select the correct answer from each drop-down menu.\nthe parent cosine function is transformed to create…

select the correct answer from each drop-down menu.\nthe parent cosine function is transformed to create function d.\n$d(x) = \\cos(2x - 1) + 5$\nto create function d, the graph of the parent cosine function undergoes these transformations:\n- horizontal shift \n- vertical shift \n- frequency

select the correct answer from each drop-down menu.\nthe parent cosine function is transformed to create function d.\n$d(x) = \\cos(2x - 1) + 5$\nto create function d, the graph of the parent cosine function undergoes these transformations:\n- horizontal shift \n- vertical shift \n- frequency

Answer

Explanation:

Step1: Analyze Horizontal Shift

The general form of a cosine function is ( y = A\cos(B(x - C)) + D ). For ( d(x)=\cos(2x - 1)+5 ), we can rewrite ( 2x - 1 ) as ( 2\left(x-\frac{1}{2}\right) ). So, the horizontal shift ( C=\frac{1}{2} ) units to the right.

Step2: Analyze Vertical Shift

In the general form, ( D ) is the vertical shift. Here, ( D = 5 ), so the vertical shift is 5 units up.

Step3: Analyze Frequency

The coefficient ( B ) affects the period (and thus frequency). The period of ( y=\cos(x) ) is ( 2\pi ), and for ( y = \cos(Bx) ), the period is ( \frac{2\pi}{|B|} ). Here, ( B = 2 ), so the period is ( \pi ), which means the frequency (number of cycles in ( 2\pi )) is 2 (since original frequency is 1 for period ( 2\pi ), now with period ( \pi ), frequency doubles).

Answer:

  • horizontal shift: (\frac{1}{2}) units to the right
  • vertical shift: 5 units up
  • frequency: doubled (or has a value of 2, depending on drop - down options, but based on analysis, the transformations are horizontal shift (\frac{1}{2}) right, vertical shift 5 up, and frequency doubled)