Quadratic Relationships

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C In the previous unit you studied a variety of number patterns called sequences.

Some of those observed patterns were either arithmetic, quadratic, cubic or geometric in nature. It is the concept of a quadratic relationship between two variables that this unit emphasizes.

C You will see that there are many real world phenomena that are in fact quadratic in nature. Consider, for example, the path of a bullet fired from a rifle or the braking distance required for a car to stop on a slippery road.

C A review of a simple quadratic sequence is in order followed by the introduction of a new and very similar quadratic relation. Let the journey begin.

Activity #1:

Ex. Study the various patterns created using a series of blocks as shown below.

Record the number of blocks required to create each pattern from 1 through 4 in the table provided.

n 1 2 3 4 5 6

t_n

a. Explain any patterns you observe between the pattern number ‘ ’ and the number of blocks ‘ ’ used to create it.

____________________________________________________________

b. Extend the table to show the number of blocks required for patterns #5 and #6.

c. Write your sequence ‘ ’ extending it to the 8^th pattern of blocks in the table on the next page. Using levels of differences show that the sequence generated is quadratic.

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d. Provide reasons why the equation aptly describes this sequence.

__________________________________________________________

e. Sketch the graph of this sequence and answer the questions that follow.

1 2 3 4 5 6 7 8

5 10 15 20 25 30 35 40 45 50 55 60 65

n t(n)

i. Describe the shape of the graph.

_____________________________

ii. Should the points on the graph be connected? Explain by making reference to your knowledge of sequences.

______________________________

iii. Why were values such as -1 and -2 not included in the table or the graph?

______________________________

iv. What special name was given to the graph of a quadratic sequence in unit I?

______________________________

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Activity #2:

C In this activity, you will investigate the equation and graph of the quadratic

relation . As you complete the exercises, think back to its similarities and differences to the equation and graph of .

Ex. Investigate the equation and graph of the relation by answering the following questions.

a. How are the equations and similar? How are they different?

__________________________________________________________

b. How does the equation suggest to you that the relationship between

‘x’ and ‘y’ is quadratic?

__________________________________________________________

c. Complete the table of values below to construct the graph of and answer the questions to the right.

x y

-3 -2 -1 0 1 2 3

1 2 3 4

-1 -2 -3 -4

1 2 3 4 5 6 7 8 9 10

x

y i. Describe the shape of the graph.

___________________

ii. Why were values such as -1 and -2 included in the

table?

___________________

iii. Could values such as and

have been used?

___________________

d. Based on your answer to the previous question, should the points on the graph be connected? Explain.

__________________________________________________________

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Activity #3:

C In this activity, you will investigate the equation and graph of the quadratic relation . As you complete the exercises, reflect on this relation’s

similarities and differences to the equation and graph of both and . Ex. Investigate the equation and graph of the relation by completing the

questions that follow.

a. How are the equations and similar? How are they different?

__________________________________________________________

b. How does the equation suggest to you that the relationship between

‘x’ and ‘y’ is quadratic?

__________________________________________________________

c. Complete the table of values below to construct the graph of and answer the questions to the right.

x y

-3 -2 -1 0 1 2 3

1 2 3 4

-1 -2 -3 -4

-1

-2

-3

-4

-5

-6

-7

-8

-9

-10

x

y i. Describe the shape of the graph.

___________________

ii. What effect did the

negative sign (-) in front of the have on the graph?

___________________

iii. Could values such as

and be

used to draw the graph?

___________________

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General Comments

C Many of the characteristics you just observed are formally noted below. Be sure to highlight and study them because they will be referenced often as the unit progresses.

C The graph of a quadratic function is a smooth continuous curve called a parabola.

C Quadratic functions always contain a term of the type x² (degree 2) where 2 is the largest power of ‘x’ that occurs.

C Quadratic functions that open up have a low point or minimum.

C Quadratic functions that open down have a high point or maximum.

C The maximum or minimum point is called the vertex.

C Notice that the graph has symmetry (looks the same on both sides). Note that the axis of symmetry must always pass through the vertex of the parabola.

Ex. Sketch the graph of the quadratic relation using the prepared table of values and grid below. Answer the questions that follow.

x y

-4 8

-3 3

-2 0

-1 -1

0 0

1 3

2 8

1 2 3 4

-1 -2 -3 -4

1 2 3 4 5 6 7 8 9

-1

x

y i. State the coordinates of the vertex for this curve.

_____________________

ii. Draw the axis of symmetry on the graph.

iii. What evidence is given in the equation to indicate that the graph is a parabola?

______________________

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Ex. Sketch the graph of the quadratic function using the prepared table of values and grid below. Answer the questions that follow.

x y

-3 -17

-2 -7

-1 -1

0 1

1 -1

2 -7

3 -17

1 2 3

-1 -2 -3

1

-1 -2 -3 -4 -5 -6 -7 -8 -9 -10 -11 -12 -13 -14 -15 -16 -17 -18

x y

i. State the coordinates of the vertex for this curve.

