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: \( y=\frac{4}{x+2}+2 \)

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