Identify Transformations with Trigonometric Graphs

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.