on CD’s). • Spreadsheet (e.g. Excel) software. • 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 a spreadsheet environment. 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 in a diagram below. • 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.
- 3. Quadratic Functions - Working with y = a(x-p)2 + q 4. Input your x-values you would use in plotting your quadratic graph beginning from cell X1 downwards. Eg. from -6 to 6 (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-p) ^2+q, 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, p, and q are stored in cells W5, W6 and W7. 6. Set the cursor over cell Y1 to note the formula. You should see: = a*(X1-p)^2+q. (The ^ symbol is used for exponents in a spreadsheet and the * symbol must be used for multiplication.) 7. Try altering the values of a, p, and q. It is difficult to see how the parabola changes because the scale automatically adjusts. Therefore, set a = 1, p = 0 and q = 0 (this will give the standard quadratic equation y = x2 from which all transformations will be made) and proceed to execution of the lesson in the lesson plan. (It is important to set up this before the lesson begins.)
- 4. Lesson Plan LESSON 1: Quadratics in Vertex Form : y = a(x - p)2 + q (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 = a(x - p)2 + q has several properties. In these activities, you will guide students examine how the shape of the parabola changes as the values of a, p, and q are modified. You will also determine how this equation will help you find the y-intercept as well as the coordinates of the vertex. You will also lead students to determine the minimum and maximum values of a parabola. Prepare students for the following activities (Activities: 1.0 - 2.2) 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 and begin the lesson by giving each group a number of quadratic function (in vertex form) on the Students’ Worksheet. In this activity ask students to indicate (by tick (√)) the features of the equations as shown on the Worksheet (without plotting or solving them). Ask the students to keep their results for discussion later in the lesson. Execution of the lesson(60min) Activity 1.0 : The shape of the parabola Prepare the graph of y = a(x-p)2 + q 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 function are altered. 1.1 Varying the value of a (set p = 0 and q = 0) • Begin with the graph of the standard function: y = ax2 on the spreadsheet and guide 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. • Get students in groups to observe and describe the transformations in the graph as we increase the
- 5. value of a systematically (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 observations. • 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. When “a” is positive it opens upwards. ii. As we continue to decrease the value of a through negative values, the parabola opens downwards. iii. When the value of a = 0 we have a line on the x-axis ( connection between quadratic and linear functions) iv. As the absolute value of a increases, the graph becomes more steeper; the parabola becomes narrower. v. As the absolute value of a decreases the parabola flattens out (widens). 1.2 Varying the value of p (set q = 0 and a = 1) Repeat the process for the activity by varying the value of p and guide students to observe and record changes on graph. Student groups will present their findings. Some discussion points could be: i. Increasing the values of p shifts the graph (vertex) horizontally to the right. (The vertex of the new parabola is located p units from the vertex of the standard function on the right.) ii. Decreasing the values of p shifts the graph (vertex) horizontally to the left. ((The vertex of the new parabola is located p units from the vertex of the standard function on the left.) iii. As the absolute value of p increases the parabola approaches a straight line (i.e. the concavity of the graph reduces). 1.3 Varying the value of q (set p = 0 and a = 1) Set p = 0 and a = 1 then vary the value of q and let students record and discuss their findings. Guide them to identify that: • The parabola moves vertically upwards when q is increasing. • The parabola moves vertically downwards when q is decreasing. 1.4 Allow students to re-visit the exercise they did at the beginning of the lesson and discuss their results. They should also do the following in the group and ask representatives to present the results: i. Write the equation for the final transformed graph: a. y = x 2 ; shift upward 3 units and shift 2 units to the right. b. y = x 2 ; shift downward 1 unit and shift 4 units to the left. ii. Sketch, on the x-y plane, the graph of the functions below, not by plotting points, but by starting with the graph of a standard function and applying transformations: a. y = 5 + (x + 3)2 b. y = - (x – 2)2 - 5 2.0 The vertex, intercepts, maximum and minimum values of a parabola. • Plot a number of quadratic graphs : ie. y = 4(x-3)2 + 1, y = -2(x+1)2 + 4, y = 4(x+3)2 – 2. • In each of the plots, guide students to record the y-intercept, the vertex and the maximum or minimum values of the parabola as shown in the table on their worksheets. (Hint: By clicking and holding the curser on these points on the graph, the coordinates will be displayed.) • Have students determine the relationship between p and q and the coordinate of the vertex • Have students determine the relationship between a, p and q and the y-intercept. • Have students determine the effect of a on the maximum or minimum point. • Have students record their observations and findings on the students’ worksheet and selected groups discuss their results with the whole class.