_____________________

iii. What is the value of the y-intercept for this curve?

Where did this value occur in the equation?

_____________________

Ex. Sketch the graph of the quadratic relation preparing your own table of values. Answer the questions that follow.

x y

-4 -3 -2 -1 0 1

2 ^-4 ^-3 ^-2 ^-1 ¹ ² ³

1 2 3 4 5 6 7 8 9

-1

-2

x

y i. State the coordinates of the vertex for this curve.

_____________________

iii. What is the value of the y-intercept for this curve?

Where did this value occur in the equation?

_____________________

Quadratic Any equation where the highest power of the variable is 2;

Relationship: that is, , where .

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Exercises: Working with the Graphs of Quadratic Relations

1. Which of the following graphs represents a quadratic relation?

a)

1 2 3 4

-1 -2 -3 -4 -5 -6 -7

1 2 3 4 5 6

-1 -2 -3 -4 -5 -6 -7 -8 -9 -10

x y

yes 9 no 9

b)

1 2 3 4

-1 -2 -3 -4 -5 -6 -7

1 2 3 4

-1

-2

-3

-4

x y

yes 9 no 9

c)

1 2 3 4 5 6 7

-1 -2 -3 -4 -5 -6 -7

1 2 3 4 5

-1

-2

-3

-4

-5

x y

yes 9 no 9 2. The graphs of several quadratic relations are presented below. For each curve:

i. Draw the axis of symmetry.

ii. State the coordinates of the vertex.

iii. State whether the vertex represents a maximum or a minimum point for the relation.

a)

1 2 3 4 5 6 7

-1 -2 -3 -4 -5 -6 -7

1 2 3 4 5

-1

-2

-3

-4

-5

x y

Vertex: _____________

Max./Min. _____________

c)

1 2 3 4 5 6 7

-1 -2 -3 -4 -5 -6 -7

1 2 3 4 5

-1

-2

-3

-4

-5

x y

Vertex: _____________

Max./Min. _____________

e)

1 2 3 4 5 6 7

-1 -2 -3 -4 -5 -6 -7

1 2 3 4 5

-1

-2

-3

-4

-5

x y

Vertex: _____________

Max./Min. _____________

b)

1 2 3 4 5 6 7

-1 -2 -3 -4 -5 -6 -7

1 2 3 4 5

-1

-2

-3

-4

-5

x y

Vertex: _____________

d)

1 2 3 4 5 6 7

-1 -2 -3 -4 -5 -6 -7

1 2 3 4 5

-1

-2

-3

-4

-5

x y

Vertex: _____________

f)

1 2 3 4 5 6 7

-1 -2 -3 -4 -5 -6 -7

1 2 3 4 5

-1

-2

-3

-4

-5

x y

Vertex: _____________

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3. Evaluate each of the following expressions without using a calculator. Show your work in the spaces provided.

a) d) g)

b) e) h)

c) f) i)

4. For each of the following quadratic relations, complete the table of values and sketch the graph. Clearly indicate on each graph the axis of symmetry and the vertex.

a)

x -3 -2 -1 0 1 2 3

y

1 2 3 4 5 6 7 8 -1

-2 -3 -4 -5 -6 -7 -8

1 2 3 4 5 6 7 8 9 10

-1 -2 -3 -4 -5 -6 -7 -8 -9 -10

x y

b)

x -4 -3 -2 -1 0 1 2

y

1 2 3 4 5 6 7 8 -1

-2 -3 -4 -5 -6 -7 -8

1 2 3 4 5 6 7 8

-1 -2 -3 -4 -5 -6 -7 -8

x y

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Graphs and Equations of Quadratic Relations (Continued)

C The previous section introduced the equations and graphs of some quadratic relations. While some of the terminology may have been new, many of the concepts should have been familiar to you from your study of quadratic sequences in Unit I.

C This section will continue your investigation of the equations and graphs of quadratic relationships. In the upcoming activities, you will use graphing calculator technology to assist you in your work.

Activity #4:

C In this activity you will use graphing calculator technology to find the equation of a quadratic relationship.

Ex. The diagram below shows two simple applications of some basic

mathematical principles. The first is that of a toy car moving down a ramp and the second is that of a toy rocket launched from ground level. Use the

diagrams to answer the questions that follow.

a. Study the two graphs below and explain why each fits the motion of the toy illustrated above it. Describe the motion and the shape of each graph.

x

y ^y

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b. For one flight of the toy rocket, a table of data was collected showing the number of seconds (s) that have passed and the height of the rocket above the ground in metres (m). The graph and data are given below.

time (s) 0 1 2 3 4

height (m) 0 15 20 15 0

c. Study the values for the height above the ground in row two of the table above. How do these values tell you that there is not a linear relationship between time and height?

