Where it climbs and where it falls, its mirror or half-turn symmetry, and its peaks and valleys — all read off the picture.
A graph is a function’s portrait, and a trained eye reads it like a sentence. Walk along the curve from left to right: where it climbs, the function is increasing; where it slides down, it is decreasing. Notice its symmetry — a mirror across the y-axis means it is even, a half-turn about the origin means it is odd. Spot its highest and lowest points and you have its maximum and minimum. These are the very reading skills the next stage, on linear functions, leans on — so we practise them here on one curve that has a little of everything.
Everything in this lesson is a way of looking. You will not compute much; you will learn to see. We met the graph of a function in 20.5 as one of its three outfits — formula, table, and graph. Now we let the graph speak for itself.
Always read a graph the way you read English — left to right. Put a fingertip on the curve at the far left and trace it toward the right, watching the height.
Said precisely: a function is increasing on a stretch when, for any two inputs there, x₁ < x₂ ⇒ f(x₁) < f(x₂) — pick the later input and you always land higher. It is decreasing when x₁ < x₂ ⇒ f(x₁) > f(x₂) — the later input lands lower. A word for “always going one way” is monotonic.
Read left to right and watch the height. Up the hill = increasing; down the hill = decreasing. The rule never depends on where the curve sits — only on which way it is heading.
Few curves go just one way forever. One stretch climbs, the next falls, then it may climb again. So we report a function’s trend as a list of monotonic intervals — the runs of x over which the heading stays the same — separated by the turning points where the curve changes direction.
Take our cubic f(x) = x³⁄12 − x. Trace it from the far left: it climbs to a crest near x = −2, slides down through the middle to a trough near x = 2, then climbs away again. So:
increasing on (−∞, −2) · decreasing on (−2, 2) · increasing on (2, ∞)
Notice how we wrote those intervals: with x-values, never y-values. A monotonic interval answers the question “for which inputs does the trend hold?”, so it lives on the x-axis. The curve happens to reach heights between −4⁄3 and 4⁄3 over the middle, but the interval of decrease is the run of x from −2 to 2 — not the run of y.
Name a monotonic interval by its x-values, not by its heights. “Decreasing on (−2, 2)” is right; “decreasing on (−4⁄3, 4⁄3)” confuses the inputs with the outputs.
Slide the point along f(x) = x³⁄12 − x. Watch the readout call out increasing, decreasing, or a turning point as you pass the crest and the trough.
Some graphs are built with a hidden balance. Two kinds matter so much they have names.
A graph that is its own mirror image across the y-axis belongs to an even function. Folding the page along the y-axis lays the left half exactly onto the right. In symbols, the height at −x equals the height at x:
f(−x) = f(x) (even)
The plainest example is f(x) = x²: since (−x)² = x², the parabola climbs identically on both sides. Whatever happens at x = 3 happens again at x = −3.
A graph carried exactly onto itself by a half-turn (180°) about the origin belongs to an odd function. Spin the whole picture a half-turn about O and it lands back on itself. In symbols, the height at −x is the opposite of the height at x:
f(−x) = −f(x) (odd)
Our cubic is odd: f(−x) = (−x)³⁄12 − (−x) = −x³⁄12 + x = −(x³⁄12 − x) = −f(x). So whatever happens at x = 2, the exact opposite happens at x = −2 — the trough at (2, −4⁄3) is matched by the crest at (−2, 4⁄3). (The half-turn about O was our k = −1 scaling back in 20.3, the same point symmetry we met for figures in 20.2.)
Most functions are neither. To test one, compute f(−x) and compare: equal to f(x) means even, equal to −f(x) means odd, and anything else means neither.
Pick a kind of symmetry, then slide x. Watch where the matched point P′ lands — and read the test f(−x) against f(x).
At a curve’s highest point its height y is as large as it gets — that height is a maximum value. At its lowest point the height is as small as it gets — a minimum. A turning point where the curve switches from climbing to falling sits at the top of a hill (a max); one that switches from falling to climbing sits at the bottom of a valley (a min).
Two flavours matter. A local max or min is the highest or lowest point in its own neighbourhood — a crest or a trough. A global max or min is the highest or lowest point everywhere on the graph.
For our cubic, plug the turning x-values into the formula:
f(−2) = (−2)³12 − (−2) = −812 + 2 = −23 + 2 = 43
f(2) = 2³12 − 2 = 812 − 2 = 23 − 2 = −43
So this cubic has a local maximum 4⁄3 at x = −2 and a local minimum −4⁄3 at x = 2. They are only local: the curve climbs forever to the right and falls forever to the left, so it has no global max or min. By contrast, f(x) = x² has a global minimum 0 at x = 0 — its single lowest point, and lowest everywhere.
