Lisa Brock

Assignment 1

Examining the Curve y = a sin(bx+c)

Examine the curve y = a sin(bx+c) for different values of a, b, and c.

### Let's start by looking at the graph of y = sin x.

Now, let's look at y = a sin x.  Begin by comparing the graph of y = sin x to graphs of y = a sin x when a is a positive number.

Each curve crosses the x-axis at the same points as y = sin x.  As a increases, the distance from the x-axis to each peak or valley increases from 1 unit to a units.

Now let's compare the graph of y = sin x to graphs of y = a sin x when a is a negative number.

Each curve still crosses the x-axis at the same points as y = sin x.  As the absolute value of a increases, the distance from the x-axis to each peak or valley increases.  These characteristics are the same as the ones apparent for positive values of a.   The negative values of a have an additional affect on the graph.  The peaks and valleys are on the opposite side of the x-axis compared to the graph of y = sin x.  y = -sin x appears to be the reflection of y = sin x.  y = -2sin x appears to be the reflection of y = 2sin x, and so on.

Therefore, a changes the distance between the peak/valley and the x-axis from 1 unit to a units.  A negative value of a will also reflect the graph over the x-axis.

The distance from the peaks and valleys to the x-axis is the same for y = sin x and y = sin(bx).  The period of y = sin x is 2p  (a complete cycle from the origin, up to the first peak, down to the first valley, and back to the x-axis).  The period of y = sin (2x) is p  or (2p)/2.  The period of y = sin (3x) is (2p)/3.  A positive values for b changes the period of the graph to (2p)/b.

Therefore, if b = 1/2, the period of the graph y = sin (1/2 x) is 4p.  Let's look at the graph to see if that is the case.

The period of y = sin (1/2 x) is 4p.

Now, let's compare the graph of y = sin x to y = sin (bx) when b is a negative number.

The distance from each peak/valley is still the same as the graph of y = sin x.  The period of the graph of y = sin (bx) is still (2p)/b.  These characteristics are the same as the ones apparent for positive values of b.   The negative values of a have an additional affect on the graph.  One cycle of y = sin x begins with a peak and ends with a valley, whereas one cycle of y = sin (bx) begins with a valley and ends with a peak.  The graph of y = sin (-2x) is the reflection of y = sin (2x) over the x-axis.  The graph of y = sin (-3x) is the reflection of y = sin (3x) over the x-axis, and so on.

Therefore, b changes the period of the graph from 2p to (2p)/b.  A negative b also reflects the graph over the x-axis.

Now, let's look at y = sin (x+c).  Begin by comparing the graph of y = sin x to graphs of y = sin (x+c) for both positive and negative values of c.

Positive Values of c

Negative Values of c

When c is positive, the graph shifts c units to the left.  When c is negative, the graph shifts c units to the right.

Now that we know the effect that a, b, and c have on the graph of y = sin x, we can predict the appearance of the graph y = a sin (bx+c).

For example, let's take the graph of y = 3 sin (2x-1).  The peaks/valleys will be 3 units from the x-axis, the period will be p, and the graph will be shifted 1 unit to the right.  Let's graph y = 3 sin (2x-1) to check.