Understanding the graphical representation of functions is crucial for mastering calculus and algebra, especially for competitive exams like JEE. Graphs provide a visual language to understand the behavior, properties, and transformations of functions. This module will guide you through the fundamental concepts of plotting and interpreting function graphs.

The foundation of graphical representation lies in the Cartesian coordinate system. It's a two-dimensional plane defined by two perpendicular lines: the horizontal x-axis and the vertical y-axis. The point where they intersect is called the origin (0,0). Any point on this plane can be uniquely identified by an ordered pair (x, y), where 'x' is the horizontal distance from the origin (abscissa) and 'y' is the vertical distance from the origin (ordinate).

A function, typically expressed as $y = f(x)$, establishes a relationship between an input value (x) and an output value (y). To graph a function, we plot a series of points (x, y) that satisfy the function's equation. For each chosen 'x' value, we calculate the corresponding 'y' value using the function's rule. These (x, y) pairs are then plotted on the Cartesian plane. Connecting these points smoothly (if the function is continuous) creates the graph of the function.

Graphs reveal essential characteristics of functions:

• Intercepts: Where the graph crosses the x-axis (x-intercepts, where $y=0$) and the y-axis (y-intercept, where $x=0$).
• Domain and Range: The set of all possible x-values (domain) and y-values (range) the function can take.
• Symmetry: Whether the graph is symmetric about the y-axis (even function, $f(-x) = f(x)$) or the origin (odd function, $f(-x) = -f(x)$).
• Monotonicity: Intervals where the function is increasing or decreasing.
• Extrema: Local maximum and minimum points.
• Asymptotes: Lines that the graph approaches but never touches.

Familiarity with the graphs of basic functions is vital for sketching more complex ones. These include:

• Linear Functions ($y = mx + c$): Straight lines.
• Quadratic Functions ($y = ax^2 + bx + c$): Parabolas.
• Cubic Functions ($y = ax^3 + bx^2 + cx + d$): S-shaped curves.
• Exponential Functions ($y = a^x$): Rapid growth or decay.
• Logarithmic Functions ($y = \log_a x$): Inverse of exponential functions.
• Trigonometric Functions (sin(x), cos(x), tan(x)): Periodic waves.

Understanding how basic function graphs are transformed (shifted, stretched, reflected) allows us to sketch graphs of more complex functions. Common transformations include:

• Vertical Shift: $y = f(x) + k$ shifts the graph of $f(x)$ up by $k$ units.
• Horizontal Shift: $y = f(x - h)$ shifts the graph of $f(x)$ right by $h$ units.
• Vertical Stretch/Compression: $y = a f(x)$ stretches the graph vertically by a factor of $a$.
• Horizontal Stretch/Compression: $y = f(bx)$ stretches the graph horizontally by a factor of $1/b$.
• Reflection: $y = -f(x)$ reflects the graph across the x-axis; $y = f(-x)$ reflects across the y-axis.