(FUN)ction Graphs

Calling all Math students…Could you answer this simple question on function graphs?  

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Correct answer, C.  See the end of this post for the explanation. 

If you got the answer straight away, well done! You’ve clearly given the sine, cosine and tangent functions the attention they deserve. But don’t panic if you got it wrong- you are not alone. The majority of ACT Math students really struggle when faced with this type of question, and this blog post has been created to help you out.  

Learning what the linear graphs of sin, cos and tan look like on the x/y plane is ESSENTIAL.

I repeat- ESSENTIAL.

It’s only once you are secure in this knowledge that you can learn how to transform them (coincidently, our blog topic for next week)! 

Lets start from the beginning…

f(θ)=sin(θ) is known as the sine function: 

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The wave repeats every 2π radians, or every 360° (i.e. it has a period of 360°). It’s maximum y value is 1, and minimum is -1. It crosses the y axis at (0,0).

f(θ)= cos (θ) is the cosine function:

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Just like the sine function, the wave has a period of 360°. It’s maximum y value is 1, and minimum is -1. However, the cosine function crosses the y axis at (0,1). See how it ‘follows’ the sine wave at 90° behind?

 f(θ)= tan(θ) is the tangent function:  

It is not a rising and falling wave like the other two- it’s y values range from negative infinity to positive infinity, and it crosses the y axes once every π radians, or 180°. However, it does pass through the origin, point (0,0), like the sine wave. 

Notice how every 180° there is an x value for which there is no apparent corresponding y value? This is because tan(90°), tan(270°), etc (and tan( π/2), tan(3π/2)) is equal to either positive or negative infinity- meaning that it is is officially undefined.

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Undeniably, it is easiest and fastest just to remember what each function looks like (i.e. the maximum and minimum y values, y-axis intercept, period etc.).
However,  if you forget what any of these functions look like, it is possible to quickly check using your calculator to create a table like the one below. You can then sketch a rough diagram to jolt your memory or clear up any confusion. 

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So now let’s go back to the first question and see how we can use our new knowledge to solve it easily.

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If we look at the sine wave and the cosine wave plotted onto the same x/y plane, we can see that the x value for their first intersection in the POSITIVE x quadrant is 45°. Therefore our answer must be C

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