Mathematics » Functions II » The Cosine Function

Functions of the Form y = cos(θ + p)

Functions of the form $$y=\cos(\theta +p)$$

We now consider cosine functions of the form $$y = \cos(\theta + p)$$ and the effects of parameter $$p$$.

Optional Investigation: The effects of $$p$$ on a cosine graph

1. On the same system of axes, plot the following graphs for $$-\text{360}\text{°} \leq \theta \leq \text{360}\text{°}$$:

1. $$y_1 = \cos \theta$$
2. $$y_2 = \cos (\theta – \text{90}\text{°})$$
3. $$y_3 = \cos (\theta – \text{60}\text{°})$$
4. $$y_4 = \cos (\theta + \text{90}\text{°})$$
5. $$y_5 = \cos (\theta + \text{180}\text{°})$$
2. Use your sketches of the functions above to complete the following table:

 $$y_1$$ $$y_2$$ $$y_3$$ $$y_4$$ $$y_5$$ period amplitude domain range maximum turning points minimum turning points $$y$$-intercept(s) $$x$$-intercept(s) effect of $$p$$

The effect of the parameter on $$y = \cos(\theta + p)$$

The effect of $$p$$ on the cosine function is a horizontal shift (or phase shift); the entire graph slides to the left or to the right.

• For $$p > 0$$, the graph of the cosine function shifts to the left by $$p$$ degrees.

• For $$p < 0$$, the graph of the cosine function shifts to the right by $$p$$ degrees.

 $$p>0$$ $$p<0$$

Example

Question

1. Sketch the following functions on the same set of axes for $$-\text{360}\text{°} \leq \theta \leq \text{360}\text{°}$$.
1. $$y_1 = \cos \theta$$
2. $$y_2 = \cos (\theta + \text{30}\text{°})$$
2. For each function determine the following:

1. Period
2. Amplitude
3. Domain and range
4. $$x$$- and $$y$$-intercepts
5. Maximum and minimum turning points

Examine the equations of the form $$y = \cos (\theta + p)$$

Notice that for $$y_1 = \cos \theta$$ we have $$p = 0$$ (no phase shift) and for $$y_2 = \cos (\theta + \text{30}\text{°})$$, $$p < 0$$ therefore the graph shifts to the left by $$\text{30}\text{°}$$.

Complete a table of values

 θ $$-\text{360}$$$$\text{°}$$ $$-\text{270}$$$$\text{°}$$ $$-\text{180}$$$$\text{°}$$ $$-\text{90}$$$$\text{°}$$ $$\text{0}$$$$\text{°}$$ $$\text{90}$$$$\text{°}$$ $$\text{180}$$$$\text{°}$$ $$\text{270}$$$$\text{°}$$ $$\text{360}$$$$\text{°}$$ $$\cos \theta$$ $$\text{1}$$ $$\text{0}$$ $$-\text{1}$$ $$\text{0}$$ $$\text{1}$$ $$\text{0}$$ $$-\text{1}$$ $$\text{0}$$ $$\text{1}$$ $$\cos(\theta + \text{30}\text{°})$$ $$\text{0.87}$$ $$-\text{0.5}$$ $$-\text{0.87}$$ $$\text{0.5}$$ $$\text{0.87}$$ $$-\text{0.5}$$ $$-\text{0.87}$$ $$\text{0.5}$$ $$\text{0.87}$$

Complete the table

 $$y_1$$ $$y_2$$ period $$\text{360}\text{°}$$ $$\text{360}\text{°}$$ amplitude $$\text{1}$$ $$\text{1}$$ domain $$[-\text{360}\text{°};\text{360}\text{°}]$$ $$[-\text{360}\text{°};\text{360}\text{°}]$$ range $$[-1;1]$$ $$[-1;1]$$ maximum turning points $$(-\text{360}\text{°};1)$$, $$(\text{0}\text{°};1)$$ and $$(\text{360}\text{°};1)$$ $$(-\text{30}\text{°};1)$$ and $$(\text{330}\text{°};1)$$ minimum turning points $$(-\text{180}\text{°};-1)$$ and $$(\text{180}\text{°};-1)$$ $$(-\text{210}\text{°};-1)$$ and $$(\text{150}\text{°};-1)$$ $$y$$-intercept(s) $$(\text{0}\text{°};0)$$ $$(\text{0}\text{°};\text{0.87})$$ $$x$$-intercept(s) $$(-\text{270}\text{°};0)$$, $$(-\text{90}\text{°};0)$$, $$(\text{90}\text{°};0)$$ and $$(\text{270}\text{°};0)$$ $$(-\text{300}\text{°};0)$$, $$(-\text{120}\text{°};0)$$, $$(\text{60}\text{°};0)$$ and $$(\text{240}\text{°};0)$$

Discovering the characteristics

For functions of the general form: $$f(\theta) = y =\cos (\theta + p)$$:

Domain and range

The domain is $$\{ \theta: \theta \in \mathbb{R} \}$$ because there is no value for $$\theta$$ for which $$f(\theta)$$ is undefined.

The range is $$\{ f(\theta): -1 \leq f(\theta) \leq 1, f(\theta) \in \mathbb{R} \}$$.

Intercepts

The $$x$$-intercepts are determined by letting $$f(\theta) = 0$$ and solving for $$\theta$$.

The $$y$$-intercept is calculated by letting $$\theta = \text{0}\text{°}$$ and solving for $$f(\theta)$$. \begin{align*} y &= \cos (\theta + p) \\ &= \cos (\text{0}\text{°} + p) \\ &= \cos p \end{align*} This gives the point $$(\text{0}\text{°};\cos p)$$.