05_Ex 1.5

Exercise Questions

Question	Link
Q1. Draw the graphs of the following equations.	1.5 Q-1
Q2. Graph the curves that has the parametric equations given below.	1.5 Q-2
Q3. Draw the graph of the functions defined below	1.5 Q-3
Q4. Find the graphical solutions of the following equations.	1.5 Q-4

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Overview

This exercise focuses on the graphical representation of equations, parametric curves, and piecewise functions. It introduces concepts such as symmetry, domain, and the behavior of equations as functions. Applications include curve sketching, identifying intercepts, and understanding continuity.

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Key Concepts

1. Symmetry: Understanding symmetry about axes and origin for different curves.
2. Domain and Range: Determining valid $x$ and $y$ values for functions.
3. Function Behavior: Identifying whether an equation defines $y$ as a function of $x$.
4. Parametric Equations: Converting parametric forms to Cartesian equations and analyzing their properties.
5. Piecewise Functions: Graphing and checking for continuity at transition points.

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Important Formulas

1. Circle: $x^2 + y^2 = r^2$

   • Radius $r$, symmetric about both axes and origin.

2. Ellipse: $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$

   • Major axis $2a$, minor axis $2b$.

3. Exponential Functions:

   • $y = e^{kx}$: Rapid growth for $x > 0$, asymptotic to $y = 0$ for $x < 0$.
   • $y = b^x$: General exponential growth behavior for $b > 1$.

4. Parametric Equations:

   • Convert $x = f(t), y = g(t)$ to Cartesian form to analyze graph.

5. Continuity:

   • Check left-hand and right-hand limits at transition points for piecewise functions.

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Tips and Tricks

1. For symmetry, substitute $(-x)$ and $(-y)$ into equations to check behavior.
2. Use a table of values for efficient graph plotting.
3. For parametric curves, eliminate the parameter $t$ to simplify analysis.
4. For exponential graphs, observe growth rates and asymptotic behavior.
5. Always check endpoints of parametric and piecewise functions for completeness.

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Summary

This exercise develops skills in sketching and analyzing graphs of various equations, understanding symmetry, and interpreting parametric and piecewise equations. It emphasizes domain and range, function behavior, and continuity.

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Reference

By Great Science Academy:

Due to each question having it’s own video, References will be provided in Exercise Questions to avoid redundancy