QUESTION 5 Given the function: $$ h(x)=-\frac{2}{x-2}+2 $$. $$ 5.1 \quad $$ Write down the equations of the asymptotes of $$ h $$. 5.2 Draw the graph of $$ h $$. Clearly show all asymptotes and intercepts with the axes. 5.3 Determine the equation of the axis of symmetry of $$ h $$ with $$ m

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QUESTION 5 Given the function: $$ h(x)=-\frac{2}{x-2}+2 $$. $$ 5.1 \quad $$ Write down the equations of the asymptotes of $$ h $$. 5.2 Draw the graph of $$ h $$. Clearly show all asymptotes and intercepts with the axes. 5.3 Determine the equation of the axis of symmetry of $$ h $$ with $$ m<0 $$.

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To analyze the function $$ h(x) = -\frac{2}{x-2} + 2 $$, we will address each part of the question step by step.

### 5.1 Asymptotes of $$ h $$

1. **Vertical Asymptote**: This occurs where the function is undefined, which happens when the denominator is zero. 
   $$
   x - 2 = 0 \implies x = 2
   $$
   So, the vertical asymptote is $$ x = 2 $$.

2. **Horizontal Asymptote**: To find the horizontal asymptote, we analyze the behavior of $$ h(x) $$ as $$ x $$ approaches infinity or negative infinity. As $$ x \to \infty $$ or $$ x \to -\infty $$, the term $$ -\frac{2}{x-2} $$ approaches 0. Thus:
   $$
   h(x) \to 0 + 2 = 2
   $$
   So, the horizontal asymptote is $$ y = 2 $$.

### Summary of Asymptotes
- Vertical asymptote: $$ x = 2 $$
- Horizontal asymptote: $$ y = 2 $$

### 5.2 Graph of $$ h $$

To graph $$ h(x) = -\frac{2}{x-2} + 2 $$, we will identify the intercepts and plot the asymptotes.

1. **Intercepts**:
   - **Y-intercept**: Set $$ x = 0 $$:
     $$
     h(0) = -\frac{2}{0-2} + 2 = -\frac{2}{-2} + 2 = 1 + 2 = 3
     $$
     So, the y-intercept is $$ (0, 3) $$.

- **X-intercept**: Set $$ h(x) = 0 $$:
     $$
     0 = -\frac{2}{x-2} + 2 \implies \frac{2}{x-2} = 2 \implies 2 = 2(x-2) \implies 2 = 2x - 4 \implies 2x = 6 \implies x = 3
     $$
     So, the x-intercept is $$ (3, 0) $$.

2. **Plotting the Graph**:
   - Draw the vertical asymptote $$ x = 2 $$ (a dashed line).
   - Draw the horizontal asymptote $$ y = 2 $$ (a dashed line).
   - Plot the intercepts $$ (0, 3) $$ and $$ (3, 0) $$.
   - The function approaches the vertical asymptote as $$ x $$ approaches 2 from the left and right, and it approaches the horizontal asymptote as $$ x $$ goes to $$ \pm \infty $$.

### 5.3 Axis of Symmetry

The function $$ h(x) $$ is a rational function and does not exhibit symmetry in the traditional sense (like even or odd functions). However, we can analyze the function for symmetry about a vertical line.

To find the axis of symmetry, we can look for a vertical line $$ x = a $$ such that $$ h(a - d) = h(a + d) $$ for some $$ d $$.

Given the structure of the function, we can see that it is symmetric about the vertical line that is halfway between the vertical asymptote and the x-intercept. The vertical asymptote is at $$ x = 2 $$ and the x-intercept is at $$ x = 3 $$. The midpoint is:
$$
a = \frac{2 + 3}{2} = 2.5
$$
Thus, the axis of symmetry is:
$$
x = 2.5
$$

### Summary of Answers
- **5.1**: Vertical asymptote: $$ x = 2 $$; Horizontal asymptote: $$ y = 2 $$.
- **5.2**: Graph includes intercepts at $$ (0, 3) $$ and $$ (3, 0) $$, with asymptotes drawn.
- **5.3**: Axis of symmetry: $$ x = 2.5 $$.
- 5.1: Vertical asymptote: $$ x = 2 $$; Horizontal asymptote: $$ y = 2 $$. 	- 5.2: Graph includes intercepts at $$ (0, 3) $$ and $$ (3, 0) $$, with asymptotes drawn. 	- 5.3: Axis of symmetry: $$ x = 2.5 $$.

QUESTION 5
Given the function: .
Write down the equations of the asymptotes of .
5.2 Draw the graph of . Clearly show all asymptotes and intercepts with the axes.
5.3 Determine the equation of the axis of symmetry of with .

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QUESTION 5 Given the function: . Write down the equations of the asymptotes of . 5.2 Draw the graph of . Clearly show all asymptotes and intercepts with the axes. 5.3 Determine the equation of the axis of symmetry of with .