Algebra

Function Graphs

Function Graphs: Level 4 Challenges

         

Given the equation 4x2+23xy+2y2=14x^2+2\sqrt{3}xy+2y^2=1.
Through what angles θ∈[0,π]\theta \in [0,\pi] should the axes be rotated so that the term xyxy is removed from the transformed equation?

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

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Tanishq Varshney by Tanishq Varshney

The graph above shows the figure created by the function x2+(y+x)2=1x^2+(y+x)^2=1.

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

Trevor Arashiro by Trevor Arashiro

For each integer n>1n>1, let F(n)F(n) be the number of solutions to the equation sin⁡x=sin⁡(nx)\sin{x} = \sin{(nx)} on the interval [0,π][0,\pi]. What is ∑n=22007\displaystyle\sum_{n=2}^{2007}F(n)F(n)?

Anuj Shikarkhane by Anuj Shikarkhane

If f(x)=x21−cos⁡xf(x)=\dfrac{x^2}{1-\cos x}, where 0<x<10<x<1. Which of the following describes the function of f(x)f(x) ?

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A Brilliant Member by A Brilliant Member

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

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Calvin Lin by Calvin Lin
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