multiple choice\n7. which one of these graphs has all of the following characteristics?\n• a range of {y |…

multiple choice\n7. which one of these graphs has all of the following characteristics?\n• a range of {y | -1 ≤ y ≤ 3, y ∈ r} • an amplitude of 2\n• a maximum value of 3 • a mid - line equation of y = 1\n• a minimum value of -1 • a period of 120°\na. graph a description b. graph b description c. graph c description d. graph d description

Answer

Explanation:

Step1: Recall amplitude - mid - line relationship

The amplitude $A$ of a periodic function is related to the mid - line $y = k$, maximum value $y_{max}$ and minimum value $y_{min}$ by the formulas $A=\frac{y_{max}-y_{min}}{2}$ and $k=\frac{y_{max} + y_{min}}{2}$. Given $y_{max}=3$ and $y_{min}=-1$, then $A=\frac{3 - (-1)}{2}=\frac{4}{2}=2$ and $k=\frac{3+( - 1)}{2}=\frac{2}{2}=1$.

Step2: Recall period formula

The period $T$ of a periodic function. We are given $T = 120^{\circ}$.

Step3: Analyze option A

The range of the function in option A is $[-3,3]$, which does not match the required range of $[-1,3]$.

Step4: Analyze option B

The mid - line of the function in option B is $y = 1$, the amplitude $A=\frac{3-(-1)}{2}=2$, the range is $[-1,3]$, the maximum value is 3, the minimum value is - 1, and the period $T = 120^{\circ}$. This option satisfies all the given characteristics.

Step5: Analyze option C

The period of the function in option C is not $120^{\circ}$.

Step6: Analyze option D

The range of the function in option D is $[-2,4]$, which does not match the required range of $[-1,3]$.