_____________________________________________________________

d. Using the space provided below the table, show, using levels of differences, that the relationship between time and distance is indeed quadratic.

e. By making reference to the graph, state the maximum height that the toy rocket reaches. What is special about this point in relation to the graph?

_____________________________________________________________

f. Why does it seem logical that the graph did not extend beyond time t = 4 seconds?

_____________________________________________________________

g. Can you easily determine an equation of the form to describe the path of the rocket? _________________________________________

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Activity #5:

C In this activity you will use technology to determine the equation of a quadratic relationship such as the one studied in the previous exercise.

C Technology can help you quickly and easily determine this information. It will require, however, that you learn to use many of the features of a graphing

calculator. In this and upcoming sections these features will be demonstrated for you.

C This text assumes that you will be using a Texas Instruments TI-83 Plus ® graphing calculator to complete the exercises. Your teacher will demonstrate how to complete each task. Note that all relevant steps required to use the technology will be recorded for you should you need to refer to them in the future.

Ex. Use calculator technology to find an equation of the form

to model the data provided below for the toy rocket flight path modeled in Activity #4. The table of data is provided below.

time (s) 0 1 2 3 4

height (m) 0 15 20 15 0

C Use the space provided below to record your own notes or comments on this process. Remember that with practice, the technology will become easier to ______________________________________________________________use.

______________________________________________________________

C Write the equation of the quadratic relation generated by the calculator.

______________________________________________________________

C A summary of the process required to complete the above has been included on the next page.

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How to use Regression to find the Equation of a Quadratic Relation Step Ø Enter the data into two lists on the calculator. This is accessed by

pressing followed by to select the menu.

Step Ù Enter the first list (‘ ’) of data into the first list (L1) and the second list of data into list two (L2).

Step Ú Press , followed by the right directional arrow key to highlight the menu.

Step Û Press the number 5 or use the down directional arrow key to highlight the selection, 5:QuadReg. This is the feature that will use the data entered into the lists from step 2 to generate the equation of the quadratic relation.

Step Ü The screen should clear showing only “QuadReg”. Press

to generate the required equation. The calculator will display the form of the required quadratic ( ) as well as the values for ‘ ’, ‘ ’ and ‘ ’. Record the equation.

Ex. The table below lists the height above the ground for a second flight of the toy rocket. Use the data to answer the questions that follow.

time (s) 0 1 2 3 4 5 6

height (m) 0 25 40 45 40 25 0

a. Using levels of differences, above, demonstrate that the data is quadratic in nature.

b. Use the regression feature of the calculator to find the equation of a quadratic relation to model this data. Write your equation below.

_____________________________________________________________

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Activity #6:

C In this activity you will use graphing technology to demonstrate that a set of data is indeed quadratic. It will not be necessary to check levels of differences as was done in the previous activity.

C In this activity you will also find the equation of the quadratic relation and store the result in the calculator’s equation editor.

Ex. A toy glider plane is released from a platform 11 metres above ground level.

The table below shows the height of the glider above the ground for the first 6 seconds of the flight.

time (s) 0 1 2 3 4 5 6

height (m) 11 6 3 2 3 6 11

Use technology to produce a scatter plot illustrating the time and height data contained in the table. Your teacher will demonstrate how a scatter plot is created. Use the space below to record your own notes regarding this process.

______________________________________________________________

C Use the grid at the right to sketch a copy of the scatter plot produced above. Label the graph appropriately.

How does the shape of the plot indicate to you that the data contained in the table is quadratic in nature?

____________________________________________

C A summary of the process required to complete the

above task is included on the next page for review purposes.

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How to Create a Scatter Plot for a Set of Data

Step Ø Enter the data into two lists on the calculator. This is accessed by pressing followed by to select the menu.

Step Ù Enter the first list (‘ ’) of data into list L1 and the second list of data into list L2.

Step Ú Press followed by to access the scatter plot menu.

Step Û Press to select Plot 1 followed by again to turn this plot on. This will allow you to graph and view it.

Step Ü Use the down directional arrow to move to “Type:” and highlight the first graph type (scatter plot) shown on the first row. Press

to select this graph type.

Step ñ Move the down directional arrow key again to the “Xlist:” and

“Ylist:” entries. They should read and respectively.

Step ò Press to view the scatter plot. Note that you may have to use the function key (select 9: ZoomStat) to properly view the scatter plot.

Ex. Construct a scatter plot for the following set of data showing the time and height measurements for a second flight of the toy glider launched from a 9 metre high platform.

time (s) 0 1 2 3 4

height (m) 9 6 5 6 9

a. Construct a scatter plot for this data. Use the grid at the right to reproduce the diagram. Label appropriately.

b. How does the scatter plot suggest that the data is quadratic in nature?

_____________________________________________________________

c. Use technology to find an equation to model the data. _________________

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d. Enter the above equation into the graphing editor ( ) and plot the curve. What do you notice about the scatter plot and the new curve just graphed?

