Third Graph Paper Original graph (M): $$ y=x^{2} $$ Graph 1: $$ y=(x-2)^{2} $$ Graph 2: $$ y=(x+4)^{2} $$ Graph3: $$ y=(x+2)^{2}+4 $$ Graph 4: $$ y=-2(x-2)^{2}-1 $$ 1.2 Use the general form of a Parabola: $$ f(x)=a(x+p)^{2}+q $$ to make conclusions using the graphs you drew in 1.1. In your conclusion you must mention the effect(s) of $$ a, p $$ and $$ q $$. (6) [30] QUESTION 2 Hyperbola - The graph of $$ g(x)=\frac{a}{x+p}+q $$ 2.1 Draw the graphs of the equations below according to the Graph Papers indicated: First Graph Paper Original Graph (M): $$ y=\frac{4}{x} $$ Graph 1: $$ y=\frac{8}{x} $$ Graph 2: $$ y=\frac{2}{x} $$ Graph 3: $$ y=-\frac{4}{x} $$ 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: $$ y=\frac{8}{x}+2 $$ Graph 4: $$ y=-\frac{8}{x}-1 $$ Third Graph Paper Original Graph (M): $$ y=\frac{4}{x} $$ Graph 1: $$ y=\frac{4}{x-2} $$ Graph 2: $$ y=\frac{4}{x+2} $$ Graph 3: $$ y=\frac{4}{x-2}-1 $$ Graph 4: $$ y=\frac{4}{x+2}+2 $$

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Let's break down the problem step by step, starting with the conclusions about the parabolas and then moving on to the hyperbolas. ### Part 1: Parabolas The general form of a parabola is given by: $$ f(x) = a(x + p)^2 + q $$ Where: - $$ a $$ affects the **direction** and **width** of the parabola. - $$ p $$ affects the **horizontal shift** (left or right). - $$ q $$ affects the **vertical shift** (up or down). #### Analyzing Each Graph 1. **Original Graph (M):** $$ y = x^2 $$ - Here, $$ a = 1 $$, $$ p = 0 $$, $$ q = 0 $$. 2. **Graph 1:** $$ y = (x - 2)^2 $$ - $$ a = 1 $$, $$ p = -2 $$, $$ q = 0 $$ - **Effect:** The graph shifts **2 units to the right**. 3. **Graph 2:** $$ y = (x + 4)^2 $$ - $$ a = 1 $$, $$ p = 4 $$, $$ q = 0 $$ - **Effect:** The graph shifts **4 units to the left**. 4. **Graph 3:** $$ y = (x + 2)^2 + 4 $$ - $$ a = 1 $$, $$ p = 2 $$, $$ q = 4 $$ - **Effect:** The graph shifts **2 units to the left** and **4 units up**. 5. **Graph 4:** $$ y = -2(x - 2)^2 - 1 $$ - $$ a = -2 $$, $$ p = -2 $$, $$ q = -1 $$ - **Effect:** The graph shifts **2 units to the right**, opens **downward** (due to $$ a < 0 $$), and shifts **1 unit down**. The parabola is also **narrower** because $$ |a| > 1 $$. ### Conclusion for Parabolas - The value of $$ a $$ determines the **direction** (upward if $$ a > 0 $$, downward if $$ a < 0 $$) and the **width** (narrower if $$ |a| > 1 $$, wider if $$ |a| < 1 $$). - The value of $$ p $$ shifts the graph **horizontally**: negative values shift left, positive values shift right. - The value of $$ q $$ shifts the graph **vertically**: negative values shift down, positive values shift up. ### Part 2: Hyperbolas The general form of a hyperbola is given by: $$ g(x) = \frac{a}{x + p} + q $$ Where: - $$ a $$ affects the **vertical stretch** or **compression**. - $$ p $$ affects the **horizontal shift** (left or right). - $$ q $$ affects the **vertical shift** (up or down). #### Analyzing Each Graph 1. **Original Graph (M):** $$ y = \frac{4}{x} $$ - Here, $$ a = 4 $$, $$ p = 0 $$, $$ q = 0 $$. 2. **Graph 1:** $$ y = \frac{8}{x} $$ - $$ a = 8 $$, $$ p = 0 $$, $$ q = 0 $$ - **Effect:** The graph is **stretched vertically**. 3. **Graph 2:** $$ y = \frac{2}{x} $$ - $$ a = 2 $$, $$ p = 0 $$, $$ q = 0 $$ - **Effect:** The graph is **compressed vertically**. 4. **Graph 3:** $$ y = -\frac{4}{x} $$ - $$ a = -4 $$, $$ p = 0 $$, $$ q = 0 $$ - **Effect:** The graph is reflected across the x-axis and stretched vertically. #### Second Graph Paper 1. **Graph 1:** $$ y = \frac{4}{x} + 2 $$ - $$ a = 4 $$, $$ p = 0 $$, $$ q = 2 $$ - **Effect:** The graph shifts **2 units up**. 2. **Graph 2:** $$ y = \frac{4}{x} - 1 $$ - $$ a = 4 $$, $$ p = 0 $$, $$ q = -1 $$ - **Effect:** The graph shifts **1 unit down**. 3. **Graph 3:** $$ y = \frac{8}{x} + 2 $$ - $$ a = 8 $$, $$ p = 0 $$, $$ q = 2 $$ - **Effect:** The graph is **stretched vertically** and shifts **2 units up**. 4. **Graph 4:** $$ y = -\frac{8}{x} - 1 $$ - $$ a = -8 $$, $$ p = 0 $$, $$ q = -1 $$ - **Effect:** The graph is reflected across the x-axis, stretched vertically, and shifts **1 unit down**. #### Third Graph Paper 1. **Graph 1:** $$ y = \frac{4}{x - 2} $$ - $$ a = 4 $$, $$ p = -2 $$, $$ q = 0 $$ - **Effect:** The graph shifts **2 units to the right**. 2. **Graph 2:** $$ y = \frac{4}{x + 2} $$ - $$ a = 4 $$, $$ p = 2 $$, $$ q = 0 $$ - **Effect:** The graph shifts **2 units to the left**. 3. **Graph 3:** $$ y = \frac{4}{x - 2} - 1 $$ - $$ a = 4 $$, $$ p = -2 $$, $$ q = -1 $$ - **Effect:** The graph shifts **2 units to the right** and **1 unit down**. 4. **Graph 4:** $$ y = \frac{4}{x + 2} + 2 $$ - $$ a = 4 $$, $$ p = 2 $$, $$ q = 2 $$ - **Effect:** The graph shifts **2 units to the left** and **2 units up**. ### Conclusion for Hyperbolas - The value of $$ a $$ affects the **vertical stretch** or **compression** and the **direction** (positive or negative). - The value of $$ p $$ shifts the graph **horizontally**: negative values shift left, positive values shift right. - The value of $$ q $$ shifts the graph **vertically**: negative **Parabolas:** - The coefficient $$ a $$ affects the direction (upward if $$ a > 0 $$, downward if $$ a < 0 $$) and the width (narrower if $$ |a| > 1 $$, wider if $$ |a| < 1 $$). - The value $$ p $$ shifts the graph horizontally: negative values shift left, positive values shift right. - The value $$ q $$ shifts the graph vertically: negative values shift down, positive values shift up. **Hyperbolas:** - The coefficient $$ a $$ affects the vertical stretch or compression and the direction (positive or negative). - The value $$ p $$ shifts the graph horizontally: negative values shift left, positive values shift right. - The value $$ q $$ shifts the graph vertically: negative values shift down, positive values shift up.

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