find the amplitude and period of the function. use the language of transformations to describe how the graph…

find the amplitude and period of the function. use the language of transformations to describe how the graph of the function is related to the graph of the parent function.\n1. $f(x)=3sin x$\n2. $g(x)= - 2.3cos(2x)$\n3. $h(x)=\frac{2}{3}sin(0.25x)$\n4. $k(x)=-\frac{5}{3}cos(\frac{2}{3}x)$

find the amplitude and period of the function. use the language of transformations to describe how the graph of the function is related to the graph of the parent function.\n1. $f(x)=3sin x$\n2. $g(x)= - 2.3cos(2x)$\n3. $h(x)=\frac{2}{3}sin(0.25x)$\n4. $k(x)=-\frac{5}{3}cos(\frac{2}{3}x)$

Answer

Explanation:

1. For (f(x)=3\sin x):

  • Amplitude:
    • The general form of a sine - function is (y = A\sin(Bx - C)+D). For (f(x)=3\sin x), where (A = 3), (B = 1), (C = 0), and (D = 0). The amplitude of a sine function (y = A\sin(Bx - C)+D) is given by (|A|). So, the amplitude of (f(x)=3\sin x) is (|3|=3).
    • Period: The period of a sine function (y = A\sin(Bx - C)+D) is (T=\frac{2\pi}{|B|}). Since (B = 1), the period (T=\frac{2\pi}{|1|}=2\pi).
    • Transformation: The graph of (y = 3\sin x) is a vertical stretch of the parent - function (y=\sin x) by a factor of 3.

2. For (g(x)=-2.3\cos(2x)):

  • Amplitude: For the function (g(x)=-2.3\cos(2x)) in the form (y = A\cos(Bx - C)+D) (where (A=-2.3), (B = 2), (C = 0), (D = 0)), the amplitude is (|A|=| - 2.3|=2.3).
  • Period: Using the formula (T=\frac{2\pi}{|B|}), with (B = 2), the period (T=\frac{2\pi}{2}=\pi).
  • Transformation: The graph of (y=-2.3\cos(2x)) is a vertical stretch by a factor of 2.3 and a reflection about the (x) - axis (due to the negative sign) of the parent - function (y = \cos x), and a horizontal compression by a factor of (\frac{1}{2}).

3. For (h(x)=\frac{2}{3}\sin(0.25x)):

  • Amplitude: For (h(x)=\frac{2}{3}\sin(0.25x)) (in the form (y = A\sin(Bx - C)+D) with (A=\frac{2}{3}), (B = 0.25=\frac{1}{4}), (C = 0), (D = 0)), the amplitude is (|A|=\left|\frac{2}{3}\right|=\frac{2}{3}).
  • Period: Using the formula (T=\frac{2\pi}{|B|}), with (B=\frac{1}{4}), the period (T=\frac{2\pi}{\left|\frac{1}{4}\right|}=8\pi).
  • Transformation: The graph of (y=\frac{2}{3}\sin(0.25x)) is a vertical stretch by a factor of (\frac{2}{3}) of the parent - function (y=\sin x) and a horizontal stretch by a factor of 4.

4. For (k(x)=-\frac{5}{3}\cos\left(\frac{2}{3}x\right)):

  • Amplitude: For (k(x)=-\frac{5}{3}\cos\left(\frac{2}{3}x\right)) (in the form (y = A\cos(Bx - C)+D) with (A =-\frac{5}{3}), (B=\frac{2}{3}), (C = 0), (D = 0)), the amplitude is (|A|=\left|-\frac{5}{3}\right|=\frac{5}{3}).
  • Period: Using the formula (T=\frac{2\pi}{|B|}), with (B=\frac{2}{3}), the period (T=\frac{2\pi}{\left|\frac{2}{3}\right|}=3\pi).
  • Transformation: The graph of (y =-\frac{5}{3}\cos\left(\frac{2}{3}x\right)) is a vertical stretch by a factor of (\frac{5}{3}) and a reflection about the (x) - axis (due to the negative sign) of the parent - function (y=\cos x), and a horizontal stretch by a factor of (\frac{3}{2}).

Answer:

  1. (f(x)=3\sin x):
    • Amplitude: (3)
    • Period: (2\pi)
    • Transformation: Vertical stretch by a factor of 3 of (y = \sin x).
  2. (g(x)=-2.3\cos(2x)):
    • Amplitude: (2.3)
    • Period: (\pi)
    • Transformation: Vertical stretch by a factor of 2.3, reflection about the (x) - axis, and horizontal compression by a factor of (\frac{1}{2}) of (y=\cos x).
  3. (h(x)=\frac{2}{3}\sin(0.25x)):
    • Amplitude: (\frac{2}{3})
    • Period: (8\pi)
    • Transformation: Vertical stretch by a factor of (\frac{2}{3}) and horizontal stretch by a factor of 4 of (y=\sin x).
  4. (k(x)=-\frac{5}{3}\cos\left(\frac{2}{3}x\right)):
    • Amplitude: (\frac{5}{3})
    • Period: (3\pi)
    • Transformation: Vertical stretch by a factor of (\frac{5}{3}), reflection about the (x) - axis, and horizontal stretch by a factor of (\frac{3}{2}) of (y=\cos x).