--- title: "Sketching a Curve from Its Factorised Form" description: "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 cros" canonical: https://lightmysky.com/learn/mathematics/sketching-a-curve-from-its-factorised-form-mt_x4tZLdH8fh source: https://lightmysky.com/learn/mathematics/sketching-a-curve-from-its-factorised-form-mt_x4tZLdH8fh.md retrieved: 2026-09-02 --- > **Agent view.** This is the Markdown twin of the page, for tools and assistants. > When to use this site, and the call that answers each job: https://lightmysky.com/agent-instructions.md > API description (OpenAPI 3.1): https://lightmysky.com/openapi.json · Authentication: https://lightmysky.com/auth.md > Pricing: https://lightmysky.com/pricing.md · Catalog: https://lightmysky.com/llms.txt · Full catalog: https://lightmysky.com/llms-full.txt > Every machine-readable file on this domain: https://lightmysky.com/.well-known/ai-catalog.json > Ask for Markdown with `Accept: text/markdown`, a `.md` address, or `?mode=agent`. # 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. Subject: Mathematics · Area: Algebra · Ages 16 to 17 Page: https://lightmysky.com/learn/mathematics/sketching-a-curve-from-its-factorised-form-mt_x4tZLdH8fh ## Ready when they can - 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 ## Lesson: Sketching a Curve from Its Factorised Form 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. *(drawing: Four checks, in this order, and the curve draws itself.)* **Example.** 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. *(drawing: Count how many times the bracket appears, then look at the axis.)* **Example.** 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. **Recap.** 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. ## Practice 24 questions on this page, each with its working shown. ## Needs first - [Polynomial Division and the Factor Theorem](https://lightmysky.com/learn/mathematics/polynomial-division-and-the-factor-theorem-mt_haL2UkGa5C) - [Sketching Quadratic Graphs from Roots and the Turning Point](https://lightmysky.com/learn/mathematics/sketching-quadratic-graphs-from-roots-and-the-turning-point-mt_RcrJIbbN2x) ## Opens up - [Forms of the Equation of a Straight Line](https://lightmysky.com/learn/mathematics/forms-of-the-equation-of-a-straight-line-mt_dhIIxdFwcp) - [Stationary Points and the Second Derivative](https://lightmysky.com/learn/mathematics/stationary-points-and-the-second-derivative-mt_VbGJEFFgfs)