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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 $f\left(\theta \right)=tan\left(2\theta \right)$ .
On the same set of axes, plot the following graphs:
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] .
$k>0$ | $k<0$ |
For $f\left(\theta \right)=tan\left(k\theta \right)$ , the domain of one branch is $\{\theta :\theta \in (-\frac{{90}^{\circ}}{k},\frac{{90}^{\circ}}{k})\}$ 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 $\left\{f\right(\theta ):f(\theta )\in (-\infty ,\infty \left)\right\}$ .
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:
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.
In the equation, $y=sin(\theta +p)$ , $p$ 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 $f\left(\theta \right)=sin(\theta +{30}^{\circ})$ .
On the same set of axes, plot the following graphs:
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] .
$p>0$ | $p<0$ |
For $f\left(\theta \right)=sin(\theta +p)$ , the domain is $\{\theta :\theta \in \mathbb{R}\}$ because there is no value of $\theta \in \mathbb{R}$ for which $f\left(\theta \right)$ is undefined.
The range of $f\left(\theta \right)=sin(\theta +p)$ is $\left\{f\right(\theta ):f(\theta )\in [-1,1\left]\right\}$ .
For functions of the form, $y=sin(\theta +p)$ , the details of calculating the intercept with the $y$ axis are given.
The $y$ -intercept is calculated as follows: set $\theta ={0}^{\circ}$
In the equation, $y=cos(\theta +p)$ , $p$ 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 $f\left(\theta \right)=cos(\theta +{30}^{\circ})$ .
On the same set of axes, plot the following graphs:
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] .
$p>0$ | $p<0$ |
For $f\left(\theta \right)=cos(\theta +p)$ , the domain is $\{\theta :\theta \in \mathbb{R}\}$ because there is no value of $\theta \in \mathbb{R}$ for which $f\left(\theta \right)$ is undefined.
The range of $f\left(\theta \right)=cos(\theta +p)$ is $\left\{f\right(\theta ):f(\theta )\in [-1,1\left]\right\}$ .
For functions of the form, $y=cos(\theta +p)$ , the details of calculating the intercept with the $y$ axis are given.
The $y$ -intercept is calculated as follows: set $\theta ={0}^{\circ}$
In the equation, $y=tan(\theta +p)$ , $p$ 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 $f\left(\theta \right)=tan(\theta +{30}^{\circ})$ .
On the same set of axes, plot the following graphs:
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] .
$k>0$ | $k<0$ |
For $f\left(\theta \right)=tan(\theta +p)$ , the domain for one branch is $\{\theta :\theta \in (-{90}^{\circ}-p,{90}^{\circ}-p\}$ because the function is undefined for $\theta =-{90}^{\circ}-p$ and $\theta ={90}^{\circ}-p$ .
The range of $f\left(\theta \right)=tan(\theta +p)$ is $\left\{f\right(\theta ):f(\theta )\in (-\infty ,\infty \left)\right\}$ .
For functions of the form, $y=tan(\theta +p)$ , the details of calculating the intercepts with the $y$ axis are given.
The $y$ -intercept is calculated as follows: set $\theta ={0}^{\circ}$
The graph of $tan(\theta +p)$ has asymptotes because as $\theta +p$ approaches ${90}^{\circ}$ , $tan(\theta +p)$ approaches infinity. Thus, there is no defined value of the function at the asymptote values.
Using your knowledge of the effects of $p$ and $k$ draw a rough sketch of the following graphs without a table of values.
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