Calculator Help Sheet

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1 Calculator Help Sheet The variable x button is: To enter an equation press: To see the graph of the equation you entered press: To find the y-intercept press: To find the x-intercepts (zeros) press: To find the maximum/minimum value press: To adjust the window to see the graph better press: Quadratic Handouts MFMP 1

2 7..1: Linear or Non- Linear Complete the tables of values and determine if the relationship is linear or non-linear. Give reasons for your answers. Figure Number 1 Number of Shaded Squares First Differences This relationship is linear or non-linear because Figure Number 1 Number of Unshaded Squares First Differences This relationship is linear or non-linear because Figure Number 1 Total Number of Squares First Differences This relationship is linear or non-linear because Quadratic Handouts MFMP

3 5.1.3: Going Around the Curve Experiment C A particular mould grows in the following way: If there is one blob of mould today, then there will be 3 tomorrow, and 6 the next day. Model this relationship using linking cubes. Purpose Find the relationship between the number of cubes in the bottom row and the total number of cubes. Hypothesis What type of relationship do you think exists between the number of cubes in the bottom row and the total number of cubes? Procedure 1. Build the following sequence of models using the cubes.. Build the next model in the sequence. Mathematical Models Complete the table, including first and second differences. Make a scatter plot and a line of best fit. Number of Cubes Total in the Number Bottom of Cubes Row First Differences Second Difference Quadratic Handouts MFMP 3

4 5.1.5: Going Around the Curve Experiment E Liz has a beautiful pond in her yard and wants to build a tower beside it using rocks. She is unsure how big she will make it and how many rocks she will need. She is particularly concerned to have the nicest rocks showing. Model the relationship comparing the length of the base to the number of visible rocks using linking cubes. Purpose Find the relationship between the number of cubes on the side of the base and the total number of unhidden cubes. Hypothesis What type of relationship do you think exists between the length of the side of the base and the number of visible cubes? Procedure 1. Build the following sequence of models using the cubes.. Build the next model in the sequence. Mathematical Models Complete the table, including first and second differences. Make a scatter plot and a line of best fit. Length of Side of Base Total Number of Unhidden Cubes First Differences Second Difference Quadratic Handouts MFMP 4

5 5.3.: Key Features of Quadratic Relations Terminology Vertex Minimum/ maximum value Definition The maximum or minimum point on the graph. It is the point where the graph changes direction. How Do I Label It? (x,y) Graph A Graph B Axis of symmetry y-intercept x-intercepts Zeros Label the graphs using the correct terminology. Graph A Graph B Quadratic Handouts MFMP 5

6 Quadratic Handouts MFMP 6

7 5.3.1: Key Features of Parabolic Graphs Student A has 30 seconds to describe the following graph to student B. Using the grid below, student B sketches the graph described by Student A. Quadratic Handouts MFMP 7

8 5.3.1: Key Features of Parabolic Graphs (continued) Student B has 30 seconds to describe the following graph to student A. Using the grid below, student A sketches the graph described by student B. Quadratic Handouts MFMP 8

9 6.3.1: Finding the y-intercept of a Quadratic Equation Name: 1. Use the graphing calculator to find the y-intercept for each of the equations: Note any patterns you see. Equation y-intercept y = x x y = x + x 8 y = x x + 6 ( 1)( ) y = x x ( 4)( 3) y = x + x + ( 3) y = x +. How can you determine the y-intercept by looking at a quadratic equation? 3. Which form of the quadratic equation is easiest to use to determine the y-intercept? Explain your choice. 4. Using your conclusion from question, state the y-intercept of each and check using a graphing calculator. Equation y = x x 8 y = x x 6 y = x + 3x + ( 4)( 1) y = x x ( )( 5) y = x x + y-intercept Does it check? Yes No 5. Explain the connection between the y-intercept and the value of y when x = 0. Quadratic Handouts MFMP 9

10 The Y-Intercept Find the y-intercept of each quadratic relation: a) y = x + 5x + 4 b) y = x + 3x 18 c) y = x 9x + 0 d) y = x 4x 3 e) y = x + x 48 f) y = ( x 4 )( x + 5) g) y = ( x + 8 )( x + ) h) y = ( x 7 )( x + 3) i) y = ( x 4)( x 6) j) y = ( x +10)( x 1) k) y = ( x +13 )( x +10) l) y = ( x +1)( x 3) m) y = ( x 7 )( x + 4) n) y = ( x + 9)( x 3) Quadratic Handouts MFMP 10