The earlier Try it already marks these peaks and valleys: slide the walker back to the crest near x = −2 or the trough near x = 2 and read the local max and min on the y-axis.
A maximum or minimum is a y-value — a height — that the function attains at a particular x. Report both: the value, and where it happens. “Local max 4⁄3 at x = −2.” The max is the 4⁄3, not the −2.
Now read the whole story off one picture. Take the hero curve and answer four questions in turn.
Here is a checklist you can run on any graph. Each row says what to look for, how to spot it, and what it tells you.
| What to look for | How to spot it | What it tells you |
|---|---|---|
| Increasing / decreasing | Trace left→right: climbing or falling? | The monotonic intervals (named by x) |
| Turning points | Where it switches direction | Crests and troughs — candidate max/min |
| Even symmetry | Fold on the y-axis — halves match? | f(−x) = f(x) |
| Odd symmetry | Half-turn about O — lands on itself? | f(−x) = −f(x) |
| Maximum / minimum | Highest / lowest point reached | The extreme y-value, at its x |
That is the reading kit. Stage 21 will hand you the simplest curves of all — straight lines — and ask exactly these questions of them. A line is increasing or decreasing depending on the sign of its slope, it is odd when it passes through the origin, and it has no turning point at all. Everything you just practised on a wiggly cubic will read off a line in a single glance.
• Read a graph left to right: climbing = increasing, falling = decreasing.
• Report the trend as monotonic intervals named by x, split at the turning points.
• Even: mirror across the y-axis, f(−x) = f(x) (like x²).
• Odd: half-turn about O, f(−x) = −f(x) (like x³⁄12 − x).
• A max / min is the highest / lowest y-value, attained at some x; it may be local or global.
• Our cubic: up–down–up, odd, local max 4⁄3 at x = −2, local min −4⁄3 at x = 2.
A graph climbs, then falls, then climbs again. What do we call a point where a climb meets a fall — and is it a high point or a low point?
A turning point — specifically a local maximum (the top of a hill), because the curve switches from increasing to decreasing there.
For y = x², is the function increasing or decreasing for x > 0? Explain by trying two inputs.
Increasing. Take x₁ = 1 and x₂ = 3: f(1) = 1 and f(3) = 9, so a bigger x gives a bigger y — the curve climbs. (For x < 0 it would be decreasing.)
Is f(x) = x² even, odd, or neither? Show the test.
Even. f(−x) = (−x)² = x² = f(x), so f(−x) = f(x) — the parabola is its own mirror across the y-axis.
Is f(x) = x³ even, odd, or neither? Show the test.
Odd. f(−x) = (−x)³ = −x³ = −f(x), so f(−x) = −f(x) — a half-turn about the origin carries the curve onto itself.
For f(x) = x³⁄12 − x, give the interval on which it is decreasing, named correctly.
(−2, 2) — the run of x between the crest at x = −2 and the trough at x = 2. Name it by x, not by the heights it passes through.
Find the minimum value of y = x² and the x where it occurs. Is it local or global?
The minimum value is 0, attained at x = 0. It is global: x² is never negative, so 0 is the lowest height anywhere.
Six questions to lock it in. Tap the answer you think is right.
This lesson is about reading, not calculating. The big idea is that a graph is a complete, glanceable description of a function: its monotonic behaviour, its symmetry, and its extreme values are all visible features. We use a single odd cubic, f(x) = x³⁄12 − x, because it shows everything at once — two turning points, a clean local max of 4⁄3 and local min of −4⁄3, and exact half-turn symmetry about the origin — while staying simple enough to evaluate by hand.
The misconceptions to watch are specific. First, students often name a monotonic interval by its y-values (“decreasing on (−4⁄3, 4⁄3)”) instead of its x-values (the correct (−2, 2)); insist that intervals of increase and decrease always describe inputs and live on the x-axis. Second, students confuse a maximum (a y-value, attained at some x) with the x where it occurs — keep saying “the max is 4⁄3, at x = −2.” Third, even and odd get swapped: even is the y-axis mirror, f(−x) = f(x); odd is the origin half-turn, f(−x) = −f(x). Have students run the algebraic test rather than guess from the picture alone.
This material aligns with the US Common Core high-school functions standards: F-IF.B.4 (interpreting key features of graphs — intervals of increase and decrease, relative maxima and minima, symmetry), F-IF.B.6 / F-IF.C.7 (reading and sketching the behaviour of a graph), and F-BF.B.3 (recognising even and odd functions from their graphs and from f(−x)). It is the launch pad for Stage 21, where these same readings are applied to linear functions.