GRAPH - CUBIC

CUBIC GRAPH : y = ax^3 

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Characteristics:
It has varying gradients.
Turing points can range from 0 to 2.
When the coefficient of B and C is 0, there will only be 1

GRAPH - SQUARE RECIPROCAL

SQUARE RECIPROCAL GRAPH : y = ax^(-2) 

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y = ax^n , n = -2

y=3x^-2

















y= -1/x^2

General characteristics of the graphs :
They have a line of symmetry at the y-axis. They are reflections of each other. They are asymptotes that never cross the x or y axis.

- Mason, Shaun and Kai En

GRAPH - QUADRATIC

QUADRATIC GRAPH : y = ax^2

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They are all parabolas (except for y=0x^2).
The graphs have both an increasing and decreasing gradient
It has a maximum or minimum point,  the points is the turning point of the graph
There is a line of symmetry parallel to the y axis

Summary of Lesson (2-4-13) (Ryan)

LESSON 1

Addition of Case 7 and 8

Case 7

y=3e^x

y=-e^x

y=e^0 (This is not exponential)
y=3e^x + 4
y= -e^x -4
y=e^-x

y = | x |

if x>0, y=x
if x<0, y=-x
if x=0, y=0

Case 8

y= 3lgx

y= -lgx (Reflection about the y-axis)

y= lnx

y= 3lgx + 4

y= -lgx - 4

y= log2 x

Case 2a

y= |-3/4x|

or

y=abs (-3/4 x)

y=|3/4x|

or

y=abs (3/4 x)

Hint: For EOY Mr Johari can convert normal graph problems to a trigonometry problem or add in different topics

Let y = f(x)

Case 1: y = -f(x)

this would be a reflection about the x-axis

Case 2: y = f(x) + c

this would be a vertical shift (moving up and down the graph)

Case 3: y = f(-x)

this would be a reflection about the y-axis

case 4: y = absolute f(x)

when x > 0 , y = f(x)

when x < 0 , y = -f(x)


LESSON 2

Prove that

1                              1
_______     +  _________ = 1
log a ab           log b ab

Answer (Courtesy of Justin) :

log a a               log b b
_______     +  _________ =  log ab a (change to base a )+ log ab b = log ab ab = 1
log a ab           log b ab

GRAPH - LOGARITHMIC

LOGARITHMIC GRAPH : y = aLg (x)

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Description:

Y-axis asymptote

y=mlg(x), when m increases, gradient increases.

y=mlg(x)+c, c is where the graph intercepts with x=1 

Done by Justin, Jemima and Crystal

GRAPH - RECIPROCAL

RECIPROCAL GRAPH : y = ax^(-1) 

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 The graph shows two curves in positive-positive region and negative-negative region. Graph also always has two asymptotes, one vertical and one horizontal.

Adding to the constant shifts the graph up and down. It also shifts the horizontal asymptote. Ie. The constant is the horizontal asymptote.

Negative coefficient of the value of x rotates the graph 90 degrees about the y axis. This is because y values is negative when it should have been positive.

 

Subtracting from the constant causes the graph to shift downwards

Subtracting a value from the x within the fraction translates the graph to the left by negative of this value. Also shifts vertical asymptote to this value.

GRAPH - EXPONENTIAL

EXPONENTIAL GRAPH : y = ae^x

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Exponential graph:

THe equation of the graph is a (e^ (bx)) +c
If b=0, the graph would be a horizontal line and intersect the y intercept at a+c
if b<0, the graph would have a decreasing gradient.
if b>0, the graph would have a increasing gradient.

The value of a would affect the position of the graph:
if a>0, the graph would be above the x axis and y intercept at a+c
if a<0, the graph would be below the x axis and y intercept at a+c

Note: For exponential graphs, c is NOT the y intercept.