11 6.3.: Quadratic Equations Name: 1. Find the y-intercept for each of the following quadratic equations given in factored form. Write the equations in standard form. Show your work. a) = ( 5)( + ) = y x x standard form: = y-intercept: b) = ( + 4)( 3) = y x x standard form: = y-intercept: c) = ( 4) = y x standard form: = y-intercept: ) y = ( x + ) d 5 = = standard form: y-intercept:. Find the y-intercept for each of the following quadratic equations: a) y = ( x + 4)( x + ) b) y = ( x 6) y-intercept: y-intercept: Quadratic Handouts MFMP 11

12 6.4.1: Finding the x-intercepts of a Quadratic Equation (continued) 1. Use the graphing calculator to find the x-intercepts for each of the following: Equation y = x 4x 1 y = x + x 8 y = x x + 6 ( 1)( ) y = x x ( 4)( 3) y = x + x + ( 3)( 5) y = x x + First x-intercept Second x-intercept. Can you determine the x-intercepts by looking at a quadratic equation? Explain. 3. Which form of the quadratic equation did you find the easiest to use when determining the x-intercepts? Explain the connection between the factors and the x-intercepts. Quadratic Handouts MFMP 1

13 Worksheet for Quadratic Handouts MFMP 13

14 6.6.1: Use Intercepts to Graph It! Given the standard form of the quadratic relation, identify the value of the sum and product needed to factor. Express the relation in factored form, identify the x-intercepts and y-intercept, and use these results to make a sketch of each parabola. Standard Form Factored Form x-intercepts y-intercept y x x = ( + 1)( + 5) A = B y = x 4x 5 C y = x + 4x 5 D y = x 6x + 5 E y = x + 7x + 6 F y = x 6x + 9 G y = x x 6 H y = x + 13x + 1 y x x 1 and 5 5 I y = x 4x 1 J y = x + x 1 Sketch of the relation Quadratic Handouts MFMP 14

15 Quadratic Handouts MFMP 15

16 6.4.3: Factoring and Making Connections to the Graphic Part B Find the x- and y-intercepts for each quadratic relation. Use the information to make the sketch on the grid provided. 1. standard form: y = x + 3x + factored form: y-intercept: first x-intercept: second x-intercept:. standard form: y = x + 5x + 4 factored form: y-intercept: first x-intercept: second x-intercept: Quadratic Handouts MFMP 16

17 6.4.3: Factoring and Making Connections to the Graphic (continued) 3. standard form: y = x + 6x + 5 factored form: y-intercept: first x-intercept: second x-intercept: 4. standard form: y = x + 4x + 4 factored form: y-intercept: first x-intercept: second x-intercept: 5. In what way is the last example different from the others? Quadratic Handouts MFMP 17

18 Handout for Quadratic Handouts MFMP 18

19 6.7.1: Investigate Relations of the Form y = ax + bx Name: 1. Obtain a graphing calculator and equation from your teacher.. Type in the equation (using the Y = button on your calculator). Key in zoom 6 to get the max and min from 10 to 10 on your window. 3. Fill in the table. Your equation A sketch of your quadratic equation (including the x- and y-intercepts) y = Coordinates of x-intercepts (, ) and (, ) and the coordinate of the y-intercept (, ) 4. In your group, sketch all four graphs. 5. Identify what is the same and what is different in these four graphs. Quadratic Handouts MFMP 19

20 6.7.1: Investigate Relations of the Form y = ax + bx (continued) 6. Fill in the following table. Standard form y = x + 4x y = x + 6x y = x x y = x 5x From the graph, identify the x-intercepts r and s r = s = r = s = r = s = r = s = Write each equation in factored form y = (x r)(x s) 7. Clear the y = screen on one of the four calculators, enter the factored form of the four equations, and graph. Are these graphs the same as the ones in your sketch? If yes, continue to question 8. If no, revise and check. Ask your teacher for assistance, if needed. 8. Can the equations in the third column of your table be simplified? Explain. 9. Record the simplified versions of your relation in factored form. Standard Form Factored Form y = x + 4x y = x + 6x y = x x y = x 5x Quadratic Handouts MFMP 0

21 6.8.1: Graphing Relationships of the Form y = x c Name: 1. Working with a partner complete the following table. Relation in standard form y- intercept Relation in factored form y = (x r) (x s) x- intercept (r) x- intercept (s) Sketch each graph. Label as A, B, C, D, or use different colours. A y = x 4 B y = x 9 C y = x 16 D y = x 5 Consider your results from question 1 and answer the following questions.. What is the same about the relations? 3. What is the same about the graphs? 4. What is the same about the vertex of each graph? 5. What do you notice about the r and s values of each relation? 6. Solve this puzzle. How can you find the y-intercept and the x-intercepts of the graph of a quadratic relation of the form y = x a? Quadratic Handouts MFMP 1

