Types of Graphs

Standard Form: Represented by the equation $y = mx + c$, where $m$ is the gradient (slope) and $c$ is the y-intercept.

Horizontal Lines: Equations in the form $y = c$ result in a flat line parallel to the x-axis, indicating a constant value regardless of $x$.

Vertical Lines: Equations in the form $x = k$ result in a vertical line parallel to the y-axis, representing a single $x$ value for all possible $y$ values.

The Parabola: Quadratic equations ($y = ax^2 + bx + c$) produce a symmetrical curve known as a parabola.

Concavity: If the coefficient $a$ is positive, the graph is 'U-shaped' (positive quadratic) with a minimum point. If $a$ is negative, it is 'n-shaped' (negative quadratic) with a maximum point.

Symmetry: Every parabola has a vertical line of symmetry that passes through its vertex (the highest or lowest point).

Cubic Graphs: Equations with $x^3$ typically have an 'S-shape' and can have up to two turning points where the curve changes direction.

Reciprocal Graphs: The basic form $y = \frac{1}{x}$ creates two distinct curved branches that never connect.

Asymptotes: Reciprocal graphs approach the axes but never touch them because division by zero is undefined ($x \neq 0$).

Identify the Power: Always look for the highest power of $x$ first; this immediately tells you the family of the graph (e.g., $x^2$ is always a parabola).

Check the Sign: The sign of the leading coefficient determines the orientation (e.g., a negative $x^3$ goes 'downhill' from top-left to bottom-right).

Plotting Precision: When drawing curves, never use a ruler to connect points; use a single, smooth freehand line to show the continuous nature of the function.

Sanity Check: If a point you calculated doesn't fit the expected shape (like a point far outside a parabola's curve), re-check your substitution and arithmetic.