Algebra Tutorial Quadratic Eqn And Parabolas

Algebra

Tutorial- Quadratic Equation

Quadratic equations are of the from: Take the example ; To solve Factorize: [To learn

Y=

y=

a+bx + c x.x

x.x + 5x + 6

x.x + 5x + 6 =0 (x +3 )(x+2) = 0

is easy.

factoring' study my earlier tutorial : Algebra Tutorial-factoring.]

The solution is: and

x+3 =0 x+2 =0

or x = -3 or x =-2

We shall see the special forms of quadratic equations which have many uses. Parabolas Take

y = c .X.X

a=0; b=0

let us plot a graph: Take c =1 Form a table: X

in the general equation. Y = x.x parabola

Y 3 2 1 0 -1 -2 -3

9

9 4 1 0 1 4 9

Let us plot this graph: This parabola has vertex at origin (o,o) and is symmetrical about Y axis. let us draw another parabola where c is negative. Y = -X.X Form a table: X Y 3 -9 2 -4 1 -1 0 0 -1 -1 -2 -4 -3 -9 Let us plot this graph.

8 7 6 5 4 3 2 1 0 -3

-2.5

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

Y = - x.x parabola 0 -1 -2 -3 -4 -5 -6 -7 -8 -9 -3

-2.5

-2

-1.5

-1

-0.5

0

0.5

1

1

what do you see now. this is also a prabola, but upside down.The apex vertex is at is [0,0] at (0,0)

To understand the construction of various parabolas and their equations, we will take up three operation Translation[along X axis]

Let me shift the parabola to the right . let the new X be X-2 The equation becomes: Let us form a table again:

To translatealong alongthe X aixis, x axis,shift I shift the apex from X

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

X

Y = (x-2) ^2

Y 5 4 3 2 1 0 -1

9

9 4 1 0 1 4 9

8 7 6 5 4 3 2 1

Let us plot the graph>

0 -1

What do you see? If I write

-0.5

0

0.5

1

1.5

2

2.5

3

3.5

0

0.5

1

The parabola is shifted to the right and the apex vertex is atisx=2 at (2,0) y=0

Y = (x+2)(x+2) X

the parabola will be shifted to the left ,with apex at x=-2 Y=0.

Y 1 0 -1 -2 -3 -4 -5

Y = (x + 2) ^ 2 parabola

9 4 1 0 1 4 9

9 8 7 6 5 4 3 2 1 0 -5

-4.5

-4

-3.5

-3

-2.5

-2

-1.5

-1

-0.5

Translation along yaxis or vertical shift It is easy to shift the parabola along the Y axis. Simply add a number toto the equation. Y = (x-2 ).(x-2) + 3 shifts the parabola above the Y =0 horizontal line by three units. Y = (x-2 ).(x-2) (x-2)+-33shifts shiftes the parabola the parabola above below the Ythe =0Y=0 horizontal horizontal line line by three by three units. units. X

Y= (x-2)^2 + 3

Y=(x-2)(x-2)+3 5 12 4 7 3 4 2 3 1 4 0 7 -1 12

12 11 10 9 8

Y=(x

7 6 5 4 3 -1

X 5 4

Y=(x-2)(x-2)-3 6 1

-0.5

0

0.5

1

1.5

2

2.5

3

3.5

Y = (x -2)^2 -3 6 5 4 3 2 1 0

4

4.5

5

Y = (x -2)^2 -3 6 5

3 2 1 0 -1

-2 -3 -2 1 6

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

0

1

2

3

4

5

We have seen een tow two operations: operations.S shifting hiftingthe to parabola the left ortoright rightand or left lifitng [along up orthe down X axis] in the and vertica shifti Scaling This operation contracts or expands the Take the basic Let us multiply

parabola.

parabola again: with a constant k Y = k x.x Y = 2 x.x

Take k= 2 Form a table: X Y=2.x.x y = x.x 3 18 9 2 8 4 1 2 1 0 0 0 -1 2 1 -2 8 4 -3 18 9

y = x.x Y = 2 x.x and Y = x.x 18 16 14 12 10 8

This parabola is narrower when compared to y=x.x

6 4 2 0 -3

Next, let

us plot the graph graphofof Y= kx.x Form a Table: X Y=0.5*x*x y=x.x 3 4.5 2 2 1 0.5 0 0 -1 0.5 -2 2 -3 4.5

where

k=1/2

-2

-1

0

1

2

9 8 7

9 4 1 0 1 4 9

6 5 4 3 2 1 0 -3

-2

-1

0

1

2

This parabola, is wider than y=x.x Now you see the effect of k on the shape.'k' is called the shape facto scaling factor. General Equation for Parabola We can now write the general

equation

for the

Parabola:

