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Function Graphs

Graphs are visual representations of functions. If you know how to read graphs, you can say a lot about a function just by looking at its graph. Learn this fine art of mathematical divining.

Level 4

         

Given the equation \(4x^2+2\sqrt{3}xy+2y^2=1\).
Through what angles \(\theta \in [0,\pi]\) should the axes be rotated so that the term \(xy\) is removed from the transformed equation?

Give the answer as the sum of all values of \(\theta\)(in degrees).

The graph above shows the figure created by the function \(x^2+(y+x)^2=1\).

Find the area of this tilted ellipse correct to 2 decimal places.

For each integer \(n>1\), let \(F(n)\) be the number of solutions to the equation \(\sin{x} = \sin{(nx)}\) on the interval \([0,\pi]\). What is \(\displaystyle\sum_{n=2}^{2007}\)\(F(n)\)?

If \(f(x)=\dfrac{x^2}{1-\cos x}\), where \(0<x<1\). Which of the following describes the function of \(f(x) \)?

If \(f(x) \) and \( g(x) \) have the same fundamental period, then so does \( f(x) + g(x) \).

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