UNIT 2Linear functionsLesson 3 of 31 Linear and Nonlinear Functions2 Rate of Change and Initial Value3 Describing Graphs: Increasing and Decreasing
Describing Graphs: Increasing and Decreasing
A graph tells a story. Read it left to right, one piece at a time.
What you'll be able to do
- Say if each piece of a graph is increasing, decreasing or constant.
- Match a story to its graph, and a graph to its story.
- Tell a steady change from one that speeds up or slows down.
- Know what a graph of distance does and doesn't show.
The big idea
Read left to right. Time moves to the right. Look at one piece of the graph at a time.
Three kinds of piece. Going up is increasing. Going down is decreasing. Flat is constant.
An example. Ty walks to the store. He waits in line. Then he walks home.
His graph. His distance from home rises, stays flat, then falls back to 0.
Steep means fast. On a distance graph, a steeper piece means faster travel.
Straight or curved. A straight piece changes at a steady rate. A curve that gets steeper is speeding up.
Not a picture. A distance graph that goes up and down is not a hill. It shows how far away someone is.
Speed is different. On a speed graph, a flat piece means a steady speed. It doesn't mean stopped.
Words you need
- Increasing
- Going up from left to right.
- Decreasing
- Going down from left to right.
- Constant
- Flat. The value stays the same.
- Steep
- Rising or falling quickly.
- Distance from home
- How far away someone is. It falls when they come back.
- Total distance
- How far someone has gone in all. It never falls.
What a piece means
| The piece | On a distance-from-home graph | On a speed graph |
|---|---|---|
| Rising | Moving away | Speeding up |
| Flat | Stopped | Steady speed |
| Falling | Coming back | Slowing down |
| Steeper | Moving faster | Changing speed faster |
Worked examples
Example 1
A graph joins (0, 0), (2, 4), (5, 4) and (7, 0). Describe each piece.
- Read left to right.
- From 0 to 2 it goes up: increasing.
- From 2 to 5 it is flat: constant.
- From 5 to 7 it goes down: decreasing.
Answer: Increasing, then constant, then decreasing
Example 2
A bus leaves a stop and drives at a steady speed. It waits at a red light. Then it drives on. Which graph shows its distance?
- Steady driving is a straight rise.
- Waiting at the light is a flat piece.
- Driving on is another rise.
- So the graph rises, stays flat, then rises: Graph A.
Answer: Graph A
Example 3
A car speeds up from a stop. Is its distance graph straight or curved?
- Each second it covers more distance than the second before.
- The rises get bigger: 1, 3, 5.
- So the graph curves upward. It gets steeper and steeper.
Answer: Increasing faster and faster
Common mistakes
Reading a distance graph as a picture of the path. A rise means moving away, not walking up a hill.
Thinking flat on a speed graph means stopped. Flat on a speed graph means a steady speed.
Thinking a falling distance graph means slowing down. Falling means getting closer. Steep means fast.
Reading right to left. Time moves to the right. Start at the left.
Practice: 20 questions
16 of 20 are word problems. For a word problem, sketch it first and mark what you know.
Tier 1 Foundations Use the rule directly
- A graph joins (0, 2), (3, 5), (5, 5) and (8, 1). Is each piece increasing, decreasing or constant?
- Ty walks to the store. He waits in line. Then he walks home. Which graph shows his distance from home: A, B or C? Describe its pieces. Word problem
- A graph of Rae's distance from home goes up, then down. Sol says Rae walked up a hill and back down. Find his mistake. Word problem
- A graph of a car's speed joins (0, 0), (2, 50), (6, 50) and (8, 0). x is minutes and y is km/h. When does the car go at a steady speed? Word problem
- A kettle heats water until it boils. Then the water stays at the same temperature. Which graph shows its temperature: A, B or C? Word problem
Tier 2 Intermediate One more step to think through
- Kai's distance graph joins (0, 0), (10, 2) and (15, 4). x is minutes and y is km. In which piece does he go faster? Word problem
- A sled's speed is 0, 2, 6 and 12 m/s at 0, 1, 2 and 3 seconds. Is the speed increasing at a steady rate? Word problem
- Nora's graph shows her distance from school. One piece falls steeply. Ben says she is slowing down there. Find his mistake. Word problem
- A graph joins (−4, −5), (−1, −1), (2, −1) and (5, 3). Describe each piece.
