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Introduction

Caitlin Holford                                                                30/03/09

Math Investigation

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 + k 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.

Middle

-11

3

15

14

1

31

-6

With these results the below graph was given:

 Graph of y = (x-h)2 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

Conclusion 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. 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.

This student written piece of work is one of many that can be found in our GCSE Miscellaneous section.

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