D. Characteristics of Functions · Rates of Change

Sketching Graphs from Descriptions of Rates of Change

D1.2 D1.3About 65 minutes

Learning goals

I can describe a graph as increasing or decreasing at an increasing, decreasing or constant rate.
I can sketch a graph from a description in words of how a quantity changes.
I can sketch height-volume graphs for containers of different shapes.
I can describe a graph in words, including where the rate of change is zero, constant or changing.

Words to know

increasing
A graph rises from left to right: y gets bigger as x gets bigger. The rate of change is positive.

decreasing
A graph falls from left to right. The rate of change is negative.

increasing at an increasing rate
Rising and getting steeper: each step adds more than the last. The graph bends upward (concave up).

increasing at a decreasing rate
Rising but getting flatter: each step adds less than the last. The graph bends downward (concave down).

concave up / concave down
A curve that bends upward like a cup (concave up) or downward like a cap (concave down).

motion sensor
A device that measures the distance to a moving object many times per second and graphs it, usually with ultrasound.

A story you can see

A snowstorm starts as a few flakes, then turns into a whiteout, then stops. The next afternoon the sun comes out and the snow slowly settles and melts. You could tell that story in words. You can also draw it: a graph of snow depth against time.

In this lesson you will turn words into graphs and graphs into words. The key is to think about the rate of change at each moment. Is the quantity going up or down? Fast or slow? Is it speeding up or slowing down?

Reading the steepness

On a graph of y against x, the steepness shows the size of the rate of change.

Steep means a large rate (fast change).
Gentle means a small rate (slow change).
Flat (horizontal) means a rate of zero.
Rising means a positive rate; falling means a negative rate.

If the rate of change is itself changing, the graph bends. There are four basic bends to know.

The four basic shapes. Read them left to right. "Increasing at an increasing rate" rises and gets steeper; "decreasing at a decreasing rate" falls but levels off.

Careful with the words

"Decreasing at a decreasing rate" means the quantity goes down and the size of the drop gets smaller: the graph falls and flattens out. A hot drink cooling on the counter does this: it cools quickly at first, then more and more slowly. Some textbooks say "the rate of decrease is decreasing" for the same idea.

Worked example

Example 1: Sketch the snowstorm

Sketch a graph of snow depth against time for this story. At 8 p.m. there is no snow. Light snow falls from 8 p.m. to 10 p.m. Heavy snow falls from 10 p.m. to 2 a.m. The snow stops at 2 a.m. The depth stays the same until 10 a.m., then the snow settles and melts more and more slowly through the afternoon.

Axes: time across (8 p.m. to the next evening), depth up.
8 p.m. to 10 p.m.: depth increasing slowly, so draw a gentle rising segment starting at 0.
10 p.m. to 2 a.m.: depth increasing quickly, so draw a steep rising segment.
2 a.m. to 10 a.m.: rate of change zero, so draw a horizontal segment.
After 10 a.m.: depth decreasing at a decreasing rate, so the graph falls and levels off (bending like the bottom-right shape above).

A gentle rise, a steep rise, a flat stretch, then a falling curve that flattens out. The exact numbers do not matter in a sketch; the shape must match the story.

Filling containers

Pour water into a container at a constant rate (say 100 mL every second). How does the height of the water change?

It depends on how wide the container is at the water level.

Where the container is wide, each 100 mL spreads out, so the height rises slowly.
Where it is narrow, the same 100 mL stacks up, so the height rises quickly.

So a cylinder gives a straight line (constant rate). A cone with its point at the bottom starts narrow, so its graph starts steep and flattens out. A flask that is wide at the bottom does the opposite.

Water poured at a constant rate. Cylinder: height rises at a constant rate. Cone, point down: height increases at a decreasing rate. Flask: height increases at an increasing rate, then rises fast at a constant rate in the narrow straight neck. (Sketches, not to scale.)

Worked example

Example 2: An hourglass-shaped vase

A vase is wide at the bottom, narrow in the middle and wide again at the top (like an hourglass). Water is poured in at a constant rate. Describe the graph of height against time.

Bottom half: the vase gets narrower as you go up, so the height rises faster and faster: increasing at an increasing rate.
At the narrow waist the graph is steepest.
Top half: the vase gets wider, so the height rises more and more slowly: increasing at a decreasing rate.
The graph never falls or goes flat while water is being added.

An S-shaped curve: it bends upward until the waist, then bends downward. The point where the bend switches (the waist) is where the height rises fastest.

Motion graphs

In a science lab, a motion sensor measures how far away you are many times per second and graphs your distance d against time t. The rate of change of distance is your velocity.

Remember from Grade 11: if you took Physics (SPH3U), you learned in Kinematics that the slope of a position-time graph is the velocity. Here we use the same idea for any rate of change.

Walking away from the sensor makes the graph rise. Walking toward it makes it fall. Standing still makes it flat. Walking faster makes it steeper. Speeding up makes it bend.

A walk in front of a motion sensor: start 0.5 m away, walk slowly away for 4 s (0.5 m/s), stand still for 3 s, then walk quickly back for 2 s (−1 m/s).