- Mo rides to a park. He rests. Then he rides home faster than he came. Graph A joins (0, 0), (20, 5), (30, 5), (50, 0). Graph B joins (0, 0), (20, 5), (30, 5), (40, 0). Which graph matches? Word problem
Tier 3 Advanced Several steps - find the hidden structure
- A bath fills for 8 minutes. Leah gets in, and the water jumps up. She soaks. She gets out, and the water drops at once. Then the bath drains. Which graph shows the water's depth? Word problem
- Graph A joins (0, 0), (4, 3), (6, 3). Graph B joins (0, 0), (4, 3), (2, 5). Each shows distance against time. Which can't show a walk? Word problem
- Runner A goes in a straight line from (0, 0) to (60, 300). Runner B joins (0, 0), (20, 150), (40, 150), (60, 300). x is seconds and y is meters. Who is ahead at 20 seconds? When are they level? Word problem
- A car's speed rises from 0 to 60 km/h, stays at 60, then falls to 0. Which graph shows its distance from the start? Word problem
- A graph joins (0, 0), (2, 6), (5, 9), (6, 13) and (9, 13). Order the pieces from the fastest rise to the slowest.
Tier 4 Challenge Stretch problems, some above grade level
- Water pours at a steady rate into a bottle. The bottle is wide at the bottom and narrow at the top. Which graph shows the water's height? Word problem
- A distance-from-home graph is a straight line going up from minute 5 to minute 10. Which speed graph matches those minutes: A, B or C? Word problem
- A seat on a big wheel starts at the bottom. The wheel turns twice. How many times does the seat's height graph change from increasing to decreasing? Word problem
- Elena walks to the library at a steady pace. She reads for 20 minutes. Then she walks home at the same pace. Graph A rises, stays flat, then falls to 0. Graph B rises, stays flat, then rises again. Maya says Graph B can't show Elena's trip. Find her mistake. Word problem
- Interval notation names a stretch of x-values by its start and end. From x = 1 to x = 4 is written (1, 4). A graph joins (−3, 2), (1, −2), (4, 5) and (6, 5). Where is it decreasing, increasing and constant? Give x-intervals. Above grade level
Answers
Tier 1 Foundations
- Increasing, then constant, then decreasing. Read left to right: going up is increasing, flat is constant, going down is decreasing.
- Graph B: increasing, then constant, then decreasing. The story needs: increasing, then constant, then decreasing. Only graph B has those pieces in that order.
- Rae walked away from home, then back toward it.. The graph is not a picture of the path. Going up means Rae moved away from home. Going down means she came back. The graph shows distance from home, not the shape of the ground.
- From minute 2 to minute 6, at 50 km/h. It is moving, not stopped.. Read left to right: going up is increasing, flat is constant, going down is decreasing.
- Graph A. The story needs: increasing, then constant. Only graph A has those pieces in that order.
Tier 2 Intermediate
- The second piece: 2 km in 5 minutes, against 2 km in 10 minutes.. Speeds in km a minute: Piece 1 0.2, Piece 2 0.4.
- No. It rises by 2, then 4, then 6. It increases faster and faster.. The changes are 2, 4, 6.
- She is moving toward school, fast.. Falling means Nora is getting closer to school. Steep means she is moving fast, not slowing down. Falling means her distance from school gets smaller. Steep means it changes quickly.
- Increasing, then constant, then increasing. A piece below the x-axis can still rise.. Read left to right: going up is increasing, flat is constant, going down is decreasing.
- Graph B. The ride home takes 10 minutes, less than the 20 minutes there.. The story needs: increasing, then constant, then decreasing. Only graph B has those pieces in that order. Its later piece is the steeper one.
Tier 3 Advanced
- Graph A. The story needs: increasing, then increasing, then constant, then decreasing, then decreasing. Only graph A has those pieces in that order. Its later piece is the steeper one.
- Graph B. It goes from time 4 back to time 2.. Graph B goes from time 4 back to time 2. Time can't run backward, so B can't show a walk.
- B at 20 seconds (150 m, against 100 m). They are level at 30 seconds, while B rests, and at 60 seconds.. (30, 150) makes both equations true: check it in y = (300/60) · x and y = 150.
- Graph A. The distance always rises: slowly, then steadily, then more slowly.. The story needs: increasing, then increasing, then increasing. Only graph A has those pieces in that order.
- Piece 3 (4 a step), piece 1 (3), piece 2 (1), piece 4 (0). Rises per step: Piece 1 3, Piece 2 1, Piece 3 4, Piece 4 0.
Tier 4 Challenge
- Graph B. The wide part needs lots of water to rise a little. The narrow top fills fast. So the height rises slowly, then faster.
- Graph A. A straight rise covers the same distance each minute. That is a steady speed, so the speed graph is flat.
- 2 times, once at each top. Count the tops. One turn has one top, so two turns have two.
- Graph B can show her total distance walked.. Graph B shows total distance walked, which never goes down. Graph A shows her distance from home. The same trip has two graphs. It depends on what the y-axis measures.
- Decreasing on (−3, 1). Increasing on (1, 4). Constant on (4, 6).. Read left to right: going up is increasing, flat is constant, going down is decreasing.
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