Types of Graphs

Quadratic graphs are generated by equations of the form $y = ax^2 + bx + c$. The resulting shape is a symmetrical curve known as a parabola, which features a single turning point called the vertex.

The orientation of the parabola depends on the coefficient $a$. If $a > 0$, the graph is u-shaped (positive quadratic) with a minimum point; if $a < 0$, the graph is n-shaped (negative quadratic) with a maximum point.

Every quadratic graph possesses a vertical line of symmetry that passes directly through the vertex. This line is always located exactly halfway between the roots (x-intercepts) of the function, if they exist.

Cubic graphs follow the general form $y = ax^3 + bx^2 + cx + d$ and typically exhibit an 'S' shape. A positive cubic ($a > 0$) starts in the bottom-left quadrant and ends in the top-right, while a negative cubic ($a < 0$) starts top-left and ends bottom-right.

Reciprocal graphs, such as $y = \frac{1}{x}$, consist of two distinct, disconnected curved branches located in opposite quadrants. These branches approach the x and y axes but never actually touch them, as division by zero is undefined.

Unlike polynomials, reciprocal graphs have asymptotes, which are lines that the curve gets infinitely close to but never intersects. For $y = \frac{1}{x}$, the asymptotes are the axes $x=0$ and $y=0$.

To plot a graph accurately, a table of values is constructed by selecting a range of x-coordinates and substituting them into the function to calculate corresponding y-coordinates. It is critical to use a sufficient number of points to capture the curve's behavior, especially near turning points.

When dealing with non-linear functions, points must be joined with a smooth, freehand curve rather than straight line segments. Using a ruler between points on a quadratic or cubic graph is a common error that fails to represent the continuous rate of change.

For reciprocal functions, the table must exclude $x=0$ because the function is undefined at that point. Plotting points on either side of zero reveals the two separate branches that characterize the reciprocal shape.

Check the Leading Coefficient: Before drawing, identify if the $x^2$ or $x^3$ term is positive or negative. This provides an immediate 'sanity check' for the general direction and shape of your plotted curve.

Handle Negatives Carefully: A frequent source of marks lost is the incorrect squaring of negative numbers (e.g., $(-3)^2 = 9$, not $-9$). Always use brackets when substituting negative values into a calculator.

Symmetry Verification: For quadratic graphs, if your plotted points do not look symmetrical about a central vertical line, re-calculate your y-values. Parabolas must be perfectly balanced on both sides of the vertex.

Smoothness over Precision: While points must be plotted accurately, examiners look for a single, continuous smooth line. Avoid 'feathering' (multiple small strokes) or using a ruler for anything other than linear graphs.