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Graphing Quadratic Functions

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Introduction

Caitlin Holford                                                                30/03/09

Math Investigation

GRAPHING QUADRATIC FUNCTIONS:

INVESTIGATION

In all of the following questions, we will investigate the equations as y = x2 and see what happens to their graphs when we add, subtract or multiply by different constants.

Firstly, we will add k to the equation, where k is a constant, giving us y = x2 + k

I chose 5 different values for k giving us:

y1 = x2 + 6

y2 = x2 + 5

y3 = x2 – 8

y4 = x2 + 22

y5 = x2 – 15

After having used these values and calculated them against x equaling -3, -2, -1, 0, 1, 2 and 3, I was able to obtain the following table:

x

y1

y2

y3

y4

y5

-3

15

14

1

31

-6

-2

10

9

-4

26

-11

-1

7

6

-7

23

-14

0

6

5

-8

22

-15

1

7

6

-7

23

-14

2

10

9

-4

26

-11

3

15

14

1

31

-6

Graph of y = x2 + kimage00.jpg

Using these values I was able to make a graph using them and y = x2 to show the difference between the two equations:


As we can see in the above graph, the vertex (turning point of a graph) is located every time on the y-axis.
...read more.

Middle

-11

3

15

14

1

31

-6

With these results the below graph was given:

Graph of y = (x-h)2


image01.jpg

We can clearly see the same parabola repeated, however, as opposed to the graph of y=x2 + k, the graphs are moved horizontally, giving us a horizontal transformation. The reason for this translation is that the value of x is being directly changed whereas with the previous equation, the value of x was simply being added or subtracted onto.

Again, the vertex is directly changed according to the value of k.

In the third part, we will see what will happen when we put these two functions x2+k and (x-h) 2 together and how this influences a graph. The two functions put together will make:

y1 = (x - 13) 2 + 6

y2 = (x - 4) 2 + 5

y3 = (x +5) 2 – 8

y4 = (x + 7) 2 + 22

y5 = (x - 19) 2 – 15

x

y1

y2

y3

y4

y5

-3

262

54

-4

38

469

-2

231

41

1

47

426

-1

202

30

8

58

385

0

175

21

17

71

346

1

150

14

28

86

309

2

127

9

41

103

274

3

106

6

56

122

241

Hand-drawn graph for (x – h) 2 + k


Coordinates for graph:

X

Y1

Y2

Y3

Y4

Y5

-3

-2

-1

0

1

2

3

Confirmation of graph with technology:

Graph for y=(x-h)2

...read more.

Conclusion

image04.jpg

The reason why some of the parabolas are negative is simply because there is a negative amount of x.

In this final part, we will investigate what happens when we put all of these functions together, giving us the equation of a (x - h) 2+ k.

y1 = 3*(x - 13) 2 + 6

y2 = 6*(x - 4) 2 + 5

y3 = -8*(x +5) 2 – 8

y4 = -26*(x + 7) 2 + 22

y5 = 17*(x - 19) 2 – 15

x

y1

y2

y3

y4

y5

-3

771

299

12

-106

8213

-2

681

221

37

-178

7482

-1

594

155

72

-266

6785

0

513

101

117

-370

6122

1

438

59

172

-490

5493

2

369

29

237

-626

4898

3

366

11

342

-778

4337

Graph for a*(x - h) 2+ k.image05.jpg

In this graph, we have a combination of all of the changes made by each constant. We have the vertical translation from adding k, the horizontal translation from subtracting h and squaring it, and finally the stretch from multiplying x by a.

Through this investigation, I have learnt how to properly graph functions and also how different values affect a graph depending on how you use them.

...read more.

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