the graph of the function (y = \tan(x)) was horizontally stretched so that its period became (10pi). which…

Explanation:

Step1: Recall period formula for tangent function

The period of the tangent function $y = A\tan(Bx - C)+D$ is $T=\frac{\pi}{|B|}$. The original period of $y = \tan(x)$ is $\pi$.

Step2: Set up equation for new - period

We know the new period $T = 10\pi$. Using the period formula $T=\frac{\pi}{|B|}$, and substituting $T = 10\pi$ into it, we get $10\pi=\frac{\pi}{|B|}$.

Step3: Solve for B

Cross - multiply the equation $10\pi=\frac{\pi}{|B|}$ to get $10\pi|B|=\pi$. Then divide both sides by $10\pi$: $|B|=\frac{1}{10}$. Since we are dealing with a horizontal stretch and the general form of a horizontal stretch/compression for $y = f(x)$ to $y = f(Bx)$ has $0 < B<1$ for a stretch, $B=\frac{1}{10}$. So the transformed function is $y=\tan(\frac{x}{10})$.

Answer:

$y = \tan(\frac{x}{10})$