Worked example

Example 3: Graph to words

A distance-time graph from a motion sensor starts at d = 3 m. It falls in a straight line to d = 1 m at t = 4 s, stays flat until t = 6 s, then rises along a curve that gets steeper and steeper until t = 9 s. Describe the walk.

0 to 4 s: falling straight line, so the walker moves toward the sensor at a constant speed: (1 − 3) ÷ 4 = −0.5 m/s, so 0.5 m/s toward it.
4 to 6 s: flat, so the walker stands still 1 m from the sensor.
6 to 9 s: rising and getting steeper, so the walker moves away and speeds up.

Walk toward the sensor at a steady 0.5 m/s from 3 m to 1 m, stop for 2 s, then walk away while speeding up.

Just for fun

Most school motion sensors work like a bat: they send out ultrasound clicks and time the echoes. So when you walk in front of one, you are being graphed by a very small robot bat.

Hands-on activity

Pour and measure

You need: a clear glass, vase or plastic bottle that is not a straight cylinder (a wine-glass-shaped plastic cup, a flower vase or a sports bottle with a narrow middle all work), a measuring cup, a ruler, water, paper and a pencil. Work over a sink or a towel.

Look at your container. Before pouring, sketch what you think the graph of water height against volume poured will look like.
Pour in 50 mL (or 100 mL for a big container). Measure the height of the water from the inside bottom with the ruler. Record it.
Repeat until the container is nearly full.
Plot height against volume. Join the points with a smooth curve.
Compare with your prediction. Where is the graph steepest? Match that to the narrowest part of the container.

Bonus (motion): in a hallway, mark 1 m, 2 m, 3 m and 4 m on the floor with tape. Have a partner call out the time every 2 s while you walk a story: slowly away, stop, quickly back. Record your position at each call and graph it.

What to look for: the steep parts of the graph match the narrow parts of the container, and the gentle parts match the wide parts. A straight-sided cylinder would give a straight line.

Your turn

Describe the shape of a graph that is "increasing at a decreasing rate".

A cup of hot chocolate cools on the counter. Its temperature drops quickly at first, then more slowly. Describe the graph of temperature against time.

Sketch (describe) the graph of distance from home against time: you walk to a friend's house at a steady pace, stay 30 min, then jog home.

Water is poured at a constant rate into a cylinder. Describe the graph of height against time.

A bottle is narrow at the bottom and wide at the top. Water is poured in at a constant rate. Describe the height-time graph.

A distance-time graph is flat from t = 2 s to t = 5 s. What is happening, and what is the velocity?

A population grows by a larger number of people each year. Which of the four basic shapes does its graph have?

More practice: warm-up

On a graph, what does "steeper" tell you about the rate of change?

What does a horizontal segment on a graph mean?

A graph falls from left to right. Is the rate of change positive or negative?

A straight-line graph has what kind of rate of change?

In a motion graph of distance from a sensor, what does a rising graph mean?

More practice: on your own

A car accelerates from a stop sign, drives at a steady speed, then brakes smoothly to stop at a red light. Describe the distance-time graph.

A bathtub is draining. The water level drops quickly at first and more slowly as the tub empties. Then it stops at 0. Describe the graph of water depth against time.

Describe the walk for this motion graph: d starts at 1 m, rises along a straight line to 4 m at t = 3 s, then falls along a straight line to 1 m at t = 9 s.

A spherical fishbowl is filled at a constant rate. Describe the height-time graph.

The number of people who have heard a rumour grows slowly, then quickly, then slowly again as almost everyone has heard it. Describe the graph.

Challenge

A container's graph of height against volume is made of two straight segments: the first is steeper than the second. Describe the container.

Two runners race 100 m. Ana runs at a constant speed and finishes in 14 s. Ben starts slowly and speeds up the whole race, and they tie. Describe both distance-time graphs on the same axes. Who is ahead at t = 7 s?

Explain why the graph of the height of a ball tossed straight up (until it is caught at the same height) is increasing at a decreasing rate and then decreasing at an increasing rate. Where is the rate of change zero?

Check your understanding

Quiz

Which description matches a graph that rises and gets flatter?

Water is poured at a constant rate into a cone with its point at the bottom. How does the height change?

On a distance-time graph from a motion sensor, a straight segment falls from 5 m to 2 m over 3 s. What is the velocity?

A hot drink cools quickly at first and then more slowly. Its temperature-time graph is:

Where is a ball thrown straight up changing height at a rate of zero?

Which container gives a straight-line graph of height against volume poured?

For parents and teachers

Big ideas (D1.2, D1.3): steepness shows how fast something changes, and the bend of a graph shows whether the rate is speeding up or slowing down. Students should be able to go from words to a sketch and from a sketch back to words.

How to help: Tell a short story ("I drove to the store, waited at a long red light, then drove home faster") and ask your teen to sketch the distance-from-home graph. Then swap roles: they sketch a graph and you tell the story it shows.

Common difficulties:

Treating a motion graph like a picture of the path (a hill on the graph does not mean walking up a hill).
Mixing up "decreasing at a decreasing rate" and "decreasing at an increasing rate". Ask: is the graph getting steeper or flatter?
Drawing the container graph backwards: wide parts give slow height changes, not fast ones.
Joining segments with gaps or vertical jumps. A distance-time graph can never be vertical (that would mean being in two places at once).