--- title: "Linear Function Graphs" description: "Recognise that a linear function produces a straight-line graph, understand the relationship between an equation of the form y = mx + c and its graphical representation, and interpret gradient and y-i" canonical: https://lightmysky.com/learn/mathematics/linear-function-graphs-mt_WBdHkc2HTf source: https://lightmysky.com/learn/mathematics/linear-function-graphs-mt_WBdHkc2HTf.md retrieved: 2026-10-19 --- > **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`. # Linear Function Graphs Recognise that a linear function produces a straight-line graph, understand the relationship between an equation of the form y = mx + c and its graphical representation, and interpret gradient and y-intercept in context Subject: Mathematics · Area: Algebra · Ages 12 to 14 Page: https://lightmysky.com/learn/mathematics/linear-function-graphs-mt_WBdHkc2HTf ## Ready when they can - Explain that changing m in y = mx + c alters the steepness and direction of the line - Identify the y-intercept of a line from its equation and from its graph - Determine whether a given equation will produce a straight line or a curve ## Lesson: Straight lines from equations Your taxi rule made a table and a graph: it started at £2 and rose £1.50 a mile. A rule like that has a name. It is a linear function, written y = mx + c, and every equation written this way makes a straight line when you graph it. The number m is the gradient. It tells you how steep the line is and whether it slopes up or down as x increases. The number c is the y-intercept. It tells you where the line crosses the y-axis, the point where x = 0. *(drawing: Each step right adds 2 to y. That steady jump is the gradient, m = 2. The first value, 3, is the y-intercept.)* **Example.** Take y = 2x + 3. Here m = 2 and c = 3. The line crosses the y-axis at 3, and for every step you move right, it climbs up by 2. You could sketch this line just from the equation: start at 3 on the y-axis, then go up 2 and right 1, again and again. *(drawing: A bigger m makes a taller bar here. Both gradients are positive, so on a real graph the bigger one is the steeper line.)* Not every equation makes a straight line. If x is squared, like in y = x² + 1, or if x is under a fraction, like y = 1/x, the graph curves instead. A straight line happens when x is on its own, to the power 1, or when there is no x at all, as in y = 3. **Recap.** y = mx + c always draws a straight line, where m sets the steepness and direction, and c sets the y-intercept. ## Practice 20 questions on this page, each with its working shown. ## Needs first - [Coordinates (age 11+)](https://lightmysky.com/learn/mathematics/coordinates-age-11-mt_hVpGOEz2kG) - [Substituting into Formulae](https://lightmysky.com/learn/mathematics/substituting-into-formulae-mt_vHzVa3SURC) - [Modelling Situations with Formulae and Graphs](https://lightmysky.com/learn/mathematics/modelling-situations-with-formulae-and-graphs-mt_z5iwdZyeDr) ## Opens up - [Plotting Linear Graphs](https://lightmysky.com/learn/mathematics/plotting-linear-graphs-mt_-3udyo6VyB) - [Proportion](https://lightmysky.com/learn/mathematics/proportion-mt_5mIcmKRCgA) - [Scatter Graphs & Correlation](https://lightmysky.com/learn/mathematics/scatter-graphs-and-correlation-mt_8atyuvPUZc) - [Choosing a Model and Naming What It Ignores](https://lightmysky.com/learn/mathematics/choosing-a-model-and-naming-what-it-ignores-mt_UtK9aauJ6e) - [Ratio Notation and Relationships](https://lightmysky.com/learn/mathematics/ratio-notation-and-relationships-mt_yK51ZnKA8m)