1. The problem asks to find the period and points of intersection with the horizontal axis (x-intercepts) for the given trigonometric functions. 2. Recall the period formulas for sine, cosine, and tangent functions: - For $y=\sin(bx)$ or $y=\cos(bx)$, the period is $\frac{2\pi}{|b|}$. - For $y=\tan(bx)$, the period is $\frac{\pi}{|b|}$. 3. We will analyze the first function only, as per instructions: Function: $k(x) = -2 \tan\left(\frac{\pi}{2} x\right)$ 4. Calculate the period: $$\text{Period} = \frac{\pi}{\left|\frac{\pi}{2}\right|} = \frac{\pi}{\frac{\pi}{2}} = 2$$ 5. Find points of intersection with the horizontal axis (where $k(x)=0$): Since $k(x) = -2 \tan\left(\frac{\pi}{2} x\right)$, zeros occur when $$\tan\left(\frac{\pi}{2} x\right) = 0$$ The tangent function is zero at integer multiples of $\pi$: $$\frac{\pi}{2} x = n\pi \implies x = 2n, \quad n \in \mathbb{Z}$$ 6. Therefore, the x-intercepts are at $x = 2n$ for all integers $n$. 7. Summary: - Period: 2 - X-intercepts: $x = 2n$, $n \in \mathbb{Z}$ This completes the solution for the first function.