For the function r(z) = -4 sin(2z) + 1, indicate the amplitude, period, and equation of the midline.

Answer 1

Hi Darian -

When given a trig function in either of the forms

y = a sin(bx) + c

y = a cos(bx) + c

we can figure out all the pieces of what you're being asked for here by making some observations and doing some calculations involving the constants in the equation.

Amplitude: amplitude can be thought of as the vertical distance from the midline of a trig function to minimum or maximum points, or also as half the vertical distance from a minimum point to a maximum point (these will be the same). In the equations above, amplitude is given by the value of the constant a, and is generally treated as a strictly positive quantity, so the amplitude here would be 4 since the coefficient in the place of a is -4.

Period: the period of a transformed trig function can be calculated by taking the normal period of the trig function (for sine or cosine the normal period is 2π) and dividing it by the value of the constant b in the equation. In this case, the period calculation looks like this since the value of b in our given equation is 2:

2π/b = 2π/2 = π

so the period of this trig function will just be pi units.

Midline: For a trig function given in the form above, the equation of the midline will always be y = c, where c is the constant at the end of the equation. Since the value of c in our given equation is 1 for this problem, the midline will have the equation y = 1.

I hope this helps! Leave a comment if you have any questions and I'll be happy to explain more, and be in touch if you'd like to schedule a session for more help with questions like this.