15. consider the function f(x)= -1/2 sin(x - π)+1. (lesson 18.3) (0.5 point) choose true or false for the…

15. consider the function f(x)= -1/2 sin(x - π)+1. (lesson 18.3) (0.5 point) choose true or false for the following statements about the parent function. statement true false the graph will vertically compress by a factor of 1/2 □ □ the graph will have an amplitude of 2 □ □ the graph will be translated π units to the left □ □ the graph will have a mid - line at y = 1 □ □ 16. consider the function f(x)=tan3(x + 3π/2) (lesson 18.3) (0.5 point each) a. the graph will be translated 3π/2 units: □ a up □ b down □ c left □ d right b. the graph will have a period of __________. write your answer in radian measure in terms of π.

15. consider the function f(x)= -1/2 sin(x - π)+1. (lesson 18.3) (0.5 point) choose true or false for the following statements about the parent function. statement true false the graph will vertically compress by a factor of 1/2 □ □ the graph will have an amplitude of 2 □ □ the graph will be translated π units to the left □ □ the graph will have a mid - line at y = 1 □ □ 16. consider the function f(x)=tan3(x + 3π/2) (lesson 18.3) (0.5 point each) a. the graph will be translated 3π/2 units: □ a up □ b down □ c left □ d right b. the graph will have a period of __________. write your answer in radian measure in terms of π.

Answer

Explanation:

Step1: Analyze vertical - compression for $f(x)=-\frac{1}{2}\sin(x - \pi)+1$

The general form of a sinusoidal function is $y = A\sin(B(x - C))+D$. For $f(x)=-\frac{1}{2}\sin(x - \pi)+1$, the coefficient of the sine function is $A =-\frac{1}{2}$. The absolute - value of $A$, $|A|=\frac{1}{2}$, so the graph vertically compresses by a factor of $\frac{1}{2}$. The statement "The graph will vertically compress by a factor of $\frac{1}{2}$" is True.

Step2: Analyze amplitude for $f(x)=-\frac{1}{2}\sin(x - \pi)+1$

The amplitude of a sinusoidal function $y = A\sin(B(x - C))+D$ is $|A|$. Here, $A =-\frac{1}{2}$, so the amplitude is $\frac{1}{2}$, not 2. The statement "The graph will have an amplitude of 2" is False.

Step3: Analyze horizontal translation for $f(x)=-\frac{1}{2}\sin(x - \pi)+1$

For the function $y = A\sin(B(x - C))+D$, the horizontal translation is given by $C$. For $f(x)=-\frac{1}{2}\sin(x - \pi)+1$, $C=\pi$, which means the graph is translated $\pi$ units to the right (not left). The statement "The graph will be translated $\pi$ units to the left" is False.

Step4: Analyze mid - line for $f(x)=-\frac{1}{2}\sin(x - \pi)+1$

For the function $y = A\sin(B(x - C))+D$, the mid - line is $y = D$. For $f(x)=-\frac{1}{2}\sin(x - \pi)+1$, $D = 1$, so the mid - line is $y = 1$. The statement "The graph will have a midline at $y = 1$" is True.

Step5: Analyze translation for $f(x)=\tan3(x+\frac{3\pi}{2})$

The general form of a tangent function is $y=\tan(B(x - C))+D$. For $f(x)=\tan3(x+\frac{3\pi}{2})=\tan(3x+\frac{9\pi}{2})$, the horizontal translation is given by $C=-\frac{3\pi}{2}$. A negative value of $C$ in $y = \tan(B(x - C))$ means the graph is translated $|C|$ units to the left. So the graph of $f(x)=\tan3(x+\frac{3\pi}{2})$ is translated $\frac{3\pi}{2}$ units to the left. The answer for 16a is C Left.

Step6: Analyze period for $f(x)=\tan3(x+\frac{3\pi}{2})$

The period of the tangent function $y = \tan(Bx)$ is $\frac{\pi}{|B|}$. For the function $f(x)=\tan3(x+\frac{3\pi}{2})$, $B = 3$. So the period is $\frac{\pi}{3}$.

Answer:

  • The graph will vertically compress by a factor of $\frac{1}{2}$: True
  • The graph will have an amplitude of 2: False
  • The graph will be translated $\pi$ units to the left: False
  • The graph will have a midline at $y = 1$: True

a. C Left b. $\frac{\pi}{3}$