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Drawing Real-Life Graphs: Distance-Time and Cost-Quantity Practice
Learn to plot and read graphs from real scenarios. Covers axes, key points, rates, intercepts, and interpretation with worked examples and four practice tasks.
Free lesson
Four exercises
Worked answers
Algebra and Function Practice
Real-life graphs turn a described situation into a picture you can read. This lesson focuses on two common types: distance-time graphs and cost-quantity graphs. You will learn a simple convention for setting up axes, plotting key points, drawing the line or curve, and answering questions about rates and intercepts. The convention is explicit so you can check your work, but valid alternatives exist: for example, you may swap which variable goes on which axis if you label clearly, or use a different scale as long as it is consistent and readable. All numerical values used here are hypothetical examples chosen for easy arithmetic. Measure your own paper and ruler if you need actual dimensions; no printer tolerances or hardware standards are assumed.
Practice materials: Printable downloads are not available for this lesson yet. Use the instructions and four exercises below.
Setting Up Axes and Choosing a Scale
Every real-life graph needs two labeled axes with units. For a distance-time graph, time usually goes on the horizontal axis and distance on the vertical axis. For a cost-quantity graph, quantity usually goes on the horizontal axis and cost on the vertical axis. This is a convention, not a rule: swapping axes is valid if you label both clearly and keep the scale consistent.
Choose a scale that makes key points easy to plot. A scale is the amount each grid square or centimeter represents. For example, if time runs from 0 to 60 minutes and you have 12 squares, 5 minutes per square works well. Label the scale on the axis, such as '1 square = 5 minutes'. Keep units consistent: do not mix minutes and hours on the same axis without converting.
- Draw two perpendicular axes and label each with the variable and unit.
- Decide the range of each variable from the scenario.
- Choose a scale that divides the range into readable steps; write it on the axis.
- Mark equal intervals and number them clearly.
Plotting Key Points and Drawing the Line or Curve
Identify the key points from the scenario: starting value, changes in rate, stops, and end value. Plot each as a dot using the scale. For constant speed or constant cost per item, the points lie on a straight line, so use a ruler to connect them. For changing rates, connect points with straight segments between changes, or a smooth curve if the scenario describes continuous change.
A stop on a distance-time graph is a horizontal segment because distance does not change while time passes. A return to start is a segment sloping back to the horizontal axis. For cost-quantity, a fixed starting fee appears as a vertical intercept, and a constant price per item gives a straight line whose slope is the price per item.
- List the key points with their coordinates and units.
- Plot each point accurately using the chosen scale.
- Connect points with a ruler for constant rates; use segments for changes.
- Label the line or curve if the scenario has more than one.
Reading Rates and Intercepts
The slope of a straight segment tells you the rate. On a distance-time graph, slope is distance divided by time, which is speed. On a cost-quantity graph, slope is cost divided by quantity, which is price per item. To find slope, pick two points on the segment, subtract the vertical values, subtract the horizontal values, and divide. Keep units: for example, meters per minute or dollars per item.
The intercept is where the graph meets an axis. The vertical intercept often represents a starting value, such as a fixed fee or an initial distance. The horizontal intercept often represents when a value reaches zero, such as returning to the start. Read intercepts directly from the axes and state their meaning in the scenario.
- Pick two clear points on the segment you are reading.
- Subtract vertical values and horizontal values separately.
- Divide vertical change by horizontal change and attach units.
- Locate intercepts on the axes and explain what they mean.
Checking Your Graph Against the Scenario
After drawing, reread the scenario and check each feature. Does the starting point match? Does a stop appear as a horizontal segment? Does a faster speed appear as a steeper slope? Does a fixed fee appear as a vertical intercept? Check that units are consistent and that the scale is labeled.
A useful check is to compute one rate from the graph and compare it with the scenario. If the scenario says 30 meters in 6 minutes, the speed is 5 meters per minute; the graph segment should rise 5 units for every 1 unit to the right on that scale. If it does not, adjust the plotting or the scale.
- Compare the starting point and end point with the scenario.
- Check each change in rate appears as a change in slope.
- Compute one rate from the graph and compare with the scenario.
- Confirm axis labels, units, and scale are clear.
Worked example
A cyclist starts 0 meters from home. She rides at a constant 200 meters per minute for 5 minutes, stops for 3 minutes, then rides back home at a constant 250 meters per minute. Draw a distance-time graph and find when she returns home.
Worked answer
Key points: (0 min, 0 m), (5 min, 1000 m), (8 min, 1000 m), and return to (12 min, 0 m). The return takes 1000 divided by 250 = 4 minutes, so she returns at 8 + 4 = 12 minutes. The graph has a rising segment from 0 to 5 minutes, a horizontal segment from 5 to 8 minutes, and a falling segment from 8 to 12 minutes ending at the horizontal axis.
