Describe a functional relationship qualitatively from its graph and sketch a graph from a verbal description

Learning goal: Describe qualitatively the functional relationship between two quantities by analyzing a graph, saying where the function is increasing or decreasing, where it is constant, and where it is linear or nonlinear. Sketch a graph that exhibits the qualitative features of a function that has been described verbally.

Qualitative means shape, not numbers. A qualitative description of a graph reports the features you can see without measuring: does the output go up, go down or stay level as the input increases; is a section a straight line or a curve; where does the behavior change? Exact values matter less than the pattern. Most graphs in this lesson show a quantity such as distance, height, temperature or volume on the vertical axis against time on the horizontal axis, but the same words apply to any pair of quantities.

Increasing, decreasing and constant. Read a graph from left to right, because the input increases in that direction. A function is increasing on an interval if the output gets larger as the input gets larger: the graph rises to the right. It is decreasing if the output gets smaller as the input gets larger: the graph falls to the right. It is constant if the output does not change: the graph is horizontal. A single function can be increasing on one interval, constant on another and decreasing on a third. Name the intervals when you describe it, for example "increasing from 0 to 10 minutes, constant from 10 to 16 minutes, decreasing from 16 to 20 minutes."

Linear and nonlinear. A section of a graph is linear if it is a straight segment, which means the rate of change is constant on that section. A section is nonlinear if it curves, which means the rate of change is itself changing. On a graph of distance from a starting point against time, for motion along a straight path away from that point, a straight rising segment means a steady speed; a curve that gets steeper means the speed is rising; a curve that flattens means the speed is dropping. (The worked model below explains why the straight-path condition is needed.) A graph can be linear in pieces and still be nonlinear as a whole if the pieces have different steepness, because the rate of change jumps where the pieces meet.

A fully worked model: reading a walk. A graph shows distance from home in meters against time in minutes for a walker on one straight road, so the distance changes only when the walker moves and the graph's steepness measures speed. From 0 to 10 minutes the graph is a straight segment rising from 0 to 600 meters. From 10 to 16 minutes it is horizontal at 600 meters. From 16 to 20 minutes it is a straight segment falling from 600 meters back to 0. The description: the function is increasing and linear on the first interval, so the walker moves away from home at a steady speed, 600 meters in 10 minutes, or 60 meters per minute. It is constant on the second interval, so the walker, being on a straight road, is not moving for 6 minutes, staying 600 meters from home. It is decreasing and linear on the third interval, so the walker returns home at a steady speed; because 600 meters are covered in only 4 minutes, that is 150 meters per minute, faster than the outward walk, which is why the final segment is steeper than the first. Steepness on a distance-time graph shows how fast the distance from home is changing, which is the walker's speed when the route is a straight line away from or toward home; the direction of the slope shows whether the distance is growing or shrinking. (Someone walking in a circle around home would keep a constant distance while moving, which is why the straight-road condition is stated.)

A misconception: the graph is not a picture of the path. A rising line on a distance-time graph does not mean the walker went uphill, and the flat section does not mean flat ground. The vertical axis measures distance, not height. A student who reads the graph as a map will describe the story wrongly. Always check what each axis measures before describing what the shape means.

Another misconception: decreasing is not the same as negative. A temperature graph can be decreasing while every value is positive, for example a cup of tea cooling from 80 °C to 30 °C. Decreasing describes the direction of change, not the sign of the values.

Nonlinear examples worth knowing. A ball thrown straight up: height increases, then decreases, and the graph is a smooth curve, steep at the start, flat at the top and steep again at the end, because the ball slows on the way up and speeds up on the way down. A cooling drink: temperature decreases along a curve that is steep at first and flattens as the drink approaches room temperature; the graph never rises. A population that doubles every hour: increasing along a curve that gets steeper and steeper, because each hour adds more than the last. In every case, "nonlinear" is justified by pointing to a changing rate of change.

