Value Tables, Function Graphs and Reading Values
A value table lists inputs and the matching outputs of a function, and plotting those pairs as points produces the graph of the function. In brief: to graph f(x) = x2 − 2x − 3, compute f(x) for several x-values, plot the points (x, f(x)) and connect them with a smooth curve. From the graph you can read values such as f(3) = 0.
Tables and graphs are the two most common ways to see what a function does. A table gives exact values at chosen points, while a graph shows the overall shape, the highs and lows and the trend. This guide explains how to build a table, how to plot and connect points, how to read values both ways and how to estimate between points.
How do you make a value table?
A value table has two rows or columns: one for x and one for f(x). Choose x-values that include both negative and positive numbers and also zero, if the domain allows. Substitute each into the formula and record the result.
For a linear function such as f(x) = 2x − 1, a few points are enough: f(−1) = −3, f(0) = −1, f(1) = 1 and f(2) = 3. The outputs rise by 2 for each step of 1 in the input, which is the constant rate of change of a line. For curves, such as parabolas or roots, choose more points, especially where the shape changes.
How do you plot a function graph step by step?
- Check the domain, so that all chosen x-values are allowed.
- Calculate a table of values.
- Draw coordinate axes with a suitable scale on both axes.
- Plot each pair (x, f(x)) as a point.
- Connect the points. Use a smooth curve if the function is continuous and the formula applies between the points.
- Extend the curve with arrows if the domain continues.
Use a ruler for lines and a smooth freehand curve otherwise. Do not connect points of a discrete data set, such as the number of students per class, because values between them have no meaning.
Worked example: a quadratic function
Graph f(x) = x2 − 2x − 3 for x from −2 to 4.
| x | Calculation | f(x) |
|---|---|---|
| −2 | 4 + 4 − 3 | 5 |
| −1 | 1 + 2 − 3 | 0 |
| 0 | 0 − 0 − 3 | −3 |
| 1 | 1 − 2 − 3 | −4 |
| 2 | 4 − 4 − 3 | −3 |
| 3 | 9 − 6 − 3 | 0 |
| 4 | 16 − 8 − 3 | 5 |
The points form a U-shaped parabola. The graph crosses the x-axis at x = −1 and x = 3, and it crosses the y-axis at (0, −3). The lowest point is (1, −4).
How do you read values from a graph?
Finding f(a): go to x = a on the horizontal axis, move vertically to the curve, then horizontally to the y-axis and read the value. In the example, f(2) = −3.
Finding x for a given output: start at the output on the y-axis, move horizontally to the curve and then down to the x-axis. A given output may correspond to more than one x. In the example, f(x) = 5 gives x = −2 and x = 4, and f(x) = 0 gives x = −1 and x = 3.
Always state the scale: if one grid square is 5 units, a reading between lines is an estimate.
How do you estimate values between table entries?
If a function is smooth, you can estimate values between two table entries. For linear behaviour, use the midpoint: halfway between f(2) = −3 and f(3) = 0 the estimate is −1.5. The true value is f(2.5) = 6.25 − 5 − 3 = −1.75, so the estimate is close but not exact. The curve bends, so a straight-line estimate is off. Closer table steps give better estimates.
Estimating beyond the table is extrapolation and is much riskier. The pattern might not continue: from f(−2) to f(4) the table says nothing about f(10), which is 77.
Where do tables and graphs come from?
Value tables are among the oldest mathematical tools. Babylonian scribes prepared tables of squares and reciprocals, and Greek and Indian astronomers computed tables of chords and sines. Graphs as pictures of relations were used by Nicole Oresme in the fourteenth century, who plotted quantities against time. The coordinate system of René Descartes and Pierre de Fermat in the seventeenth century linked algebraic formulas with curves.
Where are value tables and graphs used today?
A temperature record, a stock chart and a speedometer log are all value tables turned into graphs. Spreadsheets and graphing calculators create tables and plots automatically from a formula, but understanding the process lets you check that the plot is sensible. Experiments in science start with measured pairs and a graph that suggests the underlying function.
Which mistakes are common?
Too few points
Three points on a parabola can hide the turning point. Include points on both sides of the vertex.
Uneven scales
Use equal steps on each axis. A stretched axis can make a function look steeper or flatter than it is.
Connecting a gap
If a function is undefined at some x, such as 1/x at 0, do not draw the curve through the gap.
Sign errors in the table
Evaluate negative inputs in brackets: for x = −2, x2 is (−2)2 = 4, not −4.