There are six transformations that you can perform on the graph of a function.
Let’s look at what happens when we apply these to the function y = sin (x) 0 ≤ x ≤ 360
The results will be the same for the functions y = cos (x) and y = tan (x).
f(x) → f(x) + a {Translation with a vector of }
Sin (x) → sin (x) + 1 has the effect of moving the graph upwards by 1.
f(x) → f(x + a) {Translation with a vector of }
Sin (x) → sin (x + 90) has the effect of moving the graph left by 90°.
This transforms y = sin(x) into y = cos(x).
f(x) → af(x) {Stretch in the y-axis, scale factor a}
Sin (x) → 2sin (x) has the effect of increasing every value of y by 2.
f(x) → f(ax) {Stretch in the x-axis, scale factor 1/a}
Sin (x) → sin (2x) has the effect of multiplying every value of x by 1/2.
f(x) → -f(x) {Reflection in the x-axis }
Sin (x) → -sin (x) has the effect of reflecting the entire graph in the y-axis.
f(x) → f(-x) {Reflection in the y-axis}
Sin (x) → sin (-x) has the effect of reflecting the entire graph in the x-axis. This looks just like the previous function due to the repeating nature of the graph.
Reflecting a trig graph in the y-axis effectively translates it by 180°.
Combining transformations
You will occasionally have to combine trig transformations (this will usually be at the end of the paper and as such, a grade 9 question).
The trick with these is to chain the transformations from the base function, dealing with any functions that involve the bracket first (just like BODMAS).
Example 1:
The function f(x) = sin (x) is transformed into the function g(x).
Find the function g(x) in the form sin (ax) + b
When you get questions like this, the format of the question is set out in a deliberate order. When you have, for example, values of a and b to find, the examiner expects that you will find a before b.
Step 1: Finding the value of a
We know that we are looking at the transformation y = sin (x) → y = sin (ax) which has the effect of multiplying every value of x by 1/a.
We can see that the graph, instead of repeating once in 360°, repeats three times.
This gives a value of a = 3.
Step 2: Finding the value of b
After applying the first transformation y = sin (x) → y = sin (3x), we have the graph:
To get from this function to our final function, we can see that we have to move upwards by 1.
This gives a value of b = 1.
Our final function is therefore g(x) = sin (3x) + 1
Now it's time for some questions.