sketch two periods of the function.

Question

Sketch two periods of the the function f(x) = 3sin (x/2)+ 4, without using a graphing calculator.

label the axes or points on the graph along with what the key points (mins, maxs, points where it goes

through the midline) on the graph are

Answer 1

Forgot to mention in the video that there is no horizontal shift (phase shift) for the given function.

Here is a Desmos graph with some key points labelled.

desmos.com/calculator/9iveqfiwwu

Answer 2

I cannot draw on the computer screen, and even if I could, it would not be very accurate because I have poor fine motor skills. However, I can advise on where those "key points" are.

The maximum values of the function 3 sin (x/2) + 4 occur when x = π, 5π, 9π, 13π, etc. and share the y-value of 7 This is because sin x, the parent function, has maxima at π/2, 5π/2, etc. and a maximum value of 1 at all those places. Similarly, the minimum values of 3 sin (x/2) + 4 occur when x = 3π, 7π, 11π, etc. and share a y-value of 1. Where the function goes through the midline corresponds to a value of x for which sin x = 0, which happens at each integer multiple of π. Hence, 3 sin (x/2) + 4 goes through the midline when x = 0, 2π, 4π, 6π, etc. and there the function's value is 4.

When sketching the function, remember that at the maximum and minimum values, the function's graph has a horizontal tangent; that is, it barely touches the lines y = 1 or y = 7. In addition, two periods cover the range 0 ≤ x ≤ 8π, and 8π is a little more than 25. Also, the derivative of the function is never more than 1.5 or less than -1.5 (because the derivative of sin x is never more than 1 or less than -1), so the slope of the graph (or, rather, its tangent line) is always within those limits. The derivative reaches its maximum and minimum when the graph passes through the midine.