Graphs of trig functions  (Page 2/2)

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Functions of the form $y=tan\left(k\theta \right)$

In the equation, $y=tan\left(k\theta \right)$ , $k$ is a constant and has different effects on the graph of the function. The general shape of the graph of functions of this form is shown in [link] for the function $f\left(\theta \right)=tan\left(2\theta \right)$ .

Functions of the form $y=tan\left(k\theta \right)$

On the same set of axes, plot the following graphs:

1. $a\left(\theta \right)=tan0,5\theta$
2. $b\left(\theta \right)=tan1\theta$
3. $c\left(\theta \right)=tan1,5\theta$
4. $d\left(\theta \right)=tan2\theta$
5. $e\left(\theta \right)=tan2,5\theta$

Use your results to deduce the effect of $k$ .

You should have found that, once again, the value of $k$ affects the periodicity (i.e. frequency) of the graph. As $k$ increases, the graph is more tightly packed. As $k$ decreases, the graph is more spread out. The period of the tan graph is given by $\frac{{180}^{\circ }}{k}$ .

These different properties are summarised in [link] .

 $k>0$ $k<0$

Domain and range

For $f\left(\theta \right)=tan\left(k\theta \right)$ , the domain of one branch is $\left\{\theta :\theta \in \left(-\frac{{90}^{\circ }}{k},\frac{{90}^{\circ }}{k}\right)\right\}$ because the function is undefined for $\theta =-\frac{{90}^{\circ }}{k}$ and $\theta =\frac{{90}^{\circ }}{k}$ .

The range of $f\left(\theta \right)=tan\left(k\theta \right)$ is $\left\{f\left(\theta \right):f\left(\theta \right)\in \left(-\infty ,\infty \right)\right\}$ .

Intercepts

For functions of the form, $y=tan\left(k\theta \right)$ , the details of calculating the intercepts with the $x$ and $y$ axis are given.

There are many $x$ -intercepts; each one is halfway between the asymptotes.

The $y$ -intercept is calculated as follows:

$\begin{array}{ccc}\hfill y& =& tan\left(k\theta \right)\hfill \\ \hfill {y}_{int}& =& tan\left(0\right)\hfill \\ & =& 0\hfill \end{array}$

Asymptotes

The graph of $tank\theta$ has asymptotes because as $k\theta$ approaches ${90}^{\circ }$ , $tank\theta$ approaches infinity. In other words, there is no defined value of the function at the asymptote values.

Functions of the form $y=sin\left(\theta +p\right)$

In the equation, $y=sin\left(\theta +p\right)$ , $p$ is a constant and has different effects on the graph of the function. The general shape of the graph of functions of this form is shown in [link] for the function $f\left(\theta \right)=sin\left(\theta +{30}^{\circ }\right)$ .

Functions of the form $y=sin\left(\theta +p\right)$

On the same set of axes, plot the following graphs:

1. $a\left(\theta \right)=sin\left(\theta -{90}^{\circ }\right)$
2. $b\left(\theta \right)=sin\left(\theta -{60}^{\circ }\right)$
3. $c\left(\theta \right)=sin\theta$
4. $d\left(\theta \right)=sin\left(\theta +{90}^{\circ }\right)$
5. $e\left(\theta \right)=sin\left(\theta +{180}^{\circ }\right)$

Use your results to deduce the effect of $p$ .

You should have found that the value of $p$ affects the position of the graph along the $y$ -axis (i.e. the $y$ -intercept) and the position of the graph along the $x$ -axis (i.e. the phase shift ). The $p$ value shifts the graph horizontally. If $p$ is positive, the graph shifts left and if $p$ is negative tha graph shifts right.

These different properties are summarised in [link] .

 $p>0$ $p<0$

Domain and range

For $f\left(\theta \right)=sin\left(\theta +p\right)$ , the domain is $\left\{\theta :\theta \in \mathbb{R}\right\}$ because there is no value of $\theta \in \mathbb{R}$ for which $f\left(\theta \right)$ is undefined.

The range of $f\left(\theta \right)=sin\left(\theta +p\right)$ is $\left\{f\left(\theta \right):f\left(\theta \right)\in \left[-1,1\right]\right\}$ .

Intercepts

For functions of the form, $y=sin\left(\theta +p\right)$ , the details of calculating the intercept with the $y$ axis are given.

The $y$ -intercept is calculated as follows: set $\theta ={0}^{\circ }$

$\begin{array}{ccc}\hfill y& =& sin\left(\theta +p\right)\hfill \\ \hfill {y}_{int}& =& sin\left(0+p\right)\hfill \\ & =& sin\left(p\right)\hfill \end{array}$

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

In the equation, $y=cos\left(\theta +p\right)$ , $p$ is a constant and has different effects on the graph of the function. The general shape of the graph of functions of this form is shown in [link] for the function $f\left(\theta \right)=cos\left(\theta +{30}^{\circ }\right)$ .

