Functions and Graphs
Graphing Functions – Plotting and Reading Curves
Graphing a function means drawing every point (x, y) that satisfies its equation, turning an abstract rule into a shape you can see at a glance. Different families of functions produce distinctly different shapes – and once you can recognise a shape on sight, you can read an enormous amount of information from a graph in seconds: where a value is increasing or decreasing, where it peaks, where two quantities are equal, and how fast something is really changing.
This graph-reading skill traces back to René Descartes's 1637 unification of algebra and geometry on the Cartesian plane, and it was expanded hugely in the 18th century by Leonhard Euler, who studied and classified the behaviour of many different families of functions – exponential, logarithmic, and trigonometric among them – laying the groundwork for the systematic “families of graphs” approach used in every maths classroom today.
How to Plot Any Function: The Table of Values Method
Choose several x-values, substitute each into the equation to find the matching y-value, plot the resulting points, then join them in a smooth curve (or straight line, if the function is linear).
x = −2: y = 4 + 4 − 3 = 5.
x = −1: y = 1 + 2 − 3 = 0.
x = 0: y = 0 − 0 − 3 = −3.
x = 1: y = 1 − 2 − 3 = −4.
x = 3: y = 9 − 6 − 3 = 0.
Plotting (−2, 5), (−1, 0), (0, −3), (1, −4), (3, 0) and joining them smoothly gives a parabola with its lowest point (vertex) near (1, −4), crossing the x-axis at x = −1 and x = 3.
Recognising Function Families by Their Graph
| Equation Form | Shape | Key Feature |
|---|---|---|
| y = mx + c | Straight line | Constant gradient m; crosses y-axis at c |
| y = ax² + bx + c | Parabola | One turning point (vertex); symmetric |
| y = a · bx | Exponential curve | Horizontal asymptote at y = 0; never turns |
| y = logb(x) | Logarithmic curve | Vertical asymptote at x = 0; only defined for x > 0 |
| y = 1/x | Reciprocal (hyperbola) | Two separate branches; asymptotes on both axes |
Key Features to Look For
Intercepts: where the graph crosses the x-axis (roots) or y-axis.
Turning points: where a curve changes from increasing to decreasing, or vice versa (parabolas have exactly one).
Asymptotes: lines a curve approaches but never touches (common in exponential, logarithmic, and reciprocal graphs).
Symmetry: parabolas are always symmetric about a vertical line through the vertex.
They are inverse functions, so their graphs are mirror images of each other across the line y = x. y = 3x has a horizontal asymptote at y = 0; y = log₃(x) has a vertical asymptote at x = 0.
Real-Life Applications of Graphing
- Weather and climate: temperature-over-time and rainfall charts.
- Finance: stock price charts and portfolio growth curves.
- Fitness tracking: heart-rate and distance graphs from a smartwatch.
- Engineering and design: stress-strain curves and load capacity graphs.
- Navigation and graphics: GPS positioning and computer-generated imagery both plot coordinates on a grid, the same idea behind every graph in this section.
Key Takeaways
- Plotting a table of values and joining the points is the universal method for graphing any function.
- Linear, quadratic, exponential, logarithmic, and reciprocal functions each have a distinctive, recognisable shape.
- Intercepts, turning points, and asymptotes are the key features to read off any graph.
- Inverse functions (like exponentials and logarithms) produce graphs that mirror each other across the line y = x.