Sketching from a verbal description. To sketch a graph from a story, follow a procedure. First, label the axes with the two quantities and decide which is the input; time is usually the input. Second, mark the starting value, if the story gives one. Third, split the story into stages, and for each stage decide three things: is the quantity increasing, decreasing or constant; is the change steady (straight) or changing (curved); and how steep is it compared with the other stages? Fourth, draw each stage in turn, joining them where the story says one stage ends and the next begins. Fifth, reread the story against the sketch stage by stage.

Consider a bathtub: it fills steadily for 5 minutes, then someone gets in, so the water level rises suddenly, then the level stays the same for 15 minutes while they soak, then the plug is pulled and the tub drains, quickly at first and more slowly as it empties. The sketch: a straight rising segment for the first stage; a short, nearly vertical rise for the person getting in; a horizontal segment for the soak; then a decreasing curve that is steep at first and flattens as the level nears 0. No numbers are required on the vertical axis, but the relative heights must be right: the level after the person gets in is higher than before, and the level after draining, back at 0, is lower than every level reached after the filling began.

Describing with precision. Good qualitative descriptions name the intervals, use the words increasing, decreasing, constant, linear and nonlinear, compare steepness where it matters, and connect each feature to the situation. "It goes up then down" is a start; "it increases at a steady rate for the first 10 minutes, stays constant for 6 minutes, then decreases at a faster steady rate for 4 minutes" is a description a reader could draw from.

Worked description of a graph: A graph shows the temperature of a room over a day. From 6 a.m. to 3 p.m. the graph rises along a curve that flattens as it approaches 3 p.m.; from 3 p.m. to 10 p.m. it falls along a straight segment. Description: the function is increasing and nonlinear from 6 a.m. to 3 p.m., warming quickly in the morning and more slowly toward mid-afternoon, then decreasing and linear from 3 p.m. to 10 p.m., cooling at a steady rate. The highest temperature occurs at 3 p.m., where the graph changes from rising to falling.

Worked sketch from a story: "A cyclist rides along a straight road at a steady pace for 20 minutes, stops at a shop for 10 minutes, then rides home along the same road at a steady but slower pace." With distance from home on the vertical axis and time on the horizontal axis: a straight segment rising from the origin for 20 minutes; a horizontal segment for the next 10 minutes at the same height; then a straight segment falling to 0, drawn less steep than the first segment because the return pace is slower, so it takes more than 20 minutes to get home. Reread: steady means straight, stop means horizontal, slower means less steep.

Worked nonlinear judgement: A graph of the height of a dropped ball against time falls from the drop height to 0 along a curve that gets steeper as time goes on. The function is decreasing on the whole interval, and it is nonlinear because the rate of change is not constant: the ball covers little distance in the first moment and much more in the last. A straight line would mean the ball fell at one steady speed, which a dropped object does not do because it speeds up as it falls.

Representation task: Take the walking graph from the lesson. In a three-column record, list each interval (0 to 10, 10 to 16 and 16 to 20 minutes), the words that describe it (increasing, decreasing or constant; linear or nonlinear), and what the walker is doing. Then redraw the graph with a fourth stage: after arriving home the walker waits 3 minutes, then walks to a neighbor 200 meters away at the original outward speed. Show that the final segment should be less steep than the return segment and explain why using the two speeds.

Independent performance: Write a story of your own with at least four stages about a quantity that changes over time (a swimming pool being filled and used, a phone battery through a day, the number of people in a hall during an event). Sketch its graph with labeled axes, using at least one constant stage, at least one nonlinear stage and at least two stages with different steepness. Give the story, without the graph, to a partner, who sketches it independently; then compare the two sketches and resolve every difference by pointing to the sentence in the story that decides it. Success criteria: axes labeled; each stage's direction, straightness and relative steepness match the story; the nonlinear stage is justified by a sentence explaining why the rate of change changes; the comparison settles every difference from the text, or, if the two sketches agree, points to the sentences that made them agree.