We already know that transformations to curves, graphs or equations mean that we are doing one of four things:
Use the geogebra applet below to adjust the constants in $y=a\tan b\left(x-c\right)+d$y=atanb(x−c)+d and observe how it affects the graph. Try to answer the following questions.
The general form of the tan functions is
$f\left(x\right)=a\tan\left(bx-c\right)+d$f(x)=atan(bx−c)+d
or
$f\left(x\right)=a\tan b\left(x-\frac{c}{b}\right)+d$f(x)=atanb(x−cb)+d
Here is a summary of our transformations for $y=\tan x$y=tanx:
Dilations:
Reflection:
Period:
Vertical translations:
Phase shift:
How has the graph $y=3\tan x$y=3tanx been transformed from $y=\tan x$y=tanx?
Choose one of the following options:
Vertical dilation by a scale factor of $3$3.
Horizontal dilation by a scale factor of $3$3.
Horizontal translation by $3$3 to the right.
Vertical dilation by a scale factor of $\frac{1}{3}$13.
Select all functions that have the same graph as $y=-\tan x$y=−tanx.
$y=-\tan\left(x+135^\circ\right)$y=−tan(x+135°)
$y=-\tan\left(x+180^\circ\right)$y=−tan(x+180°)
$y=-\tan\left(x-360^\circ\right)$y=−tan(x−360°)
$y=-\tan\left(x+90^\circ\right)$y=−tan(x+90°)
Consider the function $y=\tan4x-3$y=tan4x−3.
Answer the following questions in radians, where appropriate.
Determine the $y$y-intercept.
Determine the period of the function.
How far apart are the asymptotes of the function?
State the first asymptote of the function for $x\ge0$x≥0.
State the first asymptote of the function for $x\le0$x≤0.
Graph the function.
The domain of a function is the set of all values that the independent variable (usually $x$x) can take and the range of a function is the set of all values that the dependent variable (usually $y$y) can attain.
Graphically speaking, we can determine the domain by observing the values of $x$x for which the function is defined over. We can also determine the range by observing the heights of each point on the graph.
Consider the graph of $y=\tan x$y=tanx below.
Notice that the graph of $y=\tan x$y=tanx is undefined at periodic intervals of length $\pi$π or $180^\circ$180°. We state the domain as being:
All real values of $x$x, except where $x=k\times180^\circ+90^\circ$x=k×180°+90° for any integer $k$k.
All real values of $x$x, except where $x=k\times\pi+\frac{\pi}{2}$x=k×π+π2 for any integer $k$k.
More generally speaking, we can consider the function $y=a\tan\left(bx-c\right)+d$y=atan(bx−c)+d where $a,b,c,d$a,b,c,d are constants. The horizontal transformations $b$b and $c$c will impact the domain.
Conceptually, the domain is all real values of $x$x, except for the asymptotes.
Notice that the constants $a$a and $d$d do not affect the domain of the function. This is because $a$a and $d$d relate to a vertical dilation and translation, which does not change the position of the vertical asymptotes.
The graph of $y=\tan x$y=tanx has no minimums, maximums, horizontal asymptotes or holes. In other words, the range is all real values of $y$y or $\left(-\infty,\infty\right)$(−∞,∞). The only exception is when we restrict the graph to a smaller domain as shown below.
In this case, the range is no longer all real values of $y$y but instead, $\left[0,\infty\right)$[0,∞). We include $y=0$y=0 in the range because the domain includes the value $x=0^\circ$x=0°.
For a function of the form $y=a\tan\left(bx-c\right)+d$y=atan(bx−c)+d, where $a,b,c,d$a,b,c,d are constants:
However on a restricted domain, the range can be any subset of the real numbers.
Let $f\left(x\right)=\tan x$f(x)=tanx and $g\left(x\right)=\tan2x$g(x)=tan2x.
Select the correct statement from the options below.
$f\left(x\right)$f(x) and $g\left(x\right)$g(x) have the same range, but different domains.
$f\left(x\right)$f(x) and $g\left(x\right)$g(x) have different domains and ranges.
$f\left(x\right)$f(x) and $g\left(x\right)$g(x) have the same domain, but different ranges.
$f\left(x\right)$f(x) and $g\left(x\right)$g(x) have the same domain and range.
Select the most appropriate explanation for your answer to part (a).
Multiplying $x$x by $2$2 translates a function horizontally.
Multiplying $x$x by $2$2 translates a function vertically.
Multiplying $x$x by $2$2 does not change the position of the asymptotes, and a vertical shift will not affect a range of $\left(-\infty,\infty\right)$(−∞,∞).
Multiplying $x$x by $2$2 changes the position of the asymptotes.
The graph of $f\left(x\right)=\tan\left(x-\frac{\pi}{3}\right)$f(x)=tan(x−π3) is shown.
Select the correct domain of $f\left(x\right)$f(x).
All real $x$x except when $x=\pi k+\frac{5\pi}{6}$x=πk+5π6 for all integer $k$k.
All real $x$x.
All real $x$x except when $x=\pi k+\frac{\pi}{2}$x=πk+π2 for all integer $k$k.
All real $x$x except when $x=\pi k+\frac{\pi}{6}$x=πk+π6 for all integer $k$k.
State the range of $f\left(x\right)$f(x) using interval notation.
Range: $\left(\editable{},\editable{}\right)$(,)
Let $f\left(x\right)=\tan x$f(x)=tanx and $g\left(x\right)=\tan2x$g(x)=tan2x.
Select the correct statement from the options below.
$f\left(x\right)$f(x) and $g\left(x\right)$g(x) have the same domain and range.
$f\left(x\right)$f(x) and $g\left(x\right)$g(x) have the same range, but different domains.
$f\left(x\right)$f(x) and $g\left(x\right)$g(x) have different domains and ranges.
$f\left(x\right)$f(x) and $g\left(x\right)$g(x) have the same domain, but different ranges.
Select the most appropriate explanation for your answer to part (a).
Multiplying $x$x by $2$2 does not change the position of the asymptotes, and a vertical shift will not affect a range of $\left(-\infty,\infty\right)$(−∞,∞).
Multiplying $x$x by $2$2 translates a function vertically.
Multiplying $x$x by $2$2 changes the position of the asymptotes.
Multiplying $x$x by $2$2 translates a function horizontally.