Graphing Cosine Functions

Definition: For a given point on the unit circle, the cosine 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 determined by the point’s x-coordinate.

A cosine function is in the form:

y = a cos (bx – c) + d,

where 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/= phase shift: horizontal shift, left (< 0) or right (> 0).

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

The graph of the cosine parent function:

When graphing a cosine 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 1/2
The vertical shift, d, is -1/3.

We can also see that because a < 0, the function is negative and the graph will be flipped upside-down.

Step 2: To find the period, given by 2π/b, plug b = (π/5) into the formula:

Period = 2π/b = 2π / (π/5) = 10.

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

c/b = -π/(π/5) = -5

Phase shift = -5

This tells us the graph shifts 5 radians to the left.

Step 4: Because the phase shift is -5, we know -5 will be our starting x-value. To find the corresponding y-value, plug x = -5 into the function to determine that y = -5/6. This gives us our first critical point at (-5, -5/6).

Step 5: In addition to (-5, -5/6), 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), 5/2, and continually add this to our initial x-value.

To find the y-values, we’ll need to first visualize the behavior of the graph of cosine, keeping in mind that it will be flipped upside-down (see Step 1):cos1.pngWe can see that it ascends up towards the x-axis, continues climbing up, curves back downward towards the x-axis, then dips back down 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 amplitude to the previous point if the function goes up, and subtract the amplitude it if it goes down.

Starting with (-5, -5/6) add add 5/2 to the x-value and 1/2 to the y-value:

(-5 + 5/2, -5/6 + 1/2) = (-5/2, -1/3)

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

(-5/2 + 5/2, -1/3 + 1/2) = (0, 1/6)

(0 + 5/2, 1/6 – 1/2) = (5/2, -1/3)

(5/2 + 5/2, -1/3 – 1/2) = (5, – 5/6)

Step 6: Graph the five points and connect them to form a smooth wave.

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