1.5 Q-1

Question Statement

Draw the graphs of the following equations:

1. $x^2 + y^2 = 9$
2. $\frac{x^2}{16} + \frac{y^2}{4} = 1$
3. $y = e^{2x}$
4. $y = 3^x$

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Background and Explanation

This question requires understanding the process of graphing equations, including:

• Types of Graphs: Circle, ellipse, and exponential functions.
• Symmetry: Checking symmetry with respect to axes and origin.
• Table of Values: Choosing specific values of $x$ to calculate $y$ for plotting.
• Domain and Range: Determining the feasible values of $x$ and $y$.

A general understanding of basic algebra, trigonometry, and exponential functions is helpful.

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Solution

Part 1: $x^2 + y^2 = 9$

Steps:

1. Rewriting the equation:
   Express $y$ in terms of $x$:
   $y = \pm \sqrt{9 - x^2}$

2. Key Characteristics:

   • Domain: $x \in [-3, 3]$, since $x^2 \leq 9$.
   • Range: $y \in [-3, 3]$.
   • Symmetry: The graph is symmetric about both axes and the origin (a perfect circle).

3. Plotting: Plot the points $(x, y)$ and connect them smoothly to form a circle.

4. Graph:
   

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Part 2: $\frac{x^2}{16} + \frac{y^2}{4} = 1$

Steps:

1. Rewriting the equation:
   $y = \pm 2 \sqrt{1 - \frac{x^2}{16}}$

2. Key Characteristics:

   • Domain: $x \in [-4, 4]$, since $x^2 \leq 16$.
   • Range: $y \in [-2, 2]$.
   • Symmetry: Symmetric about both axes and the origin (ellipse).
   • Major axis: Horizontal (length = 8).
   • Minor axis: Vertical (length = 4).

3. Plotting: Plot the points $(x, y)$ and connect them smoothly to form an ellipse.

4. Graph:
   

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Part 3: $y = e^{2x}$

Steps:

1. Key Characteristics:

   • Exponential growth.
   • Domain: $x \in \mathbb{R}$.
   • Range: $y > 0$.
   • Asymptote: $y = 0$ as $x \to -\infty$.

2. Plotting: Plot the points $(x, y)$ and connect them to form an exponential curve.

3. Graph:
   

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Part 4: $y = 3^x$

Steps:

1. Key Characteristics:

   • Exponential growth similar to $e^{2x}$ but with base 3.
   • Domain: $x \in \mathbb{R}$.
   • Range: $y > 0$.

2. Plotting: Plot the points $(x, y)$ and connect them to form an exponential curve.

3. Graph:
   

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Key Formulas or Methods Used

1. Circle: $x^2 + y^2 = r^2$.
2. Ellipse: $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$.
3. Exponential Functions: $y = a^x$ or $y = e^x$.

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Summary of Steps

1. Rearrange the equation to isolate $y$ or simplify plotting.
2. Determine domain and range for feasible $x$ and $y$ values.
3. Create a table for specific $x$ values and their corresponding $y$ values.
4. Plot points on a graph and connect smoothly.
5. Identify key features like symmetry, intercepts, and asymptotes.

Reference