Y = a (X - h )(X - h) + k Here the scaling factor is 'a' The parabola has the vertex at the point You have seen how this general equation is constructed. you handle parabolas. Do-it-yourself Exercises You must practise drawing different 1 y= (x -1)^2+2 2

y=

(h,k) Keep this equation always in mind

parabolas using the general

equation.

(x+1)^2 + 2

3 y= -(x+1)^2 - 3 4 y= -(x-3)^2 +3 5 y = 3(x+1)^2 +2 6 y= -2(x-1)^2 +2 The standard Equation for Parabola Well..we started with the simple basic equation for a parabola: Y = X.X Then we did three operations--translation or shifting along x axis; translation or shifting along Y axis or lifitng the parabola up or down and scaling,multiplying with a constant to change the shape of the parabola. We arrived at the general equation: Y = a (x-h)(x-h) + k With this we can plot any kind of parabola. We seem to have lost our way in these manipulative exercises-shifting and scaling. What is the standard equation for the parabola? Y = a + b X + c X.X How do we get this from the general equation of parabola given here. This is simple. Y = a (x -h)^2 + k Expanding the first term : Y = a (x.x - 2h.x + h.h) + k Y= ax.x - 2ha.x + ah.h+k Now we can see that the standarddfrom formemerges: emerges. Take the example:

y=3(x+1)^2 +2 y= 3(x.x+2x+1) + 2 y= 3x.x +6x +3+2 y= 3x.x+6x +5 Comparing with standard from:

What about the reverse process? Take the equation: y = 2x.x + 4x + 5

y= 5 + 6x+3x.x a=5 b=6

c=3

a=2

h.h=1 k=3

b=4 b=-2ha=4 h=-1 ahh=2 ahh+k=5

Now we

have founda=2 h=-1 k=3 The general equation can be written; y= 2(x+1)^2+3 Another method is completing the sqaure: y= 2xx+4x +5 y= 2(xx+2x ) +5 y=2(xx +2x +1)-2+5 y= 2(x+1)^2 +3

Therefore there is no difference between the general equation for the parabola and the standard quadratic equatio Do-it-yourself problems 1 Convert this equation n into inot general equationnand th find the and apex findandthe scaling vertex. factor for the parabola: y= xx+6x+9 y= 2x.x+8x+12

Summary We started with y=x.x and modified by translation along x axis and y axis and scaling . We also saw thethat parabola the parabola a can be ebexpressed can expressed be in in standard in the standard form ofform: quadratic equation and also as an equatio The standard

form:

Y= a =bx+cxx

The general equation for parabola: Note: This tutorial and others Send your feedback and

Y = a(x-h)(x-h) + k

are based on my extensive tutoring expereince at Palo Alto,Ca your suggestions.

Y = x.x parabola

Column G

-1.5

-1

-0.5

0

0.5

1

1.5

2

2.5

3

Y = - x.x parabola

Y

-1.5

-1

-0.5

0

0.5

1

1.5

will take up three operations.

2

2.5

3

Y = (x-2) ^2

Y

.5

1

1.5

2

2.5

3

3.5

4

4.5

5

pex at x=-2 Y=0.

+ 2) ^ 2 parabola

2

Column E

-1.5

-1

-0.5

0

0.5

1

2)^2 + 3

5

Y=(x-2)(x-2)+3

3

3.5

4

4.5

5

= (x -2)^2 -3

Y=(x-2)(x-2)-3

= (x -2)^2 -3

Y=(x-2)(x-2)-3

3

4

5

ng up or down in the vertical direction.

Y = 2 x.x and Y = x.x

Y=2.x.x y = x.x

0

1

2

3

Y=0.5*x*x y=x.x

-1

0

1

2

3

.'k' is called the shape factor.

equation always in mind

or

when

standard quadratic equation!

xis and y axis and

at Palo Altoo, California.