**Definition: **For a given point on the unit circle, the sine function has an independent variable (x) determined by the radian measure of the point’s location on the circumference of the unit circle and a dependent variable (y) determined by the point’s y-coordinate.

A sine function is in the form:

**y = a sin (bx – c) + d,**

where **a **tells us the amplitude, **b** helps determines the period,** c** helps determine the phase shift, and **d** is the vertical shift.

**|a| = amplitude:** half the distance between the minimum and maximum height, determines vertical stretch (|a| > 1), compression (0 < |a| < 1) and reflection across the x-axis (a < 0).

**2π/b = period:** horizontal length of each complete cycle, determines horizontal stretch (0 < b < 1), compression (b > 1).

**c/b = phase shift:** horizontal shift, left (< 0) or right (> 0)

**d = vertical shift,** up (d > 0) or down (d < 0).

*The graph of the sine parent function:*

When graphing a sine function, we keep track of how the following five critical points change (as plotted in the graph above).

**Example:**

Graph the function:

**Solution:**

Step 1: Using the given equation, identify important information about the graph.

Comparing the graph to the standard form, we can immediately see:

The amplitude, |a|, is 3,

The vertical shift, d, is 2.

Step 2: To find the period, given by 2π/b, plug b = 4 into the formula:

Period = 2π/b = 2π/4 = π/2.

Step 3: To find the phase shift, given by c/b, plug c = 5π/2 and b = 4 into the formula:

c/b = (5π/2)/4 = 5π/8

Phase shift = 5π/8

This tells us the graph shifts 5π/8 radians to the right.

Step 4: Because the phase shift is 5π/8, we know 5π/8 will be our starting x-value. To find the corresponding y-value, plug x = 5π/8 into the function to determine y = 2. This gives us our first critical point at **(5π/8, 2).**

Step 5: In addition to (5π/8, 2), we need to find the other four critical points.

To find the x-values of these points, we can divide the period into 4 equal parts to find the *quartile length (*how far apart the points are horizontally), π/8, and continually add that to our initial x-value.

To find the y-values, we’ll need to first visualize the behavior of the graph of sine.

We know it starts at the x-axis, ascends up, curves back down, continues to dip down below the x-axis, then climbs back up to where it started. Notice also that the vertical distance between each point is the amplitude. This means, in order to find more y-values, we can add the amplitud**e** to the previous point if the function goes up, and subtract the amplitude it if it goes down.

Starting with (5π/8, 2) add π/8 to the x-value and 3 (amplitude) to the y-value:

( 5π/8 + π/8 , 2 + 3 ) = **(3π/4 , 5)**

Repeat this step to find the next three points, keeping in mind the direction the y-values will be moving:

( 6π/8 + π/8 , 5 – 3 ) = **(7π/8 , 2)**

( 7π/8 + π/8 , 2 – 3 ) = **(π , -1)**

( π + π/8 , -1 + 3 ) = **(9π/8 , 2)**

Step 6: Graph the five points and connect them to form sine’s trademark smooth, repeating waves.

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