Sketching a Curve from Its Factorised Form
Turn a factorised polynomial into a sketch: roots give the crossings, the constant term gives the y-intercept, the highest power decides what the ends do, and a repeated factor touches instead of crossing.
What a learner can do afterwards
- Sketch y = (x + 1)(x - 2)(x - 4) with all intercepts labelled
- Explain why y = x(x - 3)² touches the axis at 3 rather than crossing it
- Predict the end behaviour of a quartic with a negative leading coefficient
1 · Read
Last stop ended with a cubic sitting in three brackets. That form is the sketch-ready one. Each bracket names a place where the curve meets the x-axis, because a product is zero exactly when one of its parts is zero. The Ridgeway crew has the drainage channel profile written as y = (x + 1)(x - 2)(x - 4), and nobody wants a table of forty values. They want the shape: where it cuts the ground, which way it leaves the page, and where it turns.
A sketch is not a plot. Four things carry it. One: the crossings, which are the values that make each bracket zero. Two: the y-intercept, which is f(0), so multiply the numbers in the brackets and keep the signs. A bracket can carry a coefficient, like (2x - 1), and at x = 0 only its number survives. Three: what the two ends do, which the highest power and its sign decide on their own. Four: whether each crossing is a crossing or a touch. Get those four right and the picture is honest, even though it is not to scale.
Sketch y = (x + 1)(x - 2)(x - 4). The crossings come straight from the brackets: -1, 2 and 4. The y-intercept is f(0) = (1)(-2)(-4) = 8. Now test the gaps by signs. Past 4 all three brackets are positive, so the curve is above the axis. Between 2 and 4 exactly one bracket is negative, so it dips below. Between -1 and 2 two are negative, so it is above again, which fits (0, 8). A number in front of the product changes f(0) but not the crossings. Between consecutive crossings the curve has to turn, so this cubic has two turning points.
A repeated factor changes what happens at its root. Take y = x(x - 3)². The bracket (x - 3) appears twice, so near x = 3 that squared part stays positive on both sides and y keeps its sign. The curve comes down to the axis at 3, touches, and turns back. At x = 0 the factor appears once, the sign does change, and the curve crosses. A bracket repeated three times, like (x + 1)³, does change sign, so the curve crosses, but it flattens on the way through. Even repeats touch, odd repeats cross.
The ends are decided by the highest power and nothing else. Multiply the leading term of each bracket, so (2x - 1) contributes 2x and a squared bracket counts twice. For y = (x + 1)(x - 2)(x - 4) that gives x³, an odd power with a positive sign, so the curve falls away on the left and climbs on the right. Now y = -(x + 3)(x - 1)²(x - 4). Its brackets give x⁴, and the minus makes it -x⁴, so both ends fall. Odd powers send the ends opposite ways, even powers send them the same way, and a minus flips whichever you decided.
Read a sketch off the factorised form. Each bracket gives a crossing, f(0) gives the y-intercept, the total power and its sign decide what both ends do, and a repeated bracket touches the axis instead of crossing it.
2 · Watch
Take it off screen
Where it sits
Where this leads
Jobs that lean on this skill. Follow one to see everything it is built on.
24 questions wait behind this lesson, each with its answer explained. Every answer feeds the sky: stars light as they are learned, and dim when it is time to come back.