Graphs Transformation

Pls click on the above pics for the full size

Also pls note that the order of TSRRST is also applicable for both the forward & backwards....meaning for eg. (backwards) to get y=f(x) from y=D + Cf (Bx + A), we will still start wif translation (T) ferst in the brackets by getting rid of A first by forming the eqn y= D + C f(Bx + A -A). Hence, the order still follows just that everything is by eliminating the form to get back y=f(x).

How to remember what sort of transformation to do?

Simple, always remember that those transformation along the x-axis or those that are in the brackets are always opposite of wat they meant. For eg in translation, y=f(x + a) means a translation of a units in the negative x-axis direction (to the left by a units // to x-axis) while y=f(x-a) means a translation of a units in the positive x-axis direction (to the right by a units // to x-axis).

For scaling involving the x-axis it is always a reciprocal of the number. For eg, y=(2x) will mean a scaling along the x-axis by 1/2 times. If its y=(Bx), it will be a scaling along the x-axis by 1/B times meaning multiply the x-coordinates by 1/B. For reflection involving the numbers in the bracket like y=(-x), it means a reflection about the y-axis. In summary, it can be noted that any transformation in the bracket is always the opposite of what is meant (for translation), a reciprocal of the number (for scaling) & about the opposing axis (for reflection). Simply put, anything in the brackets is about OPPOSITES.

For those transformation involving outside the brackets or involving the y-axis are quite straight forward because they are usually written to be exactly what it means. For example a translation of y=f(x) + a means a translation of a units in the positive y-axis direction (move up) n the reverse for y=f(x) - a means a translation of a units in the negative y-axis direction (move down). For scaling, there is NO RECIPROCAL FORM NEEDED, when it is written as y= 2f(x), it simply means scaling of factor of 2 units along the y-axis meaning just multiply the y-coordinates by 2 NOT 1/2! Only reflection still follows the general rule of being the opposite whereby y=-f(x) actually means a reflection about the x-axis. Simply put, anything out of the bracket is straight forward and is all involving the y-axis EXCEPT for REFLECTION!!!

After mastering the transformation for graphs, it becomes quite standard for questions involving graphs transformation. Just ensure that there are no carelessness.

How to remember the special angles without the need to memorise or use the calculator

In math, there are group of angles known as special angles. They are 0, 30° (p /6 ), 45 ° ( p/4), 60° (p /3) & 90° (p/2) . These angles are usually needed in math topics such as trigonometry and complex numbers. Usually a calculator can be used however some answers require candidates to give it as the exact form meaning as a fraction. In this case there is no other way than to derive the sin, cos or tan of the special angles which often means memorising. However, there is a quicker way to derive it other than memorising as shown in the table below.

The key to using this type of table is simply the order in which for sinq is from left to right which is from 0 to p/2 radians with the exception that at 0 radians...the sin must be 0. From p/6 to p/2 radians onwards, the sin for that angle will be a fraction of denominator 2 with the numerator, the square root of 1 to 4 corresponding to the respective consecutive special angles. Obviously, some of the fractions can be simplified. Now once, the sin of the number can be derived, to find cos of the special angles is the opposite or reverse order from right to left. Whatever respective numbers that you have written for the sin q from left to right is now the cos q of the angles from right to left. For example, sin 0 = cos p/2 & sin p/6 = cos p/3 . Hence, it is very simple, to derive both sin & cos of the special angles. To find the tan of the special angles, you just have to know the formula that tan q = sinq / cosq and then you can use your already find the tangent based on the sin & cos of the corresponding angles that you have derived earlier. If you are used to the table, it can saved you time and as you get familiar with the table, you can sketch the table by around a min.

Does this table works for obtuse angles or larger angles?

As long as the angles can give basic angles which are the special angles, it will work. However, you must remember the quadrant rules,in which in the 1st quadrant all the Sin, tan & cos are positive, 2nd quadrant where only the sin is positive, the 3rd quadrant where only tan is positive and the 4th quadrant where only the cos is positive. I will illustrate with an example, 135° , the basic angle here is 45° and this angle lies in the 2nd quadrant and it happens that the basic angle is also a special angle. If you want to find the cosine it will be a negative since only sin is positive in this quadrant. So, based on the table, we find the cosine corresponds to 45° in the table and then add a - in front due to quadrant rule explained earlier on. This will give you the correct exact cosine of 135°. It also works for any other angles in the other quadrants as long as the basic angles are the special angles.

This table is quite useful as you can also use it for the inverse of sin, cos & tan to find the specific special angles. So really, it is useful if you guys could master this table.