Given the function: $$ h(x)=-\frac{2}{x-2}+2 $$. $$ 5.1 \quad $$ Write down the equations of the asymptotes of $$ h $$. $$ 5.2 \quad $$ Draw the graph of $$ h $$. Clearly show all asymptotes and intercepts with the axes. $$ 5.3 \quad $$ Determine the equation of the axis of symmetry of $$ h $$ with $$ m

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Given the function: $$ h(x)=-\frac{2}{x-2}+2 $$. $$ 5.1 \quad $$ Write down the equations of the asymptotes of $$ h $$. $$ 5.2 \quad $$ Draw the graph of $$ h $$. Clearly show all asymptotes and intercepts with the axes. $$ 5.3 \quad $$ Determine the equation of the axis of symmetry of $$ h $$ with $$ m<0 $$.

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Find the vertical asymptotes of $$ -2/(x-2)+2 $$.
Function by following steps:
- step0: Find the vertical asymptotes:
  $$y=-\frac{2}{x-2}+2$$
- step1: Evaluate the limit:
  $$\lim _{xightarrow 2}\left(-\frac{2}{x-2}+2ight)$$
- step2: Evaluate the left-hand and the right-hand limits:
  $$\begin{align}&\lim _{xightarrow 2^{-}}\left(-\frac{2}{x-2}+2ight)\&\lim _{xightarrow 2^{+}}\left(-\frac{2}{x-2}+2ight)\end{align}$$
- step3: Evaluate the left-hand limit:
  $$\begin{align}&+\infty\&-\infty\end{align}$$
- step4: Limit does not exist:
  $$	extrm{The limit does not exist}$$
- step5: $$x=2$$ is a vertical asymptote$$:$$
  $$\begin{align}&x=2	extrm{ }	extrm{is a vertical asymptote}\end{align}$$
- step6: List all vertical asymptotes of the function:
  $$\begin{align}&x=2\end{align}$$
Find the horizontal asymptotes of $$ -2/(x-2)+2 $$.
Function by following steps:
- step0: Find the horizontal asymptotes:
  $$y=-\frac{2}{x-2}+2$$
- step1: Evaluate the limits $$\lim _{xightarrow +\infty}\left(yight)$$ and $$\lim _{xightarrow -\infty}\left(yight):$$
  $$\begin{align}&\lim _{xightarrow +\infty}\left(-\frac{2}{x-2}+2ight)\&\lim _{xightarrow -\infty}\left(-\frac{2}{x-2}+2ight)\end{align}$$
- step2: Calculate:
  $$\begin{align}&2\&2\end{align}$$
- step3: The finite values are horizontal asymptotes:
  $$\begin{align}&y=2\end{align}$$
Find the oblique asymptotes of $$ -2/(x-2)+2 $$.
Function by following steps:
- step0: Find the oblique asymptotes:
  $$y=-\frac{2}{x-2}+2$$
- step1: Evaluate the limits $$\lim _{xightarrow +\infty}\left(\frac{y}{x}ight)$$ and $$\lim _{xightarrow -\infty}\left(\frac{y}{x}ight):$$
  $$\begin{align}&\lim _{xightarrow +\infty}\left(\frac{-\frac{2}{x-2}+2}{x}ight)\&\lim _{xightarrow -\infty}\left(\frac{-\frac{2}{x-2}+2}{x}ight)\end{align}$$
- step2: Calculate:
  $$\begin{align}&0\&0\end{align}$$
- step3: The function has no oblique asymptote:
  $$\begin{align}&	extrm{No oblique asymptotes}\&	extrm{No oblique asymptotes}\end{align}$$
- step4: The function has no oblique asymptote:
  $$	extrm{No oblique asymptotes}$$
Find the y intercept of $$ -2/(x-2)+2 $$.
Function by following steps:
- step0: Find the y-intercept:
  $$y=-\frac{2}{x-2}+2$$
- step1: Set $$x$$=0$$:$$
  $$y=-\frac{2}{0-2}+2$$
- step2: Simplify:
  $$y=3$$
Find the x intercept of $$ -2/(x-2)+2 $$.
Function by following steps:
- step0: Find the $$x$$-intercept/zero:
  $$y=-\frac{2}{x-2}+2$$
- step1: Set $$y$$=0$$:$$
  $$0=-\frac{2}{x-2}+2$$
- step2: Swap the sides:
  $$-\frac{2}{x-2}+2=0$$
- step3: Find the domain:
  $$-\frac{2}{x-2}+2=0,x
eq 2$$
- step4: Move the constant to the right side:
  $$-\frac{2}{x-2}=0-2$$
- step5: Remove 0:
  $$-\frac{2}{x-2}=-2$$
- step6: Rewrite the expression:
  $$\frac{-2}{x-2}=-2$$
- step7: Cross multiply:
  $$-2=\left(x-2ight)\left(-2ight)$$
- step8: Simplify the equation:
  $$-2=-2\left(x-2ight)$$
- step9: Evaluate:
  $$1=x-2$$
- step10: Swap the sides:
  $$x-2=1$$
- step11: Move the constant to the right side:
  $$x=1+2$$
- step12: Add the numbers:
  $$x=3$$
- step13: Check if the solution is in the defined range:
  $$x=3,x
eq 2$$
- step14: Find the intersection:
  $$x=3$$
The equations of the asymptotes of the function $$ h(x)=-\frac{2}{x-2}+2 $$ are as follows:
- Vertical asymptote: $$ x=2 $$
- Horizontal asymptote: $$ y=2 $$
- No oblique asymptotes

The y-intercept is at $$ y=3 $$ and the x-intercept is at $$ x=3 $$.

To determine the equation of the axis of symmetry of $$ h $$ with $$ m<0 $$, we need to find the x-coordinate of the axis of symmetry. The axis of symmetry of a function is the vertical line that passes through the vertex of the parabola. For the function $$ h(x)=-\frac{2}{x-2}+2 $$, the axis of symmetry is the vertical line passing through the point where the function is symmetric. Since the function is a rational function, the axis of symmetry is the vertical line passing through the x-coordinate of the vertical asymptote, which is $$ x=2 $$.

Therefore, the equation of the axis of symmetry of $$ h $$ with $$ m<0 $$ is $$ x=2 $$.
- Vertical asymptote: $$ x=2 $$ - Horizontal asymptote: $$ y=2 $$ - No oblique asymptotes - y-intercept: $$ y=3 $$ - x-intercept: $$ x=3 $$ - Axis of symmetry: $$ x=2 $$

Given the function: .
Write down the equations of the asymptotes of .
Draw the graph of . Clearly show all asymptotes and intercepts with the axes.
Determine the equation of the axis of symmetry of with .

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