22 6.9.1 Quick Review of Factoring and Graphing For each of the following: - identify what information the equation tells you about the parabola - factor the equation and identify what the new form of the equation tells you - sketch the parabola using the information you have (make sure you plot key points!) Equation (Standard Form) 1. y = x -7x+10 Y- Intercept Factored Form Zeros Sketched Graph. y = x x 6 Equation (Factored Form) 3. y = (x )(x 5) Zeros Standard Form Y- Intercept Sketched Graph 4. y = (x-3)(x+) Quadratic Handouts MFMP

23 6.9. Matching For each equation in column A, state the y-intercept. For each equation in column B, state the zeros (or x-intercepts) Each equation in column A has a matching equation in column B. Draw an arrow Column A Y = x + 3x + y-intercept = Column B Y = (x 3)(x + ) x-intercepts = Y = x 3x 10 y-intercept = Y = (x 3)(x ) x-intercepts = Y = x + x 1 y-intercept = Y = (x + 4)(x + ) x-intercepts = Y = x 5x + 6 y-intercept = Y = (x + )(x + 1) x-intercepts = Y = x + 6x + 8 y-intercept = Y = (x 3)(x + 4) x-intercepts = Y = x x 8 y-intercept = Y = (x + 5)(x + ) x-intercepts = Y = x + 3x 4 y-intercept = Y = (x 5)(x + ) x-intercepts = Y = x x 6 y-intercept = Y = (x 4)(x + ) x-intercepts = Y = x + 7x + 10 y-intercept = Y = (x + 4)(x 1) x-intercepts = Quadratic Handouts MFMP 3

24 6.9.3 Quick Sketches of Parabolas For each quadratic function given below determine: The zeros (x-intercepts) The y-intercept With the information you find and using symmetry, graph each parabola. a) y = x + 6x + 8 b) y = x 7x + 10 y x y x c) y = x 3x 18 y x d) y = x x 30 y x Quadratic Handouts MFMP 4

25 Problem Solving with Quadratic Graphs- Interpreting Parabolas 1. The graph below shows the height, in meters, of a diver jumping off a springboard versus time, in seconds. a) What is the initial height of the jumper? What is this called in math terminology? b) Label and write the ordered pair of the vertex. What does this mean in real life? c) Label and write the ordered pairs of the roots/zeros. What do these points mean in real life?. The graph below shows the height of a toy rocket after it is launched. a) How many seconds is the rocket in the air? b) What math concept did you use to determine this? c) What is the maximum height of the rocket? d) At what time does the maximum height occur? e) At what times is the rocket 10 meters above the earth? f) Complete the table of values below with first and second differences. Time Height First Diff Second Diff g) What pattern did you observe in the second differences from the table? How does this prove that the relationship is quadratic? Quadratic Handouts MFMP 5

26 Solving Parabolas 1. a) Graph the parabola y = ( x + 3) 5 on the TI83. Sketch the graph. b) Find the value of y when x = -1. Mark this point on your sketch c) Find the y-intercept. The y-intercept occurs where the x equals zero. d) Find the value of x when y = 0.. a) Graph the parabola y = ( x + 1) + on the TI83. Sketch the graph. b) Does this parabola have a maximum or a minimum? c) What is the value of the maximum or the minimum? d) Find the height of the parabola when x = 4. e) What will the value of x be when the parabola has a height of A rocket is shot into the air off the top of a building. It s height is given by the equation h = 5t + 0t + 8, where h is the height of the rocket in metres and t is the time in seconds. a) What will the rocket's height be after 3 seconds? b) What will the rocket's height be after 4 seconds? c) What was the height of the building the rocket was shot from? 4. Examine the bridge over the river as shown in the diagram. The equation of the bridge is 1 h = x + x 10 where h is the height of the span and x is the number of metres from the shore. What is the height of the bridge 4 metres from the shore? 5. The cables of a suspension bridge form a parabola as shown. h = 1 ( x 30) 10 The equation for the span is 0 + where h is the height in metres and x is the distance in metres from the left pillar. a) What is the minimum height of the cable? b) How tall is the pillar on the right? 6. a) Tiger woods is trying to hit his wedge over a tree. The equation of the flight of his ball is 1 h = x + x 5 where h is the height of the ball in metres and x is the distance in metres away from Tiger. Will the ball make it over the tree? If so, by how much? Quadratic Handouts MFMP 6

