Question

This is a tangent graph that I need to figure out what the equation is

Answer 1

Looking at the graph, we have a periodic function, and it's period is equal to 2 units.

The graph also goes from negative infinity to positive infinity in each cycle.

So this is a tangent function. The tangent function is given by:

`\begin{gathered} f(x)=c\tan (a(x+b))+d \\ \text{Period}=(\pi)/(a) \\ horizontal\text{ shift}=b \\ \text{vertical scaling}=c \\ \text{vertical shift}=d \end{gathered}`

The curvature of the curve in each cycle changes when y = -1 (and the first curvature after zero occurs at x = 1), so the vertical shift is d = -1 and the horizontal shift is b = -1.

The period is approximately 2 units, so we have:

`\begin{gathered} (\pi)/(a)=2 \\ a=(\pi)/(2) \end{gathered}`

Now, to find the vertical scaling, let's use the approximate point (1.5, 2):

`\begin{gathered} f(x)=c\cdot\tan ((\pi)/(2)(x-1))-1 \\ 2=c\cdot\tan ((\pi)/(2)(1.5-1))-1 \\ 2=c\cdot\tan ((\pi)/(4))-1 \\ 2=c\cdot1-1 \\ c-1=2 \\ c=3 \end{gathered}`

So our function is:

`f(x)=3\cdot\tan ((\pi)/(2)(x-1))-1`