Graphs of functions.
Connect a function rule to its table, shape, roots and tangent gradients.
Plot power and exponential functions · solve equations graphically · estimate gradients
A rule becomes a table, then a graph
A function takes an input x and produces an output y using a rule. Its graph plots those matching pairs, allowing us to see many outputs at once rather than calculating them in isolation.
Build a short table by substituting chosen x-values. Plot each pair in the correct order, then join according to the function’s shape. A straight-line rule gives a straight line; a nonlinear rule usually needs a smooth curve rather than a sequence of angular joins.
Substitute the chosen x-values, plot the coordinate pairs, and join with the correct shape. Check the domain before evaluating roots or reciprocals.
The supplied families include axⁿ, and sums of up to three such terms, for n = −2, −1, −12, 0, 12, 1, 2, 3. Exponential functions such as abˣ + c form a different family: x is in the exponent.
U-shaped parabola. Roots −1 and 3; turning point (1, −4).
| x | -2 | -1 | 0 | 1 | 2 |
|---|---|---|---|---|---|
| y | 5 | 0 | -3 | -4 | -3 |
Pause and explainFor y = x², which point corresponds to x = −2?
Since (−2)² = 4, plot (−2, 4). The negative input and positive output occupy different axes.
Solve equations at intersections
A root is an input that gives output zero. On a graph, zero output lies on the x-axis, so the points where a graph meets that axis reveal solutions of f(x) = 0.
An intersection answers a related question: where do two rules give the same output for the same input? Plot both and read the x-coordinate of each meeting point. Graphical readings are usually approximate, so report an accuracy supported by the scale.
Use y = x² − 2x − 3 to solve x² − 2x − 3 = 0.
Read where the curve meets y = 0: x = −1 and 3.
Solve x² − 2x − 3 = x − 1 graphically.
Plot both graphs on the same axes. The x-coordinates of their intersections are approximately −0.56 and 3.56. Algebra checks: x² − 3x − 2 = 0.
Pause and explainWhy do intersections of y = f(x) and y = g(x) solve f(x) = g(x)?
At an intersection the two graphs have the same x-value and the same y-value, so their outputs agree.
Estimate a gradient with a tangent
A curve does not usually have one constant gradient. Its direction changes as you move along it. A tangent is a straight line matching the local direction of the curve at a chosen point.
Draw the tangent carefully, then calculate its gradient using two widely separated points on the tangent line. Those points need not lie on the curve. A larger gradient triangle reduces the effect of small reading errors.
Estimate the gradient of y = x² at x = 1.
Draw the tangent at (1, 1). Choose two widely separated points on that tangent, for example (0, −1) and (3, 5).
Use points on the tangent, not two points on the curve. The displayed tangent is an exact reference for this example; a hand-drawn estimate may vary.
For exponential growth and decay graphs, read the initial value at x = 0 and follow repeated multipliers over equal intervals. A decay model can be written 100 × 2−x = 100 × (12)x.
Pause and explainShould the two gradient points be chosen on the curve or on the tangent?
On the tangent line. The tangent’s straight-line slope estimates the curve’s gradient at the contact point.
Your turn. Read coordinates with meaning.
Distinguish three questions: plotting outputs, finding inputs where an output is zero, and estimating the gradient at a point. Each requires a different reading from the graph.
Use clear crosses, a sensible scale and smooth joins. When reading solutions, check all intersections and acknowledge the approximate accuracy of values taken from a graph.
For y = x³ + x − 4, find y when x = −1, 0 and 2.
Reveal worked answer
What equation is solved by intersections of y = x² and y = 2x + 3?
Reveal worked answer
A tangent passes through (1, 2) and (5, 14). Estimate its gradient.