Algebra Tutorial- Quadratic Equation

Quadratic equations are of the from: Y = a+bx + c x.x

Take the example ; y = x.x + 5x + 6

To solve x.x + 5x + 6 =0 is easy.Factorize: (x +3 )(x+2) = 0

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

The solution is: x+3 =0 or x = -3and x+2 =0 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 in the general equation.

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

3 92 41 10 0

-1 1-2 4-3 9

Let us plot this graph:This parabola has vertex at origin (o,o) and is symmetricalabout Y axis.let us draw another parabola where c is negative. Y = -X.XForm a table: X Y

3 -92 -41 -10 0

-1 -1-2 -4-3 -9

Let us plot this graph.

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

To understand the construction of various parabolas and their equations, we will take up three operations.

Translatio [along X axis]

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

0

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Y = x.x parabola

Column G

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

-10

-9

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-6

-5

-4

-3

-2

-1

0

Y = - x.x parabola

Y

Let me shift the parabola to the right . To translat along the x axis,shift

let the new X be X-2The equation becomes: Y = (X - 2 ).(X - 2 )

Let us form a table again: X Y5 94 43 12 01 10 4

-1 9Let us plot the graph>

What do you see? The parabola is shifted to the right and the a vertex is at (2,0)

If I write Y = (x+2)(x+2) the parabola will be shifted to the left ,with apex at x=-2 Y=0.

X Y1 90 4

-1 1-2 0-3 1-4 4-5 9

Translation along yaxis or vertical shift

It is easy to shift the parabola along the Y axis.Simply add a number t to 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) - 3 shiftes the parabola below the Y=0 horizontal line by three units.

X Y=(x-2)(x-2)+35 124 73 42 31 40 7

-1 12

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

-2 -1 0 1 2 3 4 5 6

0

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Y = (x-2) ^2

Y

-6 -5 -4 -3 -2 -1 0 1 2

0

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Y = (x + 2) ^ 2 parabola

Column E

-2 -1 0 1 2 3 4 5 6

0

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6

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14

Y= (x-2)^2 + 3

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

-2 -1 0 1 2 3 4 5 6

-4

-2

0

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6

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Y = (x -2)^2 -3

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

5 64 13 -22 -31 -20 1

-1 6

We have seeeen two operations.S hifting to the left or right and lifitng up or down in the vertical direction.

ScalingThis operation contracts or expands the parabola.

Take the basic parabola again: y = x.xLet us multiply with a constant k

Y = k x.xTake k= 2 Y = 2 x.xForm a table:X Y=2.x.x y = x.x

3 18 92 8 41 2 10 0 0

-1 2 1-2 8 4-3 18 9

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

Next, let us plot the graph of Y= kx.x where k=1/2Form a Table:X Y=0.5*x*x y=x.x

3 4.5 92 2 41 0.5 10 0 0

-1 0.5 1-2 2 4

-2 -1 0 1 2 3 4 5 6

-4

-2

0

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6

8

Y = (x -2)^2 -3

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

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

Y=2.x.x

y = x.x

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

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Y=0.5*x*x

y=x.x

-3 4.5 9

This parabola, is wider than y=x.x Now you see the effect of k on the shape.'k' is called the shape factor.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 (h,k)

You have seen how this general equation is constructed. Keep this equation always in mindyou handle parabolas.

Do-it-yourself ExercisesYou must practise drawing different parabolas using the general equation.1 y = (x -1)^2+2

2 y= (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.XThen we did three operations--translation or shifting along x axis;translation or shifting along Y axis or lifitng the parabola up or downand scaling,multiplying with a constant to change the shape of the parabola.We arrived at the general equation: Y = a (x-h)(x-h) + kWith 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.XHow do we get this from the general equation of parabola given here.This is simple. Y = a (x -h)^2 + kExpanding the first term : Y = a (x.x - 2h.x + h.h) + k

Y= ax.x - 2ha.x + ah.h+kNow we can see that the standar d form emerges.

Take the example: y=3(x+1)^2 +2y= 3(x.x+2x+1) + 2y= 3x.x +6x +3+2y= 3x.x+6x +5 y= 5 + 6x+3x.x

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

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Y=0.5*x*x

y=x.x

Comparing with standard from: a=5 b=6 c=3

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

a=2 b=4b=-2ha=4h=-1

h.h=1 ahh=2 ahh+k=5k=3

Now we have founda=2 h=-1 k=3The general equation can be written;

y= 2(x+1)^2+3Another method is completing the sqaure: y= 2xx+4x +5

y= 2(xx+2x ) +5y=2(xx +2x +1)-2+5y= 2(x+1)^2 +3

Therefore there is no difference between the general equation for the parabola and the standard quadratic equation!

Do-it-yourself problems1 Convert this equation into general equatio n and find the vertex.y= xx+6x+9y= 2x.x+8x+12

SummaryWe started with y=x.x and modified by translation along x axis and y axis andscaling .We also sawthe parabola can be e pressed in the standard form:

The standard form: Y= a =bx+cxx

The general equation for parabola: Y = a(x-h)(x-h) + k

Note: This tutorial and others are based on my extensive tutoring expereince Send your feedback and your suggestions.

To understand the construction of various parabolas and their equations, we will take up three operations.

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

0

1

2

3

4

5

6

7

8

9

10

Y = x.x parabola

Column G

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

-10

-9

-8

-7

-6

-5

-4

-3

-2

-1

0

Y = - x.x parabola

Y

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

-2 -1 0 1 2 3 4 5 6

0

1

2

3

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5

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7

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9

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Y = (x-2) ^2

Y

-6 -5 -4 -3 -2 -1 0 1 2

0

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9

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Y = (x + 2) ^ 2 parabola

Column E

-2 -1 0 1 2 3 4 5 6

0

2

4

6

8

10

12

14

Y= (x-2)^2 + 3

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

-2 -1 0 1 2 3 4 5 6

-4

-2

0

2

4

6

8

Y = (x -2)^2 -3

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

hifting to the left or right and lifitng up or down in the vertical direction.

-2 -1 0 1 2 3 4 5 6

-4

-2

0

2

4

6

8

Y = (x -2)^2 -3

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

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

0

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

Y=2.x.x

y = x.x

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

0

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2

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5

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9

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Y=0.5*x*x

y=x.x

This parabola, is wider than y=x.x Now you see the effect of k on the shape.'k' is called the shape factor. or

equation always in mind when

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

0

1

2

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Y=0.5*x*x

y=x.x

between the general equation for the parabola and the standard quadratic equation!

along x axis and y axis and

at Palo Alt o, California.