_____________________________________________________________

e. Note that it is also possible to find the equation of best fit that models the data and have the calculator automatically store the result in the graphing editor.

Your teacher will demonstrate this process. Note that an outline of the steps required to perform this task on your own is provided below.

How to Store an Equation of Best Fit in the Graphing Editor Step Ø Find the equation of best fit to model the data as outlined

previously. Stop when you have “QuadReg” displayed on the screen.

Step Ù Press the key.

Step Ú Use the right directional arrow key to highlight . Step Û Press twice. The calculator screen should now display

“QuadReg Y1". Press .

Step Ü Press . The equation should now appear in the graphing editor as well. Press to plot. Note that you may have to use the function key (select 9: ZoomStat) to properly view the scatter plot and graph.

Ex. Find an equation of best fit to model the quadratic data below and store it appropriately in the calculator’s graphing editor ( ).

time (s) 0 1 2 3 4

height (m) -5 0 11 28 51

Record the equation below.

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Exercises: Using Technology to Model Quadratic Relations

1. Which of the following situations do you think could be best modeled using a quadratic relation? Check the appropriate box to indicate your answer.

a) A baseball is thrown into the air and its height above the ground is recorded.

b) A maple tree is planted in a garden and its height is recorded every six months.

c) A car is driven on a straight stretch of road at a constant speed of 40km/h.

The position of the car from its starting point is recorded.

d) A stone is wedged in the grooves of a bicycle tire. As the bicycle moves, the height of the stone above the ground is recorded.

e) A frisbee is thrown into the air and its height above the ground is recorded.

yes 9 no 9 yes 9 no 9 yes 9 no 9 yes 9 no 9 yes 9 no 9

2. Steve hits a golf ball on a fairway. The height of the ball above the ground is recorded at 1 second intervals, as illustrated in the table below. Use this information to answer the questions that follow.

time (s) 0 1 2 3 4 5 6 7 8 9

height (m) 0 16 28 36 40 40 36 28 16 0

a) Record the heights above as a sequence.

b) Using the necessary levels of differences, show that the sequence of numbers from a) is quadratic.

c) Using graphing technology, construct a scatter plot of this data representing the path of the golf ball. Construct a simple sketch of the scatter plot in the space at the right. Label the sketch appropriately.

d) Using technology, determine the equation of best fit that would model this data. Record your equation here. ____________________

e) Use technology to determine the greatest height reached by the golf ball. ________________________

f) Using technology, determine the height of the golf ball after 8.5 seconds. ________________________

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3. A relation is defined by the equation . Use this equation to answer the question that follow.

a) Explain how you can tell from the equation that it is quadratic. _____________________________

b) What will be the shape of the corresponding graph? ___________________________________

c) Use technology to produce a sketch of the graph.

d) Use technology to determine the vertex of this curve. ___________________________________

e) Use technology to complete the table of values below for this relation.

x -6 -5 -4 -3 -2

y

4. Suppose you were asked to find the first 5 terms of the quadratic sequence . Explain in a sentence or two how you could use graphing technology to complete this task.

______________________________________________________________________________________

Use technology to find the first five terms. Write the values here:

5. Use technology to find the vertex for each of the following quadratic relations. Record each in the space provided.

a) V( , ) b) V( , ) c) V( , )

6. A relation is defined by the equation . Use this equation to answer the questions that follow.

a) Explain how you can tell from the equation that it is quadratic. _____________________________

b) What will be the shape of the corresponding graph? ___________________________________

c) Use technology to produce a sketch of the graph.

d) Use technology to determine the vertex of this curve. ___________________________________

e) Use technology to complete the table of values below for this relation.

x -3 -2 -1 0 1

y

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Using Technology to Further Investigate Quadratic Relations

C In the following exercises you will use technology to discover how key

characteristics of a quadratic relation of the form are related to the values of the coefficients or numbers ‘ ’, ‘ ’ and ‘ ’ associated with the relation.

Activity #7:

C Quadratic relations, when graphed, produce smooth curves called parabolas.

These curves open up or down. Use the exercise below to determine how the equation of the relation can be used to determine the direction of opening.

Ex. Sketch each of the following sets of quadratic relations simultaneously using technology. Note the direction of opening for each set. Make a conjecture or guess as to which part of the equation is responsible for this result.

Set A: , , open up 9 open down 9

Set B: , , open up 9 open down 9

Conclusion: _____________________________________________________

_____________________________________________________

Activity #8:

C Use the exercise below to investigate how to determine the width and steepness of the graph of a quadratic relation.

Ex. Sketch each of the following sets of quadratic relations simultaneously using technology. Note how steep or narrow each graph is. What part of each equation determines this characteristic?

Set A: , , wider 9 narrower 9

Set B: , , wider 9 narrower 9

Conclusion: _____________________________________________________

_____________________________________________________

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Activity #9:

C Perhaps one of the most important characteristics of the graph of a quadratic relation is the vertex. Recall that the vertex represents the highest or lowest point on the curve.

