QUESTION 2 Hyperbola - The gruph of $$ g(x)=\frac{a}{x+D}+q $$ 2.1 Draw the graphs of the equations below according to the Graph Papers indicated: First Graph Paper Original Graph (M): $$ y=\frac{1}{x} $$ Graph 1: $$ y=\frac{0}{x} $$ Graph 2: $$ y=\frac{2}{x} $$ Gruph 3: $$ y=-\frac{1}{x} $$ Second Graph Paper Original Graph (M): $$ y=\frac{4}{x} $$ Graph $$ 1: y=\frac{4}{x}+2 $$ Graph 2: $$ y=\frac{4}{x}-1 $$ Graph 3: $$ y=\frac{0}{x}+2 $$ Graph $$ 4: y=-\frac{8}{x}-1 $$ Third Graph Paper Original Graph (M): $$ y=\frac{1}{x} $$ Gruph I: $$ y=\frac{4}{x-2} $$ Graph 2: $$ y=\frac{1}{x+2} $$ Graph 3: $$ y=\frac{1}{x-2}-1 $$ Graph 4: $$ y=\frac{1}{x+2}+2 $$

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To draw the graphs of the given equations, we will analyze each function step by step. The functions provided are variations of hyperbolas, which can be represented in the form $$ g(x) = \frac{a}{x + D} + q $$.

### First Graph Paper

1. **Original Graph (M):** 
   $$
   y = \frac{1}{x}
   $$
   This is a standard hyperbola with vertical and horizontal asymptotes at $$ x = 0 $$ and $$ y = 0 $$.

2. **Graph 1:** 
   $$
   y = \frac{0}{x} = 0
   $$
   This is a horizontal line at $$ y = 0 $$.

3. **Graph 2:** 
   $$
   y = \frac{2}{x}
   $$
   This is a hyperbola that is vertically stretched compared to the original graph.

4. **Graph 3:** 
   $$
   y = -\frac{1}{x}
   $$
   This is a reflection of the original graph across the x-axis.

### Second Graph Paper

1. **Original Graph (M):** 
   $$
   y = \frac{4}{x}
   $$
   This is a hyperbola with vertical and horizontal asymptotes at $$ x = 0 $$ and $$ y = 0 $$.

2. **Graph 1:** 
   $$
   y = \frac{4}{x} + 2
   $$
   This graph is the original graph shifted upwards by 2 units.

3. **Graph 2:** 
   $$
   y = \frac{4}{x} - 1
   $$
   This graph is the original graph shifted downwards by 1 unit.

4. **Graph 3:** 
   $$
   y = \frac{0}{x} + 2 = 2
   $$
   This is a horizontal line at $$ y = 2 $$.

5. **Graph 4:** 
   $$
   y = -\frac{8}{x} - 1
   $$
   This graph is a reflection of the original graph stretched vertically by a factor of 2 and shifted downwards by 1 unit.

### Third Graph Paper

1. **Original Graph (M):** 
   $$
   y = \frac{1}{x}
   $$
   This is the same as before.

2. **Graph I:** 
   $$
   y = \frac{4}{x-2}
   $$
   This graph is the original graph shifted to the right by 2 units and vertically stretched.

3. **Graph 2:** 
   $$
   y = \frac{1}{x+2}
   $$
   This graph is the original graph shifted to the left by 2 units.

4. **Graph 3:** 
   $$
   y = \frac{1}{x-2} - 1
   $$
   This graph is the original graph shifted to the right by 2 units and downwards by 1 unit.

5. **Graph 4:** 
   $$
   y = \frac{1}{x+2} + 2
   $$
   This graph is the original graph shifted to the left by 2 units and upwards by 2 units.

### Summary of Graphs

- The first graph paper contains variations of the basic hyperbola $$ y = \frac{1}{x} $$.
- The second graph paper contains a hyperbola $$ y = \frac{4}{x} $$ and its transformations.
- The third graph paper contains shifts and transformations of the hyperbola $$ y = \frac{1}{x} $$.

### Graphing

To visualize these graphs, you can use graphing software or graph paper to plot the points and sketch the curves based on the transformations described. Each graph will have its own unique shape and position based on the transformations applied to the original hyperbola.
To draw the graphs of the given equations: 1. **First Graph Paper:** - **Original Graph (M):** $$ y = \frac{1}{x} $$ (standard hyperbola) - **Graph 1:** $$ y = 0 $$ (horizontal line) - **Graph 2:** $$ y = \frac{2}{x} $$ (stretched hyperbola) - **Graph 3:** $$ y = -\frac{1}{x} $$ (reflected hyperbola) 2. **Second Graph Paper:** - **Original Graph (M):** $$ y = \frac{4}{x} $$ - **Graph 1:** $$ y = \frac{4}{x} + 2 $$ (shifted up) - **Graph 2:** $$ y = \frac{4}{x} - 1 $$ (shifted down) - **Graph 3:** $$ y = 2 $$ (horizontal line) - **Graph 4:** $$ y = -\frac{8}{x} - 1 $$ (stretched and shifted down) 3. **Third Graph Paper:** - **Original Graph (M):** $$ y = \frac{1}{x} $$ - **Graph I:** $$ y = \frac{4}{x-2} $$ (shifted right and stretched) - **Graph 2:** $$ y = \frac{1}{x+2} $$ (shifted left) - **Graph 3:** $$ y = \frac{1}{x-2} - 1 $$ (shifted right and down) - **Graph 4:** $$ y = \frac{1}{x+2} + 2 $$ (shifted left and up) Each graph is a variation of the basic hyperbola $$ y = \frac{1}{x} $$ with different transformations applied.

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