--- title: "Reading solutions from a graph" description: "Use graphs of linear and quadratic functions to estimate output values for given inputs, find approximate solutions to equations, and interpret graphical information in real-world contexts" canonical: https://lightmysky.com/learn/mathematics/reading-solutions-from-a-graph-mt_Yw27nweoTj source: https://lightmysky.com/learn/mathematics/reading-solutions-from-a-graph-mt_Yw27nweoTj.md retrieved: 2026-11-12 --- > **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`. # Reading solutions from a graph Use graphs of linear and quadratic functions to estimate output values for given inputs, find approximate solutions to equations, and interpret graphical information in real-world contexts Subject: Mathematics · Area: Algebra · Ages 13 to 14 Page: https://lightmysky.com/learn/mathematics/reading-solutions-from-a-graph-mt_Yw27nweoTj ## Ready when they can - Read an approximate y-value for a given x, including the output at a graph's highest or lowest point - Find approximate solutions to an equation by identifying where a graph crosses the x-axis or another line - Interpret a real-world graph (e.g., distance–time) to answer contextual questions ## Lesson: Reading answers off a graph You have plotted coordinate pairs on axes. Now read answers from plotted information. Check axis labels, units and tick spacing first. The point grids here plot selected points, not the full line or curve between them. Read only information actually shown. A missing part gives no reading for inputs outside the picture. **Example.** A coordinate grid can plot selected points using * for a point and . for an empty position. Column labels give x; row labels give y. Read both scales, which increase rightwards and upwards. In this grid, the point in column x = 4 is on row y = 9, so input 4 gives output 9. y/x 0 2 4 9 . . * 7 . * . 5 * . . *(drawing: This shows a scale reading, not a two-axis graph. Halfway between 6 and 7 is 6.5.)* The x-axis is where y = 0. Every point where a graph crosses or touches it gives a solution to the rule set equal to zero. For y = x² - 4, the meeting points are (-2, 0) and (2, 0). Their x-coordinates give the solutions -2 and 2. A point on a distance-from-home graph gives a time and a distance, not a speed. Suppose the object travels on a straight route towards or away from home. A flat section then means it is stopped. A rising section means it moves away; a falling section means it returns. Speed comes from the magnitude of the slope, using the axis units. **Example.** To find an increase, subtract the starting output from the final output. If y = 4 at x = 1 and y = 11 at x = 3, the increase is 11 - 4 = 7. These bars compare the two readings. They do not show the original curve. Two graphs meet where they have the same output for the same input. In a shared coordinate grid, A and B can mark their separate points and * can mark a shared point. Read that point's column for the solution input. For y = x + 3 and y = 11 - x, the meeting is (4, 7). Input 4 solves x + 3 = 11 - x; output 7 is their shared value. **Example.** To find an input for a target output, start at that y-value. Move horizontally to every meeting with the graph, then vertically to the x-axis. One output may have several inputs. A meeting halfway between x = 4 and x = 5 gives x about 4.5. A written description of a meeting and its scale can also supply reading information. A minimum is the smallest output; a maximum is the largest. An upward-opening parabola's lowest point gives its minimum. A downward-opening parabola's highest point gives its maximum. Read the y-coordinate for the output value and the x-coordinate for where it happens. A graph can touch the x-axis without crossing it. That point still has output zero. Count distinct meeting points. No meeting in a limited drawing means no zero is shown there. It does not prove there are no zeros outside the drawn interval. On a temperature-time graph, a horizontal section means constant temperature. A falling section means cooling. If it becomes less steep while still falling, cooling is slower, but the temperature is still changing. Compare steepness using the same axis scales. **Recap.** Check scales and units. Read outputs from the graph, and find solutions at the inputs where graphs meet or where an output is zero. *This lesson, its practice questions and its flashcards were written for this dataset.* ## Practice 17 questions on this page, each with its working shown. ## Needs first - [Plotting Linear Graphs](https://lightmysky.com/learn/mathematics/plotting-linear-graphs-mt_-3udyo6VyB) ## Optional connections - [Quadratic Graphs](https://lightmysky.com/learn/mathematics/quadratic-graphs-mt_GZuoYaDdWd)