identify the vertical stretch or compression a and the horizontal stretch or compression b for each…

identify the vertical stretch or compression a and the horizontal stretch or compression b for each trigonometric function. then graph the function and identify the amplitude and period.\n1. $y = \\frac{1}{2}\\sin3x$\n2. $y = 2\\cos\\frac{x}{2}$\n3. $y=-3\\sin\\frac{x}{4}$\n4. $y = - 5\\cos\\pi x$

identify the vertical stretch or compression a and the horizontal stretch or compression b for each trigonometric function. then graph the function and identify the amplitude and period.\n1. $y = \\frac{1}{2}\\sin3x$\n2. $y = 2\\cos\\frac{x}{2}$\n3. $y=-3\\sin\\frac{x}{4}$\n4. $y = - 5\\cos\\pi x$

Answer

Explanation:

Step1: Recall the general form of a trig - function

The general form of a trigonometric function is $y = a\sin(bx)$ or $y=a\cos(bx)$. The amplitude is $|a|$, the period is $T=\frac{2\pi}{|b|}$, $a$ represents vertical stretch/compression and $\frac{1}{|b|}$ represents horizontal stretch/compression.

Step2: Analyze $y = \frac{1}{2}\sin(3x)$

For $y=\frac{1}{2}\sin(3x)$, $a = \frac{1}{2}$, so there is a vertical compression by a factor of $\frac{1}{2}$. $b = 3$, so there is a horizontal compression by a factor of $\frac{1}{3}$. The amplitude $A=\left|\frac{1}{2}\right|=\frac{1}{2}$, and the period $T=\frac{2\pi}{|3|}=\frac{2\pi}{3}$.

Step3: Analyze $y = 2\cos\left(\frac{x}{2}\right)$

For $y = 2\cos\left(\frac{x}{2}\right)$, $a = 2$, so there is a vertical stretch by a factor of 2. $b=\frac{1}{2}$, so there is a horizontal stretch by a factor of 2. The amplitude $A = |2|=2$, and the period $T=\frac{2\pi}{\left|\frac{1}{2}\right|}=4\pi$.

Step4: Analyze $y=-3\sin\left(\frac{x}{4}\right)$

For $y=-3\sin\left(\frac{x}{4}\right)$, $a=-3$, so there is a vertical stretch by a factor of 3 and a reflection about the $x -$axis. $b=\frac{1}{4}$, so there is a horizontal stretch by a factor of 4. The amplitude $A = |-3| = 3$, and the period $T=\frac{2\pi}{\left|\frac{1}{4}\right|}=8\pi$.

Step5: Analyze $y=-5\cos(\pi x)$

For $y=-5\cos(\pi x)$, $a=-5$, so there is a vertical stretch by a factor of 5 and a reflection about the $x -$axis. $b = \pi$, so there is a horizontal compression by a factor of $\frac{1}{\pi}$. The amplitude $A=|-5| = 5$, and the period $T=\frac{2\pi}{|\pi|}=2$.

Answer:

  1. $a=\frac{1}{2}$ (vertical compression), $b = 3$ (horizontal compression), amplitude $=\frac{1}{2}$, period $=\frac{2\pi}{3}$
  2. $a = 2$ (vertical stretch), $b=\frac{1}{2}$ (horizontal stretch), amplitude $=2$, period $=4\pi$
  3. $a=-3$ (vertical stretch and reflection), $b=\frac{1}{4}$ (horizontal stretch), amplitude $=3$, period $=8\pi$
  4. $a=-5$ (vertical stretch and reflection), $b=\pi$ (horizontal compression), amplitude $=5$, period $=2$