1. **Problem statement:** We analyze the period of sinusoidal functions of the form $f(x) = \sin(bx)$ and fill in the blanks about their periods and properties. 2. **Period of sine function:** The standard sine function $f(x) = \sin(x)$ has period $2\pi$, meaning $\sin(x + 2\pi) = \sin(x)$ for all $x$. 3. **Period of $f_1(x) = \sin(2x)$:** The period $p_1$ satisfies: $$b \cdot p_1 = 2\pi \implies 2 \cdot p_1 = 2\pi \implies p_1 = \frac{2\pi}{2} = \pi$$ So, the period of $f_1$ is $\pi$, which is \textbf{half} as long as the period of $f$. 4. **Period of $f_2(x) = \sin(0.5x)$:** Similarly, $$0.5 \cdot p_2 = 2\pi \implies p_2 = \frac{2\pi}{0.5} = 4\pi$$ So, the period of $f_2$ is $4\pi$, which is \textbf{twice} as long as the period of $f$. 5. **Relation between period $p$ and parameter $b$ in $f(x) = \sin(bx)$:** - At $x=0$, $$f(0) = \sin(b \cdot 0) = \sin(0) = 0$$ - One full period $p$ passes when $$b \cdot p = 2\pi \implies b = \frac{2\pi}{p}$$ 6. **General sine function $f(x) = a \cdot \sin(bx) + d$:** - If $a < 0$, the graph is reflected \textbf{about the x-axis}. - The function can be shifted vertically by $d$ and horizontally by phase shifts (not asked here). - The period satisfies $$b \cdot p = 2\pi$$ 7. **Determine periods for given functions:** - For $f_1(x) = \sin(4x)$: $$4 \cdot p_1 = 2\pi \implies p_1 = \frac{2\pi}{4} = \frac{\pi}{2}$$ - For $f_2(x) = \sin(\pi x)$: $$\pi \cdot p_2 = 2\pi \implies p_2 = \frac{2\pi}{\pi} = 2$$ 8. **Find $b$ for given periods:** - For $p = \frac{\pi}{4}$: $$b = \frac{2\pi}{p} = \frac{2\pi}{\pi/4} = 8$$ - For $p = 6$: $$b = \frac{2\pi}{6} = \frac{\pi}{3}$$ **Final answers:** - Die Periode von $f_1(x) = \sin(2x)$ ist \textbf{halb} so lang wie die Periode von $f(x) = \sin(x)$. - Die Periode von $f_2(x) = \sin(0.5x)$ ist \textbf{doppelt} so lang wie die Periode von $f(x) = \sin(x)$. - $f(0) = 0$. - $b = \frac{2\pi}{p}$. - Wenn $a < 0$, wird der Graph zusätzlich \textbf{an der x-Achse gespiegelt}. - Änderung der Periode: $b \cdot p = 2\pi$. - Periode von $f_1(x) = \sin(4x)$ ist $\frac{\pi}{2}$. - Periode von $f_2(x) = \sin(\pi x)$ ist $2$. - Funktionsgleichung mit Periode $p=\frac{\pi}{4}$: $f(x) = \sin(8x)$. - Funktionsgleichung mit Periode $p=6$: $f(x) = \sin\left(\frac{\pi}{3}x\right)$.