Special functions

Piece-wise functions A function is piece-wise, if it is defined over a union of domains which have different rules for each set of domain. Example: y x 0

Even/Odd functions A function is said to be even if Similarly, a function is odd if In other words, the graph of an even function is symmetric about the y-axis while the graph of an odd function is symmetrical about the origin. Example of Even Function Example of Odd Function

Can you think of other examples of even and odd functions?

Even Function Odd Function y x 0

Step functions This is an example of a step function, named for its various horizontal ‘steps’

Standard Graphs

Recall:

What do you think this graph looks like?

y = x 2 y =

What do you think this graph looks like?

What do you think this graph looks like?

In general Red is log(x) base 2. Green is ln(x) (log(x) base e). Blue is log(x) base 10. Cyan is log(x) base 0.5.

Transformation of Graphs

(1) Translation along the y-axis x y +a units x y −a units

, the graph is translated along the x-axis by a units to the left. For (2) Translation along the x-axis For, where, the graph is translated along the x-axis by a units to the right. x y a units x y

Translation along the x-axis Compare y = x 2 and y = (x−1) 2 y = x 2 y = (x–1) 2

Q1 (a)Sketch the graph of on the same pair of axes. lies on the graph of, determine, under the transformation the new coordinate of the point. (b)If the point

Q2 (a)What is the coordinates of A under the transformation (b)What is the coordinates of B under the transformation ? __________

(3) Modulus The graph of is derived from that of by reflecting the portion of the graph which lies below the x-axis y x 0

The graph of is derived from that of by reflecting the portion of the graph in the y-axis (3) Modulus x y x y

Example: Sketch the graph of Hence or otherwise, draw the graphs of and

(3) Reflection in the x-axis Reflect the whole graph in the x-axis

(4) Reflection in the y-axis Reflect the whole graph in the y-axis

(5) Modulus (Type 1) x y

1.remove left half of the graph 2.take the mirror image of right half of the graph in y-axis (6) *Modulus (Type 2)

Analysis: Whether it is the positive or negative x- value, they will have the same y-value.

y = |sin x| y = sin |x|

(7) Stretch Under the transformation, the graph is compressed horizontally / vertically OR

Difference between “stretched” and “compressed” compressed Compressed  narrower

Difference between “stretched” and “compressed” stretched Stretched  wider

Stretch/Compressed which way? OR Stretched / compressed parallel to y-axis

Stretch/Compressed? OR Stretched / compressed parallel to x-axis

Stretched along the y-axis with scale factor of 3

Compressed along x-axis with scale factor of 3 Stretched along x-axis with scale factor of 1/3 OR

Combining Transformations

Describe the transformation(s) Reflection in the x-axis

Describe the transformation(s) Translate vertically upwards by 1 unit y = 5x 3 y = 5x 3 + 1

Describe the transformation(s) Translate horizontally to the right by 3 units

Describe the transformation(s) Reflection in the y-axis

Describe the transformation(s) Translate vertically downwards by 4 units y = lg x y = lg x  4

Describe the transformation(s) Translate horizontally to the left by 2 units

Describe the transformation(s) Translate right by 2 units then translate up by 3 units

Describe the transformation(s) Reflection in the x-axis y = (x  6)(x+4) y =  (x  6)(x+4)

Q1 Describe following transformations step by step. Q2 Q3 Q4