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Juan David Varela

Maths SL

Codigo:

Zulema Leon

2010-2011

Introduction:
This portfolio has the purpose to investigate the quadratic functions and the graphs of sine functions. For doing this I’m going to graph by using to different programs: Graphmatica, and Derive, depending on the situation. I am also going to give several examples of about what I’m asked to do, for corroboratingthe answer. All this with the purpose to get to a more certain conclusion, and answers.
Investigating the Quadratic Function.
1. Sketch the graphs of:
(a). Y=X2

(b). Y=X2+3

(c). Y=X2-2

What do you notice? Can you generalize?
As we can see in the graphs, we can notice that each time we change a value in the formula, the graphs displaces in the vertical (y) axis. So we can deducethis:
y=mx+d When d <0 goes down in the graph. When d>0 it goes up in the graph.

2.
(a). Y=X2

(b). Y=(X-2)2

(c). Y=(X+3)2

What do you notice? Can you generalize?
As we saw in the graphs we can deduce that each time we change a value in the formula, the graphs displaces in the horizontal (x) axis. So we can conclude this: when a formula is written in the way (x-h) if theh>0 it goes to the left, and if h<0 the graph will displace in the x axis to the right side.

3. Where would you expect the vertex on the graph of Y=(x-4)2+5 to be? Explain why.

Following the displacement rules I would’ve expected that the vertex of the graph would be (4,5) because as the formula says: y=(x-4)2+5, the graph moved 4 places to the right in the (x) horizontal axis, andit also moved 5 up in the (y) vertical axis. Locating the vertex in the point (4,5).

4.
(a) Express x2-10x+25 in the form (x-h)2
From the general equation I could get to this answer by factorizing (completing the square) :(x-5)2
(b) Express x2-10x+32 in the form (x-h)2-g
From the general equation I could get to this answer by factorizing (completing the square): (x2-10x+__)+32-__(x2-10x+52)+32-52
(x-5)2-7.
(c) Repeat this procedure with some examples of your own.
* X2-8x+15=
(x2-8x+42)15-16=
(x2-8x+16)15-16=
(x-4)2-1
* X2-10x+2376=
(x2-10x+52)2376-52=
(x2-10x+25)2376-25=
(x-5)2+2351
* X2-12x+24=
(x2-12x+62)24-62=
(x2-12+36)24-36=
(x-6)2-12
* X2-14x+120=
(x2-14x+72)120-72=
(x2-14x+49)120-49=
(x-7)2+71
* X2-4x+8=
(x2-4x+22)8-22=(x2-4x+4)8-4=
(x-4)2+4

(d) Describe a method of writing the quadratic expression x2+bx+c in the form (x-h)2+g.
To pass an equation showed as a general form x2+bx+c to (x-h)+g. First you have to get the second term of the general form and divided in two, and then give him the power of 2. You put this value adding it with x2+bx, and you subtract this value to c, so that you can get g. afteryou’ve done that you would have this: (x2-bx+(b2)2)+c-(b2)2. Then you factorize the values inside the parenthesis and subtract the numbers outside the parenthesis.

5. Describe the shape and position of the graph of Y=(X-h)2+g. Provide an explanation for this.
The position of the parabola with depend on H and on G. The H locates the vertex in the horizontal (x) axis, and the G gives the positionof the vertex in the vertical(y) axis. This vertex gives us a point which would be the vertex, depending on the symbol before letter a the parabola would go upward or downwards.

6. Do you findings apply to the graphs of other types of functions? Can you generalize?

* y=x3+2 Polinomial

* y=3x+2 Exponential

As we can see in this graphs we can see that the rules are appliedbecause it does not matter if it is exponential, logarithm, or other, the graphs still displaces depending on what the d determines. D will always show the the displacement in the (y) vertical axis.

* y=3x+2 Exponential

* y=(x+3)2 Polinomial

As we can see again in the graphs we can also see that the rules are also applied in the x axis. It doesn’t matter if it is...
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