C You will need to know how to find the vertex because it will have many

applications in solving a variety of upcoming problems. While there are various ways to find the vertex using algebra, you will use technology only to find this special point.

Ex. Use technology to find the vertex of the quadratic relation . Note that the graph is presented below. Your teacher will demonstrate how to find the vertex using technology. Use the space below to record your own notes on how this process is completed. Note that a summary of the required keystrokes to perform the task is provided at the bottom of this page.

How to use Technology to find the Vertex of a Quadratic Relation Step Ø Enter the equation into the equation editor ( ) and graph the

relation. Note that you may have to use or commands to make the graph fit the screen.

Step Ù Press and to access the menu.

Step Ú Select “3:minimum” or “4:maximum” as required. Next use the left and right directional arrows to enter left and right boundary values then press once more. The coordinates of the vertex should appear at the bottom of the screen.

_________________________________________________

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Ex. Use technology to find the vertex of the quadratic relation . The vertex is V( , ).

C In the next example you will apply what you have learned in Activities #7 - #9 to solve an application - type problem involving quadratic relations.

Ex. An object is fired from the a hilltop such that the height ‘ ’, in metres, of the object after ‘ ’ seconds is given by the equation . Use this relation to answer the questions that follow.

a. Why do you think that this problem is quadratic in nature? Think about the description of the event: “an object is fired from a hilltop...”.

_____________________________________________________________

b. How does the equation suggest to you that the problem is quadratic in nature?

_____________________________________________________________

c. In which direction will the graph of the relation open?

Explain how you know by making reference to the equation.

_____________________________________________________________

d. Use technology to find the greatest height that the object reaches. What special point on the graph are we interested in here?

_____________________________________________________________

e. Using technology or your answer from d., determine the number of seconds required for the object to reach its greatest height.

_____________________________________________________________

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Activity #10:

C Another key characteristic of a quadratic relation is the value of the y-intercept or initial value for the relation. In this activity you will learn how to determine this value from the equation of the relation. Later you will see the significance of this value as it is called upon in a variety of application - type problems.

Ex. Study the equation and graph of each of the following quadratic relations.

What relationship appears to exist between the y-intercept of each graph and the corresponding equation ?

i.

a. Using the graph, determine the value of the y-intercept. ______

b. Study the equation which produced the graph. What do you notice?

___________________________________

ii.

a. Using the graph, determine the value of the y-intercept. ______

b. Study the equation which produced the graph. What do you notice?

___________________________________

Conclusion: _____________________________________________________

_____________________________________________________

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Activity #11:

C The graph of a quadratic relation is a parabola and all parabolas have an axis of symmetry. The axis of symmetry is simply a vertical line that divides the

parabola into two equal parts. In this activity you will learn how to determine the equation of this special line.

Ex. The graphs of two quadratic relations are given below. Draw the axis of symmetry for each relation on the graph. What special point on each graph does the axis of symmetry pass through? How can you use this point to determine the equation of the axis of symmetry?

a.

1 2 3 4 5 6 7 8

-1 -2 -3

1 2 3 4 5 6 7 8 9 10 11 12

-1 -2 -3 -4 -5 -6

x y

The vertex is: ______________

The equation of the axis of

symmetry is: __________

____

b.

1 2 3 4

-1 -2 -3 -4 -5 -6 -7

1 2 3 4 5 6

-1 -2 -3 -4 -5 -6 -7 -8 -9 -10

x y

The vertex is: ______________

The equation of the axis of

symmetry is: ______________

Ex. Use technology to determine the vertex and the equation of the axis of

symmetry for each of the following quadratic relations. State your answers in the spaces provided.

a.

Vertex: _______

A. of S.: _______

b.

Vertex: _______

A. of S.: _______

c.

Vertex: _______

A. of S.: _______

Conclusion: _____________________________________________________

_____________________________________________________

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Exercises: Investigating Quadratic Relations

1. Given the equation of each of the following quadratic relations, determine the direction of opening.

a) b) c)

2. The equations of three quadratic relations are given below. Rank them (#1, #2, and #3) from least to most steep.

a) b) c)

3. Use technology to find the vertex of each of the following quadratic relations.

a) V( , ) b) V( , ) c) V( , )

4. State the equation of the axis of symmetry for each of the quadratic relations in #3.

a) A. of. S.: __________ b) A. of S.: ___________ c) A. of S.: ___________

5. State the value of the y-intercept (initial value) for each of the following quadratic relations.

a) y-int: ______ b) y-int: ______ c) y-int: _____

6. Use the data in the table below to answer the questions that follow.

x 0 0.5 1 1.5 2

y 0 3.75 5 3.75 0

a) Using technology, construct a scatter plot to represent the data. Illustrate your plot using the grid provided at the right. Label appropriately.

b) How does the shape of the scatter plot suggest to you that the data is quadratic in nature?

