Sketch the graph of $y = an^2 x$ for $0 ext{ °} ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } 0 ext{ °} ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } 360 ext{ °}$ - Edexcel - GCSE Maths - Question 12 - 2018 - Paper 3

To sketch the graph of $y = an^2 x$, begin by identifying key characteristics of the function:

Key Points:

1. Periodicity: The function $y = an^2 x$ has a period of $180°$. Therefore, we will observe the behavior over the interval $[0, 180°]$ and then replicate it for $[180°, 360°]$.
2. Behavior at Asymptotes: The function has vertical asymptotes at $x = 90°$ and $x = 270°$ where $an x$ is undefined. This means that as $x$ approaches these points from the left and right, the function will tend towards infinity.

Graph Sketching Steps:

• Start your graph on the left at $(0, 0)$ since $an^2(0) = 0$.
• As $x$ approaches $90°$, the function will rise sharply towards infinity, indicating a vertical asymptote here.
• After $90°$, the function will drop back down to $0$ at $180°$ as $an^2(180°) = 0$.
• Reflect this behavior in the interval from $180°$ to $360°$ with another vertical asymptote at $270°$ and return back to $0$ at $360°.

Conclusion:

Your final sketch should show a repeating pattern between these key points, with clear asymptotic behavior at the relevant angles.