- 6. Conclusion (5min) A quadratic function in the form y = ax 2 + bx + k can be expressed in the standard form y = a(x-p) 2 + q by completing the squares. The graph of y is a parabola with vertex (p, q) The parabola opens upwards when a > 0 and opens down when a < 0. If a > 0, then the function has minimum value which is given by q. If a < 0, then the function has a maximum value which is q. Varying the value of q moves the graph vertically upwards when q is increasing and vertically downwards when q is decreasing. Increasing the value of p moves the vertex of the parabola to the right horizontally and to the left horizontally when p is decreasing. Ending the Lesson ( 5 min) Ask students to do the assignment and present it before the next lesson. Students’ Worksheet Introduction The graph of the function y = a(x-p) 2 + q has several properties. In these activities, you will examine how the shape of the parabola changes as the values of a, p, and q are modified. You will also determine how this function will help you find the y-intercepts, the vertex and the turning points of the parabola. 1.0 Introductory activity Identify the position of the given quadratic functions. Emphasis should be on the position of the vertex- whether it will lie on the upper right (UR), upper left (UL), lower right (LR) or lower left (LL) of the x-y axis. In each case determine weather the parabola opens upwards or downwards. Indicate as shown on the table with a tick (√). Equation UR UL LR LL upward downward y = (x-5)2 - 5 y = -2(x+1)2 + 4 y = 2-(x-4)2 1.1 The shape of the parabola As the parameters are altered in the quadratic function, observe the slides and notice the changes in the shape of the graph. Record your observations as in the questions below: Question 1.1a In which direction does the parabola open when a is positive? In which direction does it open when a is negative? Question 1.1b. What does the graph look like when a = 0?
- 7. Question 1.1c. As the absolute value of a increases, does the graph become more or less steep? Does this make the parabola appear narrower or wider? Question1.2. What effect does the value of p have on the shape of the graph? Question 1.3 What effect does the value of q have on the shape of the graph? Question 1.4a Write the equation for the final transformed graph: y = x 2 ; shift upward 3 units and shift 2 units to the right. y = x 2 ; shift downward 1 unit and shift 4 units to the left. Question 1.4b Revisit the activity you did in section 1.0 above and compare your results now to that of the results before and discuss your findings in your group Equation UR UL LR LL upward downward y = (x-5)2 - 5 y = -2(x+1)2 + 4 y = 2-(x-4)2 Question 1.4c
- 8. Sketch, on the x-y plane, the graph of the functions below, not by plotting points, but by starting with the graph of a standard function and applying transformations: y = 5 + (x + 3)2 y = - (x – 2)2 - 5 Question 2.1 In the table below, fill in the values: Equation a p q y-intercept Vertex Max/Min y = 4(x-3)2 + 1 y = -2(x+1)2 + 4 a. How does y = 4(x+3)2 - 2 the value of p in the function help you to find the vertex of the parabola? b. How does the value of q in the function help you to find the vertex of the parabola? c. What is the relationship between a, p and q and the y-intercept? d. If the vertex is a maximum point what is true of a? e. If the vertex is a minimum point what is true of a?
- 9. Question 2.2 Fill in the chart for the quadratic functions that have the following characteristics: Equation a p q y-intercept Vertex Max/Min 2 5 -2 5 -1 0 -1 0 -4 Write the equation for a quadratic function that: (a) has vertex (3, -2), opens upward and has y-intercept 7. (b) has vertex (2, 0), opens downward and has y-intercept - 8. Assignment 1. Which of the following functions will have the steepest graph? A. y = 3(x − 3) 2 + 1 B. y = 2(x − 3) 2 + 5 C. y = 2(x +6) 2 + 1 D. y = -4(x -3) 2 + 1 2. Which of the following graphs is the graph of y = 0.5x2 ?
- 10. A. Graph A B. Graph B C. Graph C D. Graph D 3. In y = a(x − h) 2 + k, h corresponds to which characteristic of the parabola? A. the y-coordinate of the vertex B. the x-coordinate of the x-intercept C. the x-coordinate of the vertex D. the y-coordinate of the y-intercept 4. How does the graph of y = 3(x − 4) 2 compare to the graph of y = 3x2 ? A. it is less steep B. it is more steep C. it is shifted 4 units to the right D. Both B and C 5. Which of the following functions has a maximum y-value of 2? A. y = 2(x + 1) 2 + 3 B. y = 3(x + 1) 2 + 2 C. y = − 3( x + 1) 2 + 2 D. y = − ( x − 2) 2 + 3 6. If the values in the table below represent points on a parabola, then which of the following must be true? A. it opens down and has no x-intercepts B. it opens down and has two x-intercepts C. it opens up and has one x-intercept D. it opens up and has two x-intercepts