Analyzing Cosine Transformations

00:00 6 marks

For the function \( y = 3\cos(2x – \frac{\pi}{3}) \), determine its key transformation parameters and sketch its graph over one period.

Part a

5 minutes 3 marks

Question

Determine the amplitude, period, and horizontal shift for the function \( y = 3\cos(2x – \frac{\pi}{3}) \).

Hint Hint

Hint

Recall, the amplitude is given by the coefficient 3, the period by \(\frac{2\pi}{b}\), and the horizontal shift by solving \(2x – \frac{\pi}{3} = 0\).

Answer Answer
Mark Scheme Mark Scheme
Explanation Explanation
Formula Booklet

Answer Answer

Complete Answer

Amplitude = 3, Period = \(\pi\), Horizontal shift = \(\frac{\pi}{6}\) to the right.

Mark Scheme Mark Scheme

Answer Explanation Answer Explanation

Formula Booklet

Part b

5 minutes 3 marks

Question

Sketch the graph of \( y = 3\cos(2x – \frac{\pi}{3}) \) over one period, clearly marking the key points derived in part (a) using verbal descriptions.

Hint Hint

Hint

Plot the maximum, minimum, and zero-crossings based on the amplitude, period, and horizontal shift. Describe these points clearly in text.

Answer Answer
Mark Scheme Mark Scheme
Explanation Explanation
Formula Booklet

Answer Answer

Complete Answer

Verbal description: The graph starts at its maximum value of 3 at \(x = \frac{\pi}{6}\). It then decreases to cross the x-axis, reaches its minimum value of -3, increases to cross the x-axis again, and returns to its maximum value of 3, completing one period at \(x = \frac{\pi}{6} + \pi = \frac{7\pi}{6}\). Key points are: Maximum at \((\frac{\pi}{6}, 3)\), first zero crossing at \((\frac{5\pi}{12}, 0)\), minimum at \((\frac{2\pi}{3}, -3)\), second zero crossing at \((\frac{11\pi}{12}, 0)\), and period end at \((\frac{7\pi}{6}, 3)\).

Mark Scheme Mark Scheme

Answer Explanation Answer Explanation

Formula Booklet