Determine a sequence of transformations which maps the graph of $y = \cos \theta$ onto the graph of $y = 3\cos \theta + 3\sin \theta$ - AQA - A-Level Maths Pure - Question 5 - 2017 - Paper 2

To get the function $y = 3\cos \theta + 3\sin \theta$ into the required form, we identify:

• $R = \sqrt{3^2 + 3^2} = \sqrt{18} = 3\sqrt{2}$
• Using the formula $\tan \alpha = \frac{b}{a} = \frac{3}{3} = 1$, we find $\alpha = \frac{\pi}{4}$.

Thus, the transformation can be written as:

$y = 3\sqrt{2} \cos\left(\theta - \frac{\pi}{4}\right)$

The transformations can be broken down as follows:

1. Stretch in the y-direction: The factor of $3\sqrt{2}$ indicates that the graph is stretched vertically by a factor of $3\sqrt{2}$.
2. Phase shift: The term $\theta - \frac{\pi}{4}$ shows that the graph is shifted to the right by an angle of $\frac{\pi}{4}$.

Thus, the sequence of transformations is: a vertical stretch followed by a rightward phase shift.