Given: \( f(x)=2 \tan x \quad \) and \( g(x)=\frac{1}{2} \tan x \quad, \quad x \in\left[0^{\circ} ; 360^{\circ}\right] \) 3.1 Sketch the graphs of \( f \) and \( g \) on the same set of axes. 3.2 If there is a point with coordinates \( \left(45^{\circ} ; a\right) \), in the graphs of \( f, g \) and \( y=\tan x \), Write down the value of \( a \) in the graphs of \( f, g \) and \( y=\tan x \). 3.3 The graphs are in the form \( y=a \tan x \) : Compare and comment on the values of \( a \) in the graphs of \( f . g \) and \( y=\tan x \).

**3.1 Sketch of the Graphs** - The graph of \( y = \tan x \) is the standard tangent function with vertical asymptotes at \( x = 90^\circ, 270^\circ \) (and every \( 180^\circ \) period), and it passes through the origin. - For \( f(x)=2\tan x \), the graph has the same period and asymptotes as \( \tan x \) but is vertically stretched by a factor of 2. This means for a given \( x \), the value of \( f(x) \) is twice the corresponding value of \( \tan x \). - For \( g(x)=\frac{1}{2}\tan x \), the graph is similar in form but vertically compressed by a factor of \(\frac{1}{2}\); for a given \( x \), \( g(x) \) is half of \( \tan x \). When sketching on the same axes: - All three graphs share the same vertical asymptotes. - The graph of \( f(x)=2\tan x \) is above that of \( \tan x \) when \( \tan x \) is positive and below when \( \tan x \) is negative. - Conversely, the graph of \( g(x)=\frac{1}{2}\tan x \) is below \( \tan x \) when \( \tan x \) is positive and above when \( \tan x \) is negative (since it is a contraction, the distances from the \( x\)-axis are reduced). **3.2 Value of \( a \) at \( x=45^\circ \)** - For \( y=\tan x \): \[ \tan 45^\circ = 1 \quad \Rightarrow \quad a = 1. \] - For \( f(x)=2\tan x \): \[ f(45^\circ) = 2\tan 45^\circ = 2(1)=2 \quad \Rightarrow \quad a = 2. \] - For \( g(x)=\frac{1}{2}\tan x \): \[ g(45^\circ) = \frac{1}{2}\tan 45^\circ = \frac{1}{2}(1)=\frac{1}{2} \quad \Rightarrow \quad a = \frac{1}{2}. \] **3.3 Comparison and Comment on the Values of \( a \) in \( y=a\tan x \)** - In the graph \( y=\tan x \), the coefficient \( a \) is 1. This is the standard tangent graph without any vertical scaling. - In \( f(x)=2\tan x \), the coefficient \( a \) is 2. This indicates a vertical stretch by a factor of 2 compared to the standard \( \tan x \) graph. - In \( g(x)=\frac{1}{2}\tan x \), the coefficient \( a \) is \(\frac{1}{2}\). This represents a vertical compression by a factor of \(\frac{1}{2}\) as compared to the standard \( \tan x \) graph. The value of \( a \) controls the steepness of the graph: - A larger \( a \) (as in \( f(x) \)) makes the graph steeper. - A smaller \( a \) (as in \( g(x) \)) makes the graph flatter. - When \( a = 1 \), the graph is the standard tangent function.