Sketch the graph of $y = an^2x$ for $0 eq x eq 360$. - Edexcel - GCSE Maths - Question 12 - 2018 - Paper 3

To sketch the graph of $y = an^2x$, follow these steps:

Identify Key Features

1. Periodicity: The tangent function has a period of $180^ ext{o}$, so the square of the tangent function will also have the same periodicity.

2. Asymptotes: The graph of $an^2x$ will have vertical asymptotes at $x = 90^ ext{o}$ and $x = 270^ ext{o}$, where $an x$ is undefined.

Determine Values

3. Points to Plot: For intervals, evaluate the function at key points:
   • At $x = 0^ ext{o}$, $y = an^2(0) = 0$.
   • At $x = 45^ ext{o}$, $y = an^2(45) = 1$.
   • At $x = 90^ ext{o}$, there's a vertical asymptote.
   • At $x = 180^ ext{o}$, $y = an^2(180) = 0$.
   • At $x = 225^ ext{o}$, $y = an^2(225) = 1$.
   • At $x = 270^ ext{o}$, there's another vertical asymptote.
   • At $x = 360^ ext{o}$, $y = an^2(360) = 0$.

Graph the Function

4. Graphing: Start plotting the points you calculated above. Draw the curve to approach the asymptotes at $90^ ext{o}$ and $270^ ext{o}$, and make sure the graph has a parabolic shape opening upwards between each interval bounded by asymptotes.

Final Touches

5. Label the Axes: Make sure to clearly label your axes and any critical points for clarity.