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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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 your 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. For $$ h(x) $$:
   $$
   x - 2 = 0 \implies x = 2
   $$
   Thus, 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. The term $$ -\frac{2}{x-2} $$ approaches 0 as $$ x $$ goes to infinity. Therefore:
   $$
   h(x) \to 0 + 2 = 2 \quad \text{as } x \to \infty
   $$
   Thus, 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. **Graphing**:
   - Draw the vertical asymptote at $$ x = 2 $$ (dashed line).
   - Draw the horizontal asymptote at $$ y = 2 $$ (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 infinity.

### 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 find a line of symmetry if we consider the transformation of the function.

The function can be rewritten as:
$$
h(x) = -\frac{2}{x-2} + 2
$$
This function is symmetric about the vertical line that is halfway between the vertical asymptote and the x-intercept. The x-intercept is at $$ x = 3 $$ and the vertical asymptote is at $$ x = 2 $$.

The axis of symmetry can be found as follows:
- The midpoint between $$ x = 2 $$ and $$ x = 3 $$ is:
$$
x = \frac{2 + 3}{2} = \frac{5}{2} = 2.5
$$

Since we are looking for $$ m < 0 $$, we can consider the line $$ x = 2.5 $$ as a vertical line of symmetry. However, since the function is not symmetric in the traditional sense, we can conclude that the function does not have a standard axis of symmetry.

### Final Summary
- **Asymptotes**: $$ x = 2 $$ (vertical), $$ y = 2 $$ (horizontal)
- **Intercepts**: $$ (0, 3) $$ (y-intercept), $$ (3, 0) $$ (x-intercept)
- **Axis of symmetry**: The function does not have a traditional axis of symmetry, but the vertical line $$ x = 2.5 $$ can be considered as a point of reflection for the behavior of the function around the vertical asymptote.
The vertical asymptote is $$ x = 2 $$, the horizontal asymptote is $$ y = 2 $$, the y-intercept is $$ (0, 3) $$, the x-intercept is $$ (3, 0) $$, and the function does not have a traditional axis of symmetry, but it can be considered symmetric about the line $$ x = 2.5 $$.

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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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 .