_____________________________________________________________

c) Using technology, find an equation to model or fit the data. Record your equation below.

_____________________________________________________________

d) Use your equation from part c) to find the value of ‘ ’ when ‘ ’ is 1.75.______________________

e) Use the table feature to find the value of ‘ ’ when ‘ ’ is 5. ________________________________

7. Use the data in the table below to find the quadratic equation of best fit. Write your equation below.

x 0 3 6 9 12

y 0 54 72 54 0

The equation of the relation is: ___________________________________

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8. An object is fired upwards from the top of a building such that its height ‘ ’ in metres above the ground ‘ ’ seconds after launch is given by the quadratic relation . Use this information to answer the questions that follow.

a) Use technology to find the vertex of this relation. Record your answer here: ____________________

b) Study the coordinates of the vertex. Use it to determine the maximum height the object reaches. Record your answer here: _________

c) Use the vertex to determine the number of seconds that must pass for the object to achieve this height.

Record your answer here: _________

d) State the equation of the axis of symmetry for this curve. __________

9. For each of the following quadratic relations examine the equation and the value of ‘ ’ to determine if the vertex will be a maximum or a minimum.

a)

vertex is a ____________

b)

vertex is a ____________

c)

vertex is a ____________

10. Match each definition on the left with the appropriate term on the right.

a) The graph of any quadratic relation is called a ____.

b) The largest exponent in any quadratic relation is the number ____.

c) The low/high point of the graph of a quadratic relation is called the ____.

d) All parabolas have an axis of ____. This is a vertical line that divides the graph into two equal parts.

e) If a quadratic relation has a graph that opens down then the coefficient or number in front of the x² term has to be ____.

f) If a quadratic relation has a graph that opens up then the coefficient or number in front of the x² term has to be ___?

g) If a quadratic relation has a graph that opens down then the vertex is the high point or the ____.

h) If a quadratic relation has a graph that opens up then the vertex is the low point or the ____.

i. positive ii. maximum iii. negative

iv. parabola

v. minimum

vi. vertex

vii. symmetry

viii. two

11. For each quadratic relation below determine i.) the vertex, ii.) the equation of the axis of symmetry, iii.) the initial value and iv.) the direction of opening .

a) b) c)

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Using Quadratic Relations to Solve Problems

C Quadratic relations have many useful applications for problem solving. The previous sections have hinted at this.

C In this section you will investigate this idea in more detail solving a variety of problems that involve quadratic relations. We begin with some examples from the field of science.

Activity #12:

C In this activity you will use quadratic relationships to help solve a number of physics problems.

Ex. A stone is dropped from a bridge into a stream. The height, , in metres

above the stream seconds after the moment of release is given by the relation .

a. Using technology, complete the table of values below showing the height of the stone above the stream.

time (s) 0 1 2 3 4

height (m)

b. Construct a sketch of the relation.

c. Why does it make sense for the table and graph to start at time seconds?

_____________________________________

1 2 3 4

10 20 30 40 50 60 70 80

time (t) height (h)

d. From what height was the stone dropped? ____

e. What happens to the stone at time seconds? _____________________

f. Use technology to find the height of the stone after seconds. ______

g. Use technology to find the time that has elapsed when the stone is at a height of 68.75 metres. ________________________________________________

h. Use technology to find the time that has elapsed when the stone is at a height of 18.75 metres. ________________________________________________

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Ex. A snowball is tossed towards a target so that its path is given by

.

a. If ‘ ’ is the height of the snowball above the ground (metres) and ‘ ’ is the horizontal distance (in metres) the snowball has traveled since release,

determine the maximum height that the snowball reaches and its distance from the release point. What special point provides both answers here?

________________________________________________________________

b. From what height was the snowball released? What special name is given to this value?

_______________________________________________________________

c. If the target is located a horizontal distance of 4 metres from the release point, use technology to determine the height of the bull’s-eye above the ground.

________________________________________________________________

Ex. Steve is taking some batting practice with a tennis ball in a gymnasium. The first ball follows the path where ‘ ’ is the time in seconds and ‘ ’ is the height of the tennis ball in metres.

a. If the gymnasium ceiling is 10 metres high, use technology to determine if the ball hits the ceiling.

________________________________________________________________

b. How many seconds pass before the ball returns to the gymnasium floor?

________________________________________________________________

c. From what height was the ball hit?

________________________________________________________________

d. Determine the height of the ball after 4 seconds.

________________________________________________________________

e. For how long has the ball traveled when it is at a height of 7.625 meters .

________________________________________________________________

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Using Quadratic Relations to Make Predictions

C Quadratic relationships frequently occur in the world of business, agriculture and engineering. For example...

“ a business person might want to find the best price at which to sell a product in order to maximize profits or to minimize losses.

‘ a farmer might wish to know how to cultivate his land in order to maximize its usage.

