--- title: "Sketching Quadratic Graphs from Roots and the Turning Point" description: "Sketch a parabola from its roots, y-intercept, line of symmetry and turning point, reading the turning point off the completed square form." canonical: https://lightmysky.com/learn/mathematics/sketching-quadratic-graphs-from-roots-and-the-turning-point-mt_RcrJIbbN2x source: https://lightmysky.com/learn/mathematics/sketching-quadratic-graphs-from-roots-and-the-turning-point-mt_RcrJIbbN2x.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 Quadratic Graphs from Roots and the Turning Point Sketch a parabola from its roots, y-intercept, line of symmetry and turning point, reading the turning point off the completed square form. Subject: Mathematics · Area: Algebra · Ages 15 to 16 Page: https://lightmysky.com/learn/mathematics/sketching-quadratic-graphs-from-roots-and-the-turning-point-mt_RcrJIbbN2x ## Ready when they can - Read the turning point straight from (x + p)² + q - Mark roots, y-intercept and line of symmetry on a sketch - Use the sketch to say for which x values the curve sits above the axis ## Lesson: Sketching Quadratic Graphs from Roots and the Turning Point You can already factorise to find a quadratic's roots, and you can use the discriminant to predict how many times the curve meets the x-axis. This stop puts those facts on paper: a sketch of the parabola. Four things pin the curve down: the roots, the y-intercept, the line of symmetry, and the turning point. Find those four, and the curve can only be in one place. *(drawing: Four points, then the curve. No grid paper needed.)* The shape of the curve is decided by the x squared coefficient. Positive means a U shape with a minimum turning point, negative means an upside-down U with a maximum. The line of symmetry splits the curve in half: it sits halfway between the two roots, and its equation is x = -b over 2a. The y-intercept is the value of y when x is 0, which is just c. And the discriminant from last stop tells you how many times the curve meets the x-axis: twice, once, or not at all. **Example.** Sketch y = x squared - 6x + 5. First the roots: x squared - 6x + 5 factorises to (x - 1)(x - 5), so the curve crosses the x-axis at 1 and 5. The y-intercept is c, which is 5. The line of symmetry is halfway between the roots, at x = 3. For the turning point, complete the square: x squared - 6x + 5 is (x - 3) squared - 4, so the turning point is (3, -4). Mark those points, draw a U through them, and the sketch is done. **Example.** The completed square form hands the turning point to you directly. In (x + p) squared + q, the square is smallest at 0, so the turning point is (-p, q), and the line of symmetry is x = -p. Take y = (x + 2) squared - 9. The turning point is (-2, -9) and the line of symmetry is x = -2. Expand to check the rest: y = x squared + 4x - 5, so the y-intercept is -5, and the roots come from (x + 2) squared = 9, giving x + 2 = 3 or -3, so the roots are 1 and -5. **Example.** Now a curve that opens the other way. Sketch y = -x squared + 4x + 5. The x squared coefficient is -1, so it is an upside-down U with a maximum. The roots: factorise -(x squared - 4x - 5), which is -(x - 5)(x + 1), so the roots are 5 and -1. The y-intercept is 5. The line of symmetry is halfway, at x = 2, and completing the square gives -(x - 2) squared + 9, so the maximum is (2, 9). The curve sits above the x-axis between the roots, from -1 to 5. **Tip.** A sketch is not a graph. You do not need grid paper or every point: the roots, the y-intercept, the line of symmetry, and the turning point are enough to pin the curve down. If the roots are surds, mark them approximately and say so. The question is testing whether you know where the key points are, not whether your drawing is neat. **Recap.** Find the roots, the y-intercept, the line of symmetry and the turning point, then draw the curve through them: the turning point reads straight off (x + p) squared + q, and the curve sits above the axis on the side where the U opens. ## Practice 24 questions on this page, each with its working shown. ## Needs first - [The Discriminant and the Number of Roots](https://lightmysky.com/learn/mathematics/the-discriminant-and-the-number-of-roots-mt_uj95V0aLVS) - [Quadratic Graphs](https://lightmysky.com/learn/mathematics/quadratic-graphs-mt_GZuoYaDdWd) ## Opens up - [Simultaneous Equations with One Quadratic](https://lightmysky.com/learn/mathematics/simultaneous-equations-with-one-quadratic-mt_D89ql2sJvc) - [Sketching a Curve from Its Factorised Form](https://lightmysky.com/learn/mathematics/sketching-a-curve-from-its-factorised-form-mt_x4tZLdH8fh)