Question

A sine function has the following key features: Period = 4 Amplitude = 3 Midline: y=−1 y-intercept: (0,-1) The function is not a reflection of its parent function over the x-axis. Use the sine tool to graph the function. The first point must be on the midline and the second point must be a maximum or minimum value on the graph closest to the first point.

Answer 1

Answer:

• See the graphs attached and the explanation below

Explanation:

The most simple sine function, considered the parent function, is:

     $y=sin(x)$

That function has:

• Midline, also known as rest or equilibrium position: y = 0
• Minimum: - 1
• Maximum: 1
• Amplitude: the distance between a minimum or a maximum and the midline = 1
• period: the interval of repetition of the function = 2π

The more general sine function is:

             $y=Asin(Bx+C)+D$

That function has:

• Midline: y = D (it is a vertical shift from the parent function)
• Minimum: - A + D
• Maximum: A + D
• Amplitude: A
• period: 2π/B
• phase shift: C (it is a horizontal shift of the from the parent function)

Now, you have to draw the sine function with the given key features:

• Period = 4 ⇒ 2π/B = 4 ⇒ B = π/2

• Amplitude, A = 3

• midline y = - 1 ⇒ D = - 1

• y-intercept = (0, -1)

Substitute the know values and use the y-intercept to find C:

       

            $y=3sin(2x/\pi+C)-1$

Substitute (0, -1)

             

              $-1=3sin(0+C)-1\\ \\ 3sin(C)=0\\ \\ sin(C)=0\\ \\ C=0$

Hence, the function to graph is:

              $y=3sin(\pi x/2)-1$

To draw that function use this:

• Maxima: 3(1) - 1 = 3 - 1 = 2, at x = 1 ± 4n (n = 0, 1, 2, 3, ...)
• Minima: 3(-1) - 1 = - 3 - 1 = -4
• y-intercept: (0, - 1)
• x-intercepts: the solutions to 0 = 3sin(πx/2) = - 1
• first point of the midline: (0, -1) it is the same y-intercept

With that you can understand the graphs attached.