‘ an engineer might wish to design an oil pipeline that will maximize the flow of crude oil.

C In this section you will investigate similar applications.

Ex. The manager of an arena is planning a concert. A survey has told him that if he increases the ticket price, then fewer people will attend the event. The survey also helped the manager predict the revenue he would obtain for the ticket prices shown. He would like to use this information to determine what ticket price will yield the most (maximum) money or revenue.

Ticket Price ( ) $1.00 $2.00 $3.00 $4.00 $5.00 Revenue ( ) $1000 $1800 $2400 $2800 $3000

a. Write the revenues shown as a sequence in the space provided below.

b. Use levels of differences to show, (in the space below the sequence), that the relationship between ticket price and revenue is quadratic.

c. Use technology to determine the equation of best fit for the data. _________

d. How does the equation tell you that this relation has a maximum? _________

______________________________________________________________

e. Use technology to predict the maximize revenue and the corresponding ticket price. What special point provides both answers?

______________________________________________________________

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Ex. A farmer has 80 metres of fencing material with which to build a rectangular pen to house some animals. Because the pen will be built along a wall only three lengths of fencing will be required.

What width should he use so that the pen will have the maximum possible area? What is the maximum area?

a. How many sides of the pen will be constructed with the fencing material? ___

b. By making reference to the diagram above, how many of the sides are

“widths” and how many sides are “lengths”?

_______________________________________________________________

c. Write a formula that relates the width, , and length, , of a rectangle to its area, ? ______________________________________________________

d. Using your work from above, complete the table below illustrating the relationship among the length, width and area.

width (m) 1 2 3 4 5

length (m) area (m²)

e. Use levels of differences to show that there is a quadratic relationship between the width and the area. Use the table below.

width (m) 1 2 3 4 5

area (m²)

f. Use the graphing calculator to find the equation of best fit. _____________

__

g. Using technology, determine the maximum area for the pen and the width and length that will provide it.

_______________________________________________________________

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Exercises: Using Quadratic Relations to Solve Problems

1. For each quadratic relation below, determine the minimum value for ‘ ’ and the value of ‘ ’ at which it occurs. What special point will help you find both values?

a)

The vertex is: ________

The min. y is: ________

When x is: ________

b)

The min. y is: ________

When x is: ________

c)

The min. y is: ________

When x is: ________

2. For each quadratic relation below, determine the maximum value for ‘ ’ and the value of ‘ ’ at which it occurs. What special point will help you find both values?

a)

The max. y is: ________

When x is: ________

b)

The max. y is: ________

When x is: ________

c)

The max. y is: ________

When x is: ________

a) Use technology to find the vertex for this relation. __________________________________________

b) What is the maximum height the object reaches? __________________________________________

c) After how many seconds does the object reach its maximum height? ___________________________

d) From what height was the object launched? ________________________________________________

4. Steve is practising kicking a football into the air. His first kick follows the path while a second kick follows the path , where ‘ ’ is the distance in metres from where the ball was kicked and ‘ ’ is the height of the ball in meters.

a) Use the given equations to complete the table below.

Equation Vertex Max. Height (y) Dist. From Start (x) Initial Value

V( , ) y = when x =

b) Which ball was kicked higher? By how much? _______________________________________

c) From what height was each ball kicked? _______________________________________

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5. A farmer has 100 metres of fencing material he will use to build a rectangular barrier around his pumpkin patch. Because the pumpkin patch is adjacent to an existing wall, only three lengths of fencing will be required. What width should he use to yield the maximum possible area? What is the maximum area?

a) How many sides of the pumpkin patch must be surrounded with the fencing material? ______

b) Making reference to the diagram above, how many of the sides are “widths” and how many sides are

“lengths”?

___________________________________________________________________________________

c) Write a formula that relates the width, , and length, of a rectangle to its area, ?

___________________________________________________________________________________

d) Complete the table below illustrating the relationship among the length, width and area.

width (m) 1 2 3 4 5

length (m) area (m²)

e) Use levels of differences to show that there is a quadratic relationship between the width and the area.

Use the table below.

width (m) 1 2 3 4 5

area (m²)

f) Use the graphing calculator to find the equation of best fit. ___________________________________

g) What length and width will yield the maximum area? What is the maximum area?

___________________________________________________________________________________

a) Use technology to find the vertex for this relation. _________________________________________

b) What is the maximum height the object reaches? _________________________________________

c) After how many seconds does the object reach its maximum height? __________________________

d) State the equation of the axis of symmetry for this curve. ___________________________________

e) Determine the height from which the object was launched. ___________________________________

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7. The Town’s business manager is planning a Trade Fair and is wondering what price to charge participants at the door. A survey has told her that if she increases the ticket price, then fewer people will attend the event.

The survey also helped the manager project the total revenue for several ticket prices as shown. What ticket price will yield the maximum revenue?