Four practice exercises
Exercise 1
A walker starts 0 meters from a bench. He walks away at 60 meters per minute for 4 minutes, rests for 2 minutes, then walks back to the bench at 80 meters per minute. Plot the key points on a distance-time graph and state the time he returns to the bench.
Show answer and check
Key points: (0, 0), (4, 240), (6, 240), and return at (9, 0). He returns at 9 minutes.
Why: Distance after 4 minutes is 60 times 4 = 240 meters. The rest keeps distance at 240 meters from 4 to 6 minutes. The return takes 240 divided by 80 = 3 minutes, so 6 + 3 = 9 minutes.
Check: Compute the return time from the graph segment: vertical change 240 meters divided by horizontal change 3 minutes gives 80 meters per minute, matching the scenario.
Exercise 2
A market stall charges a fixed 5 dollars to set up and 2 dollars per item sold. Plot a cost-quantity graph for 0 to 6 items. Find the cost for 4 items and the slope.
Show answer and check
Key points: (0, 5), (1, 7), (2, 9), (3, 11), (4, 13), (5, 15), (6, 17). Cost for 4 items is 13 dollars. Slope is 2 dollars per item.
Why: Cost equals 5 plus 2 times quantity. At quantity 4, cost is 5 + 8 = 13. The vertical intercept is 5, and each additional item adds 2 dollars, so slope is 2.
Check: Pick two points such as (2, 9) and (4, 13): vertical change 4 divided by horizontal change 2 gives 2 dollars per item.
Exercise 3
A car travels at 90 kilometers per hour for 2 hours, stops for 1 hour, then travels at 60 kilometers per hour for 1 hour. Draw a distance-time graph using hours and kilometers. Find the total distance and the time of the final point.
Show answer and check
Key points: (0, 0), (2, 180), (3, 180), (4, 240). Total distance is 240 kilometers at 4 hours.
Why: First segment: 90 times 2 = 180 kilometers. Stop keeps distance at 180 kilometers from 2 to 3 hours. Final segment: 60 times 1 = 60 kilometers, so 180 + 60 = 240 kilometers at 4 hours.
Check: Check the final segment slope: vertical change 60 kilometers divided by horizontal change 1 hour gives 60 kilometers per hour, matching the scenario.
Exercise 4
A phone plan costs a fixed 10 dollars per month plus 0.50 dollars per gigabyte used. Plot a cost-quantity graph for 0 to 8 gigabytes. Find the cost for 6 gigabytes and explain the vertical intercept.
Show answer and check
Key points: (0, 10), (2, 11), (4, 12), (6, 13), (8, 14). Cost for 6 gigabytes is 13 dollars. The vertical intercept is 10 dollars, the fixed monthly fee.
Why: Cost equals 10 plus 0.50 times gigabytes. At 6 gigabytes, cost is 10 + 3 = 13. The intercept at quantity zero is the fixed fee before any data is used.
Check: Compute slope from (2, 11) and (6, 13): vertical change 2 divided by horizontal change 4 gives 0.50 dollars per gigabyte, matching the scenario.
Common mistakes
- Plotting time on the vertical axis without labeling, then reading speed upside down. Label both axes with variable and unit. If you swap axes, recompute slope as horizontal change divided by vertical change for speed.
- Using different units on one axis, such as mixing minutes and hours. Convert all values to one unit before plotting, and write the unit on the axis.
- Drawing a stop as a sloped line on a distance-time graph. A stop means distance does not change while time passes, so draw a horizontal segment.
- Ignoring a fixed starting fee on a cost-quantity graph. Plot the fixed fee as the vertical intercept at quantity zero, then add the per-item cost to find other points.
Self-check rubric
- Axes are drawn, labeled with variable and unit, and the scale is stated.
- Key points are plotted accurately and consistently with the chosen scale.
- Segments or curves match the described rates, stops, and changes.
- At least one rate is computed correctly with units and checked against the scenario.
- Intercepts are identified and explained in the context of the scenario.
- Valid alternative choices, such as swapped axes with clear labels or a different consistent scale, are accepted if interpretation remains correct.
Questions and answers
Do I always put time on the horizontal axis?
It is a common convention, but not a rule. You may swap axes if you label both clearly and adjust how you read slope. For speed, slope is distance change divided by time change; if axes are swapped, divide the other way.
How do I show a change in speed?
Use separate straight segments with different slopes. A steeper segment means a faster speed. Connect the segments at the time the speed changes.
What does a horizontal segment mean on a cost-quantity graph?
It means cost does not change as quantity increases, which is unusual for a purchase scenario. Check the scenario; it may represent a capped price or a free additional item.
Can I use a curve instead of straight segments?
Yes, if the scenario describes continuous change, such as slowing down gradually. For constant rates, straight segments are clearer and easier to check.