Lesson 5: Linear Relations. Distance-Time Graphs
Lesson 5: Linear Relations. Distance-Time Graphs
Read and describe distance-time graphs and the stories they tell.
In this lesson you will learn about linear relations and what they mean. One of the most common examples of a linear relation is the relationship between distance and time when a distance changes over time at a constant rate.
A linear relation is represented by a straight-line – a relationship between two variables that change at a constant rate. Linear relations can be expressed in 4 different ways: a graph, a table of values, a word description and an equation.
In this lesson we will compare two objects moving and discuss the relationship between distance, time and speed while comparing the graphs representing the motion patterns of the two objects.
You will learn why the equation D = vt (or D = st) is a direct proportion and how knowing the time and speed one can determine the distance traveled.
Linear Relations Game
Key ideas: Linear Relations. Distance-Time Graphs
A relation connects two variables, such as time and distance. A relation is linear when one variable changes at a constant rate as the other changes. Its graph is a straight line, and in a table of values with equal steps the first differences are constant.
A linear relation can be shown four ways: a table of values, a graph, an equation, and words.
On a distance-time graph:
- The slope (steepness) is the speed: a steeper line means faster.
- A horizontal segment means the object is stopped (distance is not changing).
- A segment going down means the object is moving back toward the starting point.
- At constant speed v, distance = speed × time: d = vt. This is a direct variation: the graph passes through (0, 0).
- If the object starts some distance d0 from the reference point, d = vt + d0. This is a partial variation: the graph starts on the vertical axis above 0.
Worked examples
Example 1
A cyclist’s distance from home is shown in the table. Is the relation linear? Write an equation and find the distance after 2.5 h.
Time (h): 0, 1, 2, 3 Distance (km): 0, 15, 30, 45
- First differences: 15, 15, 15. They are constant, so the relation is linear.
- The rate is 15 km/h and the graph starts at (0, 0): d = 15t (direct variation).
- d = 15(2.5) = 37.5 km
Example 2
Maya walks 600 m from home to a store in 10 min, stays 5 min, then walks home in 15 min. Describe each part of her distance-time graph.
- 0 to 10 min: line rising to 600 m. Speed = 600 ÷ 10 = 60 m/min.
- 10 to 15 min: horizontal line at 600 m: she is stopped.
- 15 to 30 min: line falling back to 0 m. Speed = 600 ÷ 15 = 40 m/min. The line is less steep, so she walks home more slowly.
Example 3
A bus starts 20 km from Toronto and drives away at 80 km/h. Write an equation and find its distance after 1.5 h.
- It starts at 20 km, so this is a partial variation: d = 80t + 20.
- d = 80(1.5) + 20 = 140 km from Toronto.
Common mistakes
- Reading the graph as a picture of the route (a “hill” on the graph is not a hill on the road).
- Thinking a line going down means slowing down; it means returning toward the start.
- Deciding a relation is linear from only two points; check the first differences across the whole table.
- Mixing units (minutes and hours) when calculating speed.
Questions students ask
How can I tell from a table if a relation is linear?
If the x-values go up in equal steps and the first differences of y are all the same, the relation is linear.
What is the difference between direct and partial variation?
A direct variation starts at 0 (y = mx); a partial variation has a starting value (y = mx + b, with b ≠ 0).
Check your understanding
Try the quiz for this lesson. You’ll see right away if each answer is correct.