Ticket Price ( ) $1.00 $2.00 $3.00 $4.00 $5.00

Revenue ( ) $500 $1300 $1900 $2300 $2500

a) Write the revenues shown as a sequence in the space provided below.

b) Use levels of differences to explain why the relationship between ticket price and revenue is quadratic.

c) Using technology, determine the equation of best fit for the data. _____________________________

d) How does the equation tell you that this relation has a maximum value? _________________________

e) Use technology to predict the ticket price that will yield a maximum revenue. ______________________

f) What will the maximum revenue be if this ticket price is charged? _____________________________

8. The perimeter of a rectangle is 36 centimetres. Use this information to answer the questions that follow.

a) Complete the table below showing the possible areas of the various rectangles formed for the lengths and widths given.

width (cm) 1 2 3 4 5 6

length (m) 17 16 15

area (cm²) 17 32 45

b) Graph the relationship between the width, , and the area, , on the grid provided to the right.

c) Find the equation of best fit for the data. ____________________

d) Explain how this equation tells you that the graph will have a maximum value.

__________________________________________________________

e) What width yields the maximum area? ________________________

f) What is the maximum possible area? __________________________

g) Use your equation from c) to determine when the area will be zero.

__________________________________________________________

1 2 3 4 5 6

5 10 15 20 25 30 35 40 45 50 55 60 65 70 75

width (x) area (y)

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Solving Quadratic Equations

C Quadratic relationships were the focus of the last section. There you studied how to identify these relationships, how to sketch the graph and how to determine many of the curve’s characteristics using the equation of the relationship only. More recently, a variety of problems were solved using principles developed about quadratic relationships and their graphs.

C This section focuses on solving quadratic equations which will enable you to solve various other problems that are quadratic in nature.

C In particular, you will investigate how to solve such equations using two key techniques. They are:

“ Solving by graphing

“ Solving using a formula

C We will begin by reviewing the concept of solving an equation using linear examples only.

Activity #13:

C In this activity you will review how to solve linear equations.

Ex. What number when added to 3 will equal 10? Of course, the answer is 7. The diagram at the right illustrates the question and answer in the form of a balanced scale. On the left of the scale we have 3+7 while on the right we have 10. The balanced scale illustrates that both expressions are equal.

Ex. How could the above equation have been written

using algebra? If the number we are looking for is given a variable name such as ‘ ’, then the equation would read as . Explain below how

algebra was used to solve this equation.

________________________________

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Ex. Explain how algebra was used to solve the equation below.

________________________________

Defⁿ: A value that solves an equation (such as 4 in the example above) is called a solution or root of the equation.

Ex. Solve each linear equation.

a.

c.

e.

b.

d.

f.

C How many solutions or roots did each linear equation have? _____________

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Investigating the Roots of Quadratic Equations

C Now that a review of solving linear equations has been completed, it is time to turn our attention to finding the roots of a quadratic equation. We begin with some review and an introduction to the topic.

Activity #14:

C In this activity an introduction to quadratic equations and solutions is presented.

Ex. Study each relation below and answer the questions that follow.

a. How do you know that each relation is quadratic?

______________________________________________________________

b. Which one represents the graph of a quadratic function? Why?

______________________________________________________________

c. Which one represents a quadratic sequence? Why?

______________________________________________________________

d. Why do you suppose represents a quadratic equation?

______________________________________________________________

e. Given , substitute the value 2 for ‘x’. Repeat the exercise

substituting . What do you notice? ____________________________

(let ) (let )

Defⁿ: The expression is called a quadratic equation.

Value(s) for ‘x’ that satisfy this equation are called solutions or roots.

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Ex. Identify each of the following equations as linear, quadratic or neither.

a. Type:________________

b. Type:________________

c. Type:________________

d. Type:________________

e. Type:________________

f. Type:________________

Ex. Which of the following equations have as a solution? Use check marks (T) to indicate your answers in the spaces provided.

a. Yes ____ No____

b. Yes ____ No____

c. Yes ____ No____

d. Yes ____ No____

Ex. Algebraically determine which (if any) of the following quadratic equations has as a root. Show your work.

a. b. c.

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Exercises: Finding Roots of Equations

1. Solve each linear equation. Show all steps.

a)

b)

c)

d)

e)

f)

g)

h)

2. Identify each equation below as i) linear, ii) quadratic or iii) neither. Indicate your answer in the space provided.

a)

_____________

b)

_____________

c)

_____________

d)

_____________

3. Algebraically determine which of the following quadratic equations has as a root. Place a check mark (T) next to those that do. Be sure to show your work!

a) “ b) “ c) “ d) “

4. Algebraically determine which of the following quadratic equations has as a root. Place a check mark (T) next to those that do. Show your work.

a) “ b) “ c) “ d) “

5. Algebraically determine which of the following quadratic equations has as a root. Place a check mark (T) next to those that do. Show your work.

a) “ b) “ c) “ d) “