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  • Level: GCSE
  • Subject: Maths
  • Word count: 2937

In the following coursework, will investigate the gradient functions using the formula y=ax^n, where a is a constant and n is a number.

Extracts from this document...

Introduction

Gradient functions

In the following coursework, will investigate the gradient functions using the formula y=ax^n, where a is a constant and n is a number.

a

n

Y=ax^n

1

1

x

2

1

2x

3

1

3x

4

1

4x

5

1

5x

a

n

Y=ax^n

1

2

x

2

2

4x

3

2

6x

4

2

8x

5

2

10x

a

n

Y=ax^n

1

3

3x^2

2

3

6x^2

3

3

9x^2

4

3

12x^2

5

3

15x^2

a

n

Y=ax^n

1

4

4x^3

2

4

8x^3

3

4

12x^3

4

4

16x^3

5

4

20x^3

I will  plot the graphs of the functions above and I will find their gradient using the formula  gradient=increase in y-axis /increase in x-axis.

Straight line graphs

Straight line graphs are graphs with the equation y=mx+c or y=ax^1,where is stand for the gradient and c is the y- intercept.

Gradient calculations

  1. y=x graph

Gradient of A= increase in y -axis/increase in x-axis

                        = 2/2

                        =1

Gradient of B= increase in y-axis/increase in x-axis

                        = 2/2

                        =1

2. y=2x graph

Gradient of D= increase in y-axis/increase in x-axis

                        = 4/2

                        =2

Gradient of E= increase in y-axis/increase in x-axis

                        = 4/2

                        =2

Gradient of F= increase in y-axis/increase in x-axis

                        = 4/2

                        =2

3. y=-2x graph

Gradient of G= increase in y-axis/increase in x-axis

                        = -4/2

                        =-2

Gradient of H= increase in y-axis/increase in x-axis

                        = -4/2

                        =-2

Gradient of H = increase in y-axis/increase in x-axis

                        = -4/2

                        =-2

Gradient of I = increase in y-axis/increase in x-axis

                        = -4/2

                        =-2

4. y=3x graph

Gradient of J= increase in y-axis/increase in x-axis

                        = 3/1

                        =3

Gradient of K = increase in y-axis/increase in x-axis

                        = 3/1

                        =3

Gradient of L = increase in y-axis/increase in x-axis

                        = 3/1

                        =3

5. y=-3x graph

Gradient of M = increase in y-axis/increase in x-axis

                        = -3/1

                        =-1

...read more.

Middle

2.01

8.08

2.02

8.16

8

2.001

8.008

2.00

8.016

8

2.0001

8.008

2.0002

8.0016

8

3.y=1/2x^2

Gradient when x= 1 = increase in y-axis/increase in x-axis

                        = 1/1.1

                        =0.80

                        =1

Gradient when x= 2 = increase in y-axis/increase in x-axis

                        = 1/1.

                        =1.60

                        =2

Gradient when x= 3 = increase in y-axis/increase in x-axis

                        = 4.5/1.1

                        =4.3

                        =3

Therefore for y=1/2x^2: -

Gradient when x= 1= 0.45

Gradient when x= 2= 1.60

Gradient when x= 3= 4.3

Small increment method

At x= 2

x1

y1

x2

y2

G.F(y2-y1/x2-x1)

2.01

2.02

2.04

2.04

2

2.001

2.002

2.004

2.004

2

2.001

2.0002

2.0004

2.004

2

Conclusion

function

Gradient

y= x^2

2x

y= 2x^2

4x

y= 1/2x^2

x

From going over the calculations I conclude that the gradient of the quadratic graphs depends on the point the tangent drawn at and therefore these graphs have different gradients at different points. For example in the y= x^2 function when x= 1 the gradient is 4 but when x= 2 the gradient is 8. This shows that Quadratic function graphs have different gradients at different points.

Gradient of Quadratic functions depends on the point the tangent is drawn at.

Cubic graphs

Cubic graphs are graphs with the equation y= ax^3 or y= ax^3=bx+c.

Gradient calculations

1.y= x^3 graph

Gradient when x= 1 = increase in y-axis/increase in x-axis

                        = 3.4/1.1

                        =3.09

                        =3

Gradient when x= 2 = increase in y-axis/increase in x-axis

                        = 12/1

                        =12

Gradient when x= 3 = increase in y-axis/increase in x-axis

                        = 27/1

                        =27

Therefore for y= x^3: -

Gradient when x= 1= 3.09

Gradient when x= 2= 12

Gradient when x= 3= 27

Small increment method

At x= 2

x1

y1

x2

y2

G.F(y2-y1/x2-x1)

2.01

12.12

2.02

12.24

12

2.001

12.012

2.002

12.024

12

2.0001

12.0012

2.0002

12.0024

12

2.y=-x^3 graph

Gradient when x= 1 = increase in y-axis/increase in x-axis

                        = 3.4/-1.1

                        =-3.09

                        =-3

Gradient when x= 2 = increase in y-axis/increase in x-axis

                        = 12/-1

                        =-12

Gradient when x= 3 = increase in y-axis/increase in x-axis

                        = 27.5/1

                        =27.5

Therefore for y=- x^3: -

Gradient when x= 1= -3

Gradient when x= 2= -12

Gradient when x= 3= 28

Small increment method

At x= 2

x1

y1

x2

y2

G.F(y2-y1/x2-x1)

