# Trigonometry - How to figure out translation of a graph

I have been wracking my brain and can not figure this out so I've broken down and am asking this question on here in hopes that someone can show me how to do this.

I need to figure out if this sine graph has been translated left or right and by how many units.

I can't figure it out from the information provided by the graph. I'm stuck on this idea that I need to figure out the period but I can't do that because I can't determine what the x-values of the min. and max. are. Well, it's looks like the x-value of the min. is 0 but I can't figure out the max.

At any rate, anyone know how to do this?

• It could've been shifted left or right, but it doesn't matter. – Simply Beautiful Art Feb 20 '17 at 1:15
• Given that you know the graph was merely translated, not changed in any other way, the minimum is enough information. If you really want to know the maximum you can figure it out from the minimum. – David K Feb 20 '17 at 1:32
• Note that the positive maximum occurs a little past $3$ and the next minimum, though off the edge appears to be a bit more than $6$. This supports the idea that the period of this function has not been modified, which your question also seems to suggest, since it only talks about translation. The amplitude clearly is still $1$, so that hasn't changed either. As you've noted, there is a minimum at $0$. Where is the nearest minimum of $\sin$? That will tell you how far it has been translated and in which direction. – Paul Sinclair Feb 20 '17 at 3:49

Period = distance between two peaks or maxima. It appears the peaks or maxima are at a little more than $\pm 3$ so $x =\pm \pi \approx \pm 3.14$ would be a reasonable guess. Then period = $\pi - (- \pi) = 2\pi$, unchanged from the basix function sin$(x)$.
Sin$(x)$ = 0 at x = 0, and sin$(x)$ is increasing at $x = 0.$
The two points visible on this graph which fit both criteria are at x close to or a little above 1.5, and $x$ close to -4.6 or -4.7. Since it appears we are dealing with radian measure, a period of $2 \pi$ and maxima and minima at multiples of pi, then a reasonable assumption would be that $x = 0$ and $f(x)$ is increasing ($f'(x)$ positive) at $x = \pi/2 \approx 1.57$ and at $x = - 3 \pi/2 \approx -4.71$
Thus the graph could be sin$(x)$ shifted $\pi/2$ to the right OR $3 \pi/2$ to the left. (Since the period is $2\pi$ adding any multiple of $2\pi$ to the phase shift will give a correct result.)
The function would be $f(x) = \sin (x - \pi/2)$ OR $f(x) = \sin (x + 3\pi/2)$