27 7. A submarine is engaged in a dive-and-surface maneuver. The depth of the submarine is given by the equation d = ( t 3) 16, where d is the depth in metres and t is the time in minutes. a) Graph the parabola representing the dive on the TI83. b) What is the maximum depth the submarine dives to? c) After how many minutes will the submarine be at maximum depth? d) What will the depth be after 1 minute? e) What was the depth when the maneuver began? f) When will the submarine be exactly 15 metres deep? g) How many minutes will it take for the submarine to surface? 8. A helicopter takes off from a helipad on top of a building and flies with the equation h = ( t 9) + 100, where h is the height above the ground and t is the time in minutes. a) Graph the flight of the helicopter on the TI83. b) What is the height of the helicopter after 3 minutes? c) What is the height of the helicopter after 14 minutes? d) How long will it take for the helicopter to be at exactly 51 metres above the ground? e) How high was the building the helicopter took off from? f) What time will it take for the helicopter to land? Quadratic Handouts MFMP 7

28 6.10. Applying Quadratic Relationships There are many relationships that turn out to be quadratic. One of the most common is the relationship between the height of something (or someone) flying through the air and time. 1. A football player kicks a ball of a football tee. The height of the ball, h, in metres after t seconds can be modelled using the formula: h = -5t + 0t. a) Graph the relationship using your graphing calculator. Remember that you need to set your window settings. Record the window settings you used. b) Sketch your graph in the window at right. Make sure to label your axes. c) You can get the table of values for this relationship. Press ` to access the table setup screen. Make sure your screen looks like the one given. d) Now press ` and %. Fill in the window with the values you see. e) For which times does the height not make sense? Why? f) What is the initial height of the ball? g) Where do you look in the table and the graph to determine the answer? h) Why does this make sense? Quadratic Handouts MFMP 8

29 6.10. Applying Quadratic Relationships (continued) i) What is the maximum height of the ball? j) Where do you look in the table and the graph to determine the answer? k) When does the ball hit the ground? l) Where do you look in the table and the graph to determine the answer? m) When is the ball more 10 m above the ground? (You may need to give approximate answers.) Playing Football on Mars. The force of gravity on Mars is less than half that on Earth. A ball thrown upward can be modelled using h = -t + 15t + where h is the height in m and t is the time in seconds. a) Graph the relationship using your graphing calculator. Remember that you need to set your window settings. Record the window settings you used. You may need to play around with the settings until you see the full graph. b) What is the initial height of the ball? c) Explain what this means. d) What is the maximum height of the ball? e) When does the ball reach its maximum height? f) When does the ball hit the ground? g) When is the ball more 0 m above the ground? (You may need to give approximate answers.) h) If the same ball was thrown upward on the Earth, how would you expect the relationship to change? i) The force of gravity on Jupiter is much greater than on the Earth. If the same ball was thrown upward on Jupiter, how would you expect the relationship to change? Quadratic Handouts MFMP 9

30 6.11. Review of Quadratics 1. Linear vs Quadratic A linear relation forms a graph with a. A quadratic relation forms a graph with shape of a. Below is a table of value, determine if this relation is linear, quadratic or neither: Time (x) Distance (y) First Differences Second Difference The relation is: A linear relation has the equal and the equal to. A quadratic relation has the equal. Quadratic Handouts MFMP 30

31 6.11. Review of Quadratics (Continued). Features of a Quadratic Graph A quadratic relation can be seen when a ball is thrown in the air and the height is measured versus time. A sketch of this graph might look: A quadratic relation can be seen when a duck flies into the water, catches a fish and flies back out: This parabola is facing and the vertex is a. This parabola is facing and the vertex is a. The key features of a quadratic graph are: Quadratic Handouts MFMP 31

32 6.11. Review of Quadratics (Continued) 3. Forms of A Quadratic - Standard Form: y =ax + bx + c The y-intercept is the or term. We can change factored into standard form by. There are three methods of expanding:,, and Example 1: Find the y-intercept, x-intercepts of the following quadratic: y = (x + 4)(x 5) Method 1: Tiles Method : Table Method 3: Algebra (FOIL) The y-intercept is: The x-intercepts are: and Example : Expand the following: y = x(3x 4) Quadratic Handouts MFMP 3

33 6.11. Review of Quadratics (Continued) 4. Forms of A Quadratic - Factored Form: y =(x r)(x s) There are two methods of factoring: algebra tiles and algebra Example: factor y = x + 3x + using tiles 1. Product/Sum Form: Factor y = ax + bx + c In this case r s = c and r + s = b Find the x-intercepts of the following: y = x + 6x + 8 y = x 3x 18. Common Factoring: y = ax + bx Let us factor: y = x + 3x This can be written as: Based on this r s = r + s = The factored form is y = or Factor as state the x and y intercepts of: y = x + 6x y = x x Difference of Squares: y = ax b y = x 4 can be written as y = r s = r + s = factored form y = x - 16 y = x 00 Quadratic Handouts MFMP 33

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