18. (a) Sketch the graph of $y = \cos x + 1$ for $0^{\circ} \leq x \leq 720^{\circ}$ - OCR - GCSE Maths - Question 18 - 2018 - Paper 6

To sketch the graph of the function $y = \cos x + 1$, we start by recognizing key features of the cosine function:

1. Basic Shape: The cosine function oscillates between -1 and 1; therefore, $\cos x + 1$ will oscillate between 0 and 2.

2. Period: The standard period of $\cos x$ is $360^{\circ}$, so the behavior of the graph will repeat itself after each $360^{\circ}$. Hence, when sketching for $0^{\circ} \leq x \leq 720^{\circ}$, we will sketch two full cycles.

3. Key Points:

   • At $0^{\circ}$, $y = \cos(0) + 1 = 1 + 1 = 2$.
   • At $90^{\circ}$, $y = \cos(90) + 1 = 0 + 1 = 1$.
   • At $180^{\circ}$, $y = \cos(180) + 1 = -1 + 1 = 0$.
   • At $270^{\circ}$, $y = \cos(270) + 1 = 0 + 1 = 1$.
   • At $360^{\circ}$, $y = \cos(360) + 1 = 1 + 1 = 2$.
   • The same points repeat at $x = 360^{\circ}$ and $x = 720^{\circ}$.

4. Graphing: Start the graph at $(0, 2)$ and follow the points derived above, ensuring that the curve is smooth, reaches its maximum at $x = 0^{\circ}$, $360^{\circ}$, and has a minimum at $180^{\circ}$.

5. Labeling: Clearly label the x-axis and y-axis with appropriate values, ensuring the range of y covers $0$ to $2$ accordingly.