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Write the equation of the function, involving a sine or cosine, whose graph is shown here.

enter image description here

I think I need to use this form:

f(t) = A cos(Bx) + C

I found the amplitude (1) because that's the distance from the axis.

I found the period (2) because $$ 1.75\pi -0.75\pi = \pi $$

The vertical shift I think is (-1) because it is the midpoint from the top to the bottom of the wave.

f(t) = cos(2x) -1

I don't think this is right. Any help please?

N. F. Taussig
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marksawnson
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  • Share your thought process for finding the shift and the width – tryst with freedom Feb 21 '22 at 23:54
  • It's not right. You can see this by evaluating $f(0)=0$ which does not match the graph. Can you figure out what you did wrong from here? – CyclotomicField Feb 22 '22 at 00:09
  • The amplitude is half the distance between the maximum and minimum values of the function, so your amplitude is incorrect. Once you correct the amplitude, try drawing the graph of your equation and compare it with the given graph. What do you notice? – N. F. Taussig Feb 22 '22 at 00:22
  • @N.F.Taussig You are right - the amplitude is (-1) because it is not measured from the x-axis to the top of the wave. – marksawnson Feb 22 '22 at 00:33
  • The maximum value of the function is $1$; the minimum value is $-3$. Hence, the amplitude is $|A| = \dfrac{1 - (-3)}{2} = \dfrac{4}{2} = 2$. – N. F. Taussig Feb 22 '22 at 00:49
  • @N.F.Taussig Oh that makes sense. But another guy said that the amplitude is -2? – marksawnson Feb 22 '22 at 00:52
  • Notice that I wrote $|A| = 2$. He wrote that $A = -2$ since the sine curve opens downwards to the immediate right of $x = 0$. – N. F. Taussig Feb 22 '22 at 00:59

1 Answers1

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Let the function be $f(x) = a\sin(bx)+c$. Then the period $T = \pi = \dfrac{2\pi}{b}\implies b = 2$. Also: $f(0) = -1\implies c = -1, f(-\frac{\pi}{4}) = 1\implies a = -2\implies f(x) = -2\sin(2x)-1$. You can check it works.

Wang YeFei
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