2.01

-12.12

2.02

-12.24

-12

2.001

-12.012

2.002

-12.024

-12

2.0001

-12.0012

2.0002

-12.0024

-12

3.y= -x^3+1 graph

Gradient when x= 1 = increase in y-axis/increase in x-axis

                        = 2.5/-1

                        =-2.5

                        =-3

Gradient when x= 2 = increase in y-axis/increase in x-axis

                        = 12/-1

                        =-12

Gradient when x= 3 = increase in y-axis/increase in x-axis

                        = 26.5/-1

                        =-26.5

                        =-27

Therefore for y= -x^3+1: -

Gradient when x= 1= -2.5

Gradient when x= 2= -12

Gradient when x= 3= -26.5

Small increment method

At x= 2

x1

y1

x2

y2

G.F(y2-y1/x2-x1)

2.01

-12.12

2.02

-12.24

-12

2.001

-12.012

2.002

-12.024

-12

2.0001

-12.0012

2.0002

-12.0024

-12

4.y= 0.5x^3 graph

Gradient when x= 1 = increase in y-axis/increase in x-axis

                        = 1.5/1.1

                        = 1.36

                        = 1.5

Gradient when x= 2 = increase in y-axis/increase in x-axis

                        = 6/1

                        = 6

Gradient when x= 3 = increase in y-axis/increase in x-axis

                        = 13.5/1

                        = 13.5

Therefore for y= 0.5x^3: -

Gradient when x= 1= 1.36

Gradient when x= 2= 6

Gradient when x= 3= 13.5

Small increment method

At x= 2

x1

y1

x2

y2

G.F(y2-y1/x2-x1)

2.01

6.06

2.02

6.12

6

2.001

6.006

2.002

6.012

6

2.0001

6.0006

2.0002

6.00012

6

5.y= -0.5x^3 graph

Gradient when x= 1 = increase in y-axis/increase in x-axis

                        = 1.5/-1.1

                        = -1.36

                        = -1.5

Gradient when x= 2 = increase in y-axis/increase in x-axis

                        = 6/-1

                        = -6

Gradient when x= 3 = increase in y-axis/increase in x-axis

                        = 13.5/-1

                        = -13.5

Therefore for y= 0.5x^3: -

Gradient when x= 1= -1.36

Gradient when x= 2= -6

Gradient when x= 3= -13.5

Small increment method

At x= 2

x1

y1

x2

y2

G.F(y2-y1/x2-x1)

2.01

-6.06

2.02

-6.12

-6

2.001

-6.006

2.002

-6.012

-6

2.0001

-6.0006

2.0002

-6.00012

-6

...read more.

Conclusion

G.F(y2-y1/x2-x1)

2.01

64.96

2.02

65.93

97

2.001

64.096

2.002

65.093

97

2.0001

64.00962

2.0002

65.0093

97

Generalization

function

gradient

Straight line graphs(y=mx+c)

The coefficient of x

Quadratic(y=ax^2)

2x

Cubic (y=ax^3)

3x^2

Combined (y=x^2+x^3)

2x+3x^2

y=x^4

4x^3

As we can have seen earlier, the gradient straight line graph is equal to the coefficient of x. For example the gradient of y=x is 1 as the coefficient of x is 1. The gradient of Quadratic graphs depends on the point the gradient from. For example in y=x^2 graph the gradient when x=1 is 2 but when x=2 it’s 4. The cubic and y=x^4 graphs also the point that the gradient from. For example in the y=x^3 when x=1 the gradient is 3,but when x=2 the gradient is 12. From this I have come to conclusion that: -

Y=ax^2

 When a=1  and  n=2

y=nx^n-1

When n=3

y=4x^3-1

y=4x^3

When n=5

y=5x^5-1

y=5x^4

y=nx^n-1

Sin, Cos And Tan graphs

They are trigonometric functions with a unique formulas such as y= cosx and

y=sinx.

The gradient of all trigonometry graphs can be calculated using the formula  Gradient= d(trignomtricfunction)/dx .For example: -

x

Approx. gradient of y= sinx ie. Gradient=d(sinx)/dx

Value

y=sinx

y=cosx

0

0

1

1

90

0

As we can see in the combined graph of  y=sinx +  y= cosx as well as the table above, when x=0 in the y=sinx graph  y=0 too but when x=0 in the  y=cos x the y=1.But when x=90 in  y=sin x graph  y=1 and when x=90 in the  y=cos x graph  y=0  this shows that the Gradient of  y=sin x= Gradient of  y=cos x.

Bibilography

  1.    Core Maths text book for a-levels.
  2.     Heinemin Exdecel Maths  book

...read more.

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