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.

\(p>0\)

\(p<0\)

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

Sketch the cosine graphs

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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 \((\text{0}\text{°};\cos p)\).

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