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
What a learner can do afterwards
- 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
1 · Read
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.
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 | * | . | . |
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.
These cells list the zero-output points. They do not draw the curve.
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.
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.
These calculations check the stated meeting. They are not a graph.
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.
This scale shows the input reading, not the original graph.
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.
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.
These additional questions, readings and flashcards were written for this dataset.
More to read
Following the reading route
A graph pairs inputs with outputs. Check what each axis measures before reading a point. A coordinate pair such as (4, 10) means input 4 gives output 10.
For a known input, move vertically to the graph and then horizontally to the output axis. For a known output, reverse the route. Move horizontally to every meeting with the graph, then vertically to the input axis. Several inputs can share one output.
Use the printed scale. Halfway between ticks labelled 10 and 12 represents 11. One quarter of that interval above 10 represents 10.5. Grid squares do not always represent whole units.
Which coordinate answers the question?
The x-axis has output zero. A point where a graph crosses or touches that axis gives an input that solves the graph's rule set equal to zero. Read the x-coordinate for that solution.
For a minimum or maximum output, read the y-coordinate instead. An upward-opening parabola has a lowest point. A downward-opening parabola has a highest point. At a lowest point (5, -3), the minimum output is -3 and it occurs at input 5.
A short drawing may leave other features outside the picture. Seeing no zero in a limited interval does not prove that the rule has no zeros elsewhere.
Reading changes with units
A rising graph has increasing outputs as inputs increase. A falling graph has decreasing outputs. A horizontal section has unchanged outputs.
Subtract the starting output from the final output to find the change. Outputs changing from 8 to 3 give 3 - 8 = -5, a decrease of 5.
Slope compares output change with input change. Its units come from the axes. On a temperature-time graph, degrees per minute describe temperature change. They do not describe travel speed. Compare steepness only after checking the scales, since rescaling an axis changes how steep the same data look.
Reading a meeting in context
Suppose a rental graph has hours on its horizontal axis and cost in pounds on its vertical axis. The point (2, 14) means two hours cost £14. It does not tell you that the hourly charge is £14.
To read the hours for a budget, begin at the budget on the cost axis. Move across to the graph, then read the matching input. The required part must be drawn.
If two pricing graphs meet at (3, 18), both prices are £18 for three hours. Input 3 solves the equation between the pricing rules. Output 18 is their shared cost.
2 · Watch
En anglaisTake it off screen
Where it sits
Learn first
Optional connections
This opens up
Nothing builds on it yet.
Where this leads
Jobs that lean on this skill. Follow one to see everything it is built on.
17 questions wait behind this lesson, each with its answer explained. Every answer feeds the sky: stars light as they are learned, and dim when it is time to come back.