--- title: "What a 17-year-old learns" description: "17 topics are written for age 17, across 1 subjects." canonical: https://lightmysky.com/learn/age/17 source: https://lightmysky.com/learn/age/17.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`. # What a 17-year-old learns 17 topics are written for age 17, across 1 subjects. Topics are placed by what they need, not by school year, so this is a starting point rather than a syllabus. Page: https://lightmysky.com/learn/age/17 ## Mathematics - [Domain and Range of a Function](https://lightmysky.com/learn/mathematics/domain-and-range-of-a-function-mt_b_4zoHz8Jc): Describe the set of inputs a function accepts and the set of outputs it can return, and say what forces a restriction: a division, a square root, or the situation being modelled. - [Polynomial Division and the Factor Theorem](https://lightmysky.com/learn/mathematics/polynomial-division-and-the-factor-theorem-mt_haL2UkGa5C): Divide a polynomial by a linear expression and read the remainder, then use the fact that a root gives a factor to break a cubic apart. - [Sketching a Curve from Its Factorised Form](https://lightmysky.com/learn/mathematics/sketching-a-curve-from-its-factorised-form-mt_x4tZLdH8fh): 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. - [Forms of the Equation of a Straight Line](https://lightmysky.com/learn/mathematics/forms-of-the-equation-of-a-straight-line-mt_dhIIxdFwcp): Move between y = mx + c, y - y₁ = m(x - x₁) and ax + by + c = 0, and produce the equation of a line from a point and a gradient or from two points. - [Parallel and Perpendicular Lines in Coordinates](https://lightmysky.com/learn/mathematics/parallel-and-perpendicular-lines-in-coordinates-mt_xNEmm139Ka): Use equal gradients for parallel lines and a gradient product of -1 for perpendicular ones, and apply both to find missing lines, midpoints and distances. - [The Equation of a Circle](https://lightmysky.com/learn/mathematics/the-equation-of-a-circle-mt_YARqmi_ioe): Read a circle's centre and radius from (x - a)² + (y - b)² = r², and recover them from an expanded equation by completing the square in both variables. - [Lines Meeting Circles: Tangents and Chords](https://lightmysky.com/learn/mathematics/lines-meeting-circles-tangents-and-chords-mt_ELA4IzIVdv): Find where a line cuts a circle by substitution, use the discriminant to tell a tangent from a chord or a miss, and use the right angle between a tangent and the radius. - [The Gradient of a Curve as a Limit](https://lightmysky.com/learn/mathematics/the-gradient-of-a-curve-as-a-limit-mt_LAlhcwVjln): See that a curve has a different gradient at every point, and that the gradient of a chord settles on a single value as the second point slides in. That limiting value is the gradient of the tangent. - [Differentiation from First Principles](https://lightmysky.com/learn/mathematics/differentiation-from-first-principles-mt_pdfPztEjw_): Write the chord gradient as [f(x + h) - f(x)] / h, simplify it algebraically, and let h approach zero to get the derivative. This is the definition every rule later rests on. - [The Power Rule for Differentiating Polynomials](https://lightmysky.com/learn/mathematics/the-power-rule-for-differentiating-polynomials-mt_l7f4j44bR3): Differentiate any sum of powers of x with the rule that xⁿ becomes nxⁿ⁻¹, including negative and fractional powers once the term is rewritten as a power. - [The Derivative as a Rate of Change](https://lightmysky.com/learn/mathematics/the-derivative-as-a-rate-of-change-mt_RqyLE3jrAW): Read dy/dx as how fast one quantity changes per unit of another, with the units to match, so a derivative answers questions about cost per item, litres per second or metres per second. - [Tangents and Normals to a Curve](https://lightmysky.com/learn/mathematics/tangents-and-normals-to-a-curve-mt_BzMw7IT6kp): Find the equation of the tangent at a point on a curve from the derivative there, and the normal from the perpendicular gradient. - [Increasing and Decreasing Functions](https://lightmysky.com/learn/mathematics/increasing-and-decreasing-functions-mt_KOrXxQvBrZ): Use the sign of the derivative to say where a curve rises and where it falls, which turns a question about shape into an inequality in x. - [Stationary Points and the Second Derivative](https://lightmysky.com/learn/mathematics/stationary-points-and-the-second-derivative-mt_VbGJEFFgfs): Solve f'(x) = 0 to locate maximum, minimum and inflection points, then use the sign of f''(x) to decide which is which. - [Optimisation with Calculus](https://lightmysky.com/learn/mathematics/optimisation-with-calculus-mt_9I0ql39Lzw): Turn a worded problem into one function of one variable, use a constraint to remove the second variable, then differentiate to find the best value and justify that it is a maximum or minimum. - [Integration as the Reverse of Differentiation](https://lightmysky.com/learn/mathematics/integration-as-the-reverse-of-differentiation-mt_edtk3ArxRk): Recover a function from its derivative by raising the power and dividing, and carry the constant of integration that differentiation destroyed. Use a known point to pin that constant down. - [Kinematics with Calculus](https://lightmysky.com/learn/mathematics/kinematics-with-calculus-mt_2kNspOOoDw): When the acceleration is not constant, differentiate displacement to get velocity and velocity to get acceleration, and integrate back the other way, using an initial condition to fix each constant.