The basics of graphing f(x) = Asin(bx+c) + D

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Here is f(x) = sinx from x = -6 to 6.

Height (amplitude) = 1 (from 1 to -1)

Period = 2pi

f(0) = 0

f(pi/2) =1

f(-pi/2) = -1

Here is f(x) = 2sinx from x = -6 to 6.

Height (amplitude) = 2 (from 2 to -2)

Period = 2pi

f(0) = 0

f(pi/2) =2

f(-pi/2) = -2

The thing that is changed is the height or amplitude. f(x) = Asinx oscillates from A to -A. Choosing a negative A makes the graph start at 0 and move downward instead of upward.

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Here is f(x) = sin2x from x = -6 to 6.

Height (amplitude) = 1 (from 1 to -1)

Period = pi

f(0) = 0

f(pi/4) =1

f(-pi/4) = -1

The thing that is changed is the period. f(x) = sinbx has period 2pi/b.

To repeat faster than 2pi, choose |b| > 1. For slower repeats |b| < 1. Choosing a negative b makes the graph start at 0 and move downward instead of upward.

Here is f(x) = sin(x + pi/4) from x = -6 to 6.

Height (amplitude) = 1 (from 1 to -1)

Period = 2pi

f(-pi/4) = 0

f(pi/4) =1

f(-3pi/4) = -1
What changes here are the x positions of the midpoint and the maximum and minimum values. f(x) = sin(x + c) reaches 0 at x = -c, the high point at x = pi/2 - c and the low point at x = -pi/2 - c.

Here is f(x) = sinx + pi/4 from x = -6 to 6.

Height (amplitude) = 1 (from 1+pi/4 to -1+pi/4)

Period = 2pi

f(0) =pi/4

f(pi/2) =1+pi/4

f(-pi/2) = -1+pi/4

What changes here are the y positions of the midpoint and the maximum and minimum values. Instead of oscillating between 1 and -1, f(x) = sinx + D goes back and forth between D+1 to D-1.


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To review.

f(x) = Asin(bx + c) + D

The constant A changes the height or amplitude of the graph, rising to |A| above the middle y value, which is the constant D, and falling to -|A| below the middle y value. Negative values of A cause the graph to go downward to the right of bx + c = 0 instead of upward.

The constant b changes the distance in x between the consecutive highest points, also known as the period of the function.  The standard period is 2pi, and if |b| ≠ 1, the period changes to 2pi/|b|. The constant c changes where on the x-axis the sine function reaches its middle value. This is called phase shift. The point of the middle value moves to bx + c  = 0, so positive values cause the graph to shift left and negative values cause the graph to shift right.