Drawing Graphs from Tables

To generate a table, specific values of $x$ are substituted into the function. For example, in a quadratic function like $y = ax^2 + bx + c$, each $x$ value must be processed through the operations of squaring, multiplication, and addition to find the corresponding $y$.

The step size refers to the constant interval between consecutive $x$ values. A smaller step size provides more data points, which is essential for accurately capturing the curvature of non-linear graphs like parabolas or circles.

When calculating values manually, it is critical to use brackets for negative numbers, especially when squaring (e.g., $(-3)^2 = 9$, whereas $-3^2 = -9$). Modern scientific calculators often feature a 'Table Mode' that automates this process by asking for a start value, end value, and step size.

Each pair from the table is plotted as a coordinate point $(x, y)$ on a Cartesian grid. The horizontal position is determined by the $x$-axis, and the vertical position by the $y$-axis.

Accuracy is paramount; points should typically be plotted within half of the smallest grid square to ensure the resulting graph is mathematically valid. If a point appears as an 'outlier' that breaks the expected pattern, it usually indicates a calculation error in the table.

Before plotting, the axes must be scaled appropriately. The scale should be consistent (equal intervals) and large enough to make the graph easy to read while ensuring all points from the table fit within the drawing area.

The method for connecting points depends on the nature of the function. Linear functions (where the highest power of $x$ is 1) result in a straight line and should be connected using a ruler.

Non-linear functions, such as quadratics ($x^2$) or cubics ($x^3$), produce curves. These must be joined with a single, smooth, freehand line that passes through every plotted point without using a ruler between individual segments.

For curves, the shape should be predictable; for instance, a quadratic graph (parabola) should be symmetrical. If the points do not form a smooth flow, the calculations in the table should be re-verified.

It is vital to distinguish between discrete data and continuous functions. In a continuous function, every possible $x$ value has a corresponding $y$, justifying the connection of points with a line or curve.

Always check the symmetry of your table for quadratic functions. If the $y$-values do not repeat in a mirrored pattern around the vertex, there is likely a calculation error, particularly with negative inputs.

Verify the axis scales before you start plotting. Examiners often use different scales for the $x$ and $y$ axes (e.g., 1 unit per cm on $x$, but 5 units per cm on $y$); misreading these is a common source of lost marks.

When asked to solve an equation using a graph you have drawn, draw a horizontal or vertical line to the curve and read the coordinates carefully. Ensure your final answer matches the required level of accuracy based on the grid's smallest divisions.