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

On the same set of axes, plot the following graphs:

1. $a\left(\theta \right)=cos\left(\theta -{90}^{\circ }\right)$
2. $b\left(\theta \right)=cos\left(\theta -{60}^{\circ }\right)$
3. $c\left(\theta \right)=cos\theta$
4. $d\left(\theta \right)=cos\left(\theta +{90}^{\circ }\right)$
5. $e\left(\theta \right)=cos\left(\theta +{180}^{\circ }\right)$

Use your results to deduce the effect of $p$ .

You should have found that the value of $p$ affects the $y$ -intercept and phase shift of the graph. As in the case of the sine graph, positive values of $p$ shift the cosine graph left while negative $p$ values shift the graph right.

These different properties are summarised in [link] .

 $p>0$ $p<0$

Domain and range

For $f\left(\theta \right)=cos\left(\theta +p\right)$ , the domain is $\left\{\theta :\theta \in \mathbb{R}\right\}$ because there is no value of $\theta \in \mathbb{R}$ for which $f\left(\theta \right)$ is undefined.

The range of $f\left(\theta \right)=cos\left(\theta +p\right)$ is $\left\{f\left(\theta \right):f\left(\theta \right)\in \left[-1,1\right]\right\}$ .

Intercepts

For functions of the form, $y=cos\left(\theta +p\right)$ , the details of calculating the intercept with the $y$ axis are given.

The $y$ -intercept is calculated as follows: set $\theta ={0}^{\circ }$

$\begin{array}{ccc}\hfill y& =& cos\left(\theta +p\right)\hfill \\ \hfill {y}_{int}& =& cos\left(0+p\right)\hfill \\ & =& cos\left(p\right)\hfill \end{array}$

Functions of the form $y=tan\left(\theta +p\right)$

In the equation, $y=tan\left(\theta +p\right)$ , $p$ is a constant and has different effects on the graph of the function. The general shape of the graph of functions of this form is shown in [link] for the function $f\left(\theta \right)=tan\left(\theta +{30}^{\circ }\right)$ .

Functions of the form $y=tan\left(\theta +p\right)$

On the same set of axes, plot the following graphs:

1. $a\left(\theta \right)=tan\left(\theta -{90}^{\circ }\right)$
2. $b\left(\theta \right)=tan\left(\theta -{60}^{\circ }\right)$
3. $c\left(\theta \right)=tan\theta$
4. $d\left(\theta \right)=tan\left(\theta +{60}^{\circ }\right)$
5. $e\left(\theta \right)=tan\left(\theta +{180}^{\circ }\right)$

Use your results to deduce the effect of $p$ .

You should have found that the value of $p$ once again affects the $y$ -intercept and phase shift of the graph. There is a horizontal shift to the left if $p$ is positive and to the right if $p$ is negative.

These different properties are summarised in [link] .

 $k>0$ $k<0$

Domain and range

For $f\left(\theta \right)=tan\left(\theta +p\right)$ , the domain for one branch is $\left\{\theta :\theta \in \left(-{90}^{\circ }-p,{90}^{\circ }-p\right\}$ because the function is undefined for $\theta =-{90}^{\circ }-p$ and $\theta ={90}^{\circ }-p$ .

The range of $f\left(\theta \right)=tan\left(\theta +p\right)$ is $\left\{f\left(\theta \right):f\left(\theta \right)\in \left(-\infty ,\infty \right)\right\}$ .

Intercepts

For functions of the form, $y=tan\left(\theta +p\right)$ , the details of calculating the intercepts with the $y$ axis are given.

The $y$ -intercept is calculated as follows: set $\theta ={0}^{\circ }$

$\begin{array}{ccc}\hfill y& =& tan\left(\theta +p\right)\hfill \\ \hfill {y}_{int}& =& tan\left(p\right)\hfill \end{array}$

Asymptotes

The graph of $tan\left(\theta +p\right)$ has asymptotes because as $\theta +p$ approaches ${90}^{\circ }$ , $tan\left(\theta +p\right)$ approaches infinity. Thus, there is no defined value of the function at the asymptote values.

Functions of various form

Using your knowledge of the effects of $p$ and $k$ draw a rough sketch of the following graphs without a table of values.

1. $y=sin3x$
2. $y=-cos2x$
3. $y=tan\frac{1}{2}x$
4. $y=sin\left(x-{45}^{\circ }\right)$
5. $y=cos\left(x+{45}^{\circ }\right)$
6. $y=tan\left(x-{45}^{\circ }\right)$
7. $y=2sin2x$
8. $y=sin\left(x+{30}^{\circ }\right)+1$

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