A. Quadratic Functions · Quadratic Functions

Collecting and Graphing Data That Can Be Modelled by a Quadratic

A3.1About 75 minutes

Learning goals

I can plan an investigation that produces quadratic data.
I can collect primary data with tools such as video, and find secondary data.
I can graph data in a scatter plot and draw a curve of best fit.
I can use differences and the shape of the graph to decide whether a quadratic model is reasonable.

Words to know

primary data
Data you collect yourself by measuring, counting or recording.

secondary data
Data collected by someone else, such as a published table.

independent variable
The variable you choose or control (horizontal axis).

dependent variable
The variable you measure (vertical axis).

scatter plot
A graph of data points (x, y) that are not joined.

curve of best fit
A smooth curve that follows the pattern of the data as closely as possible.

second differences
Differences of the first differences; roughly constant for quadratic data with equal x-steps.

Where do quadratic models come from?

So far the functions in this unit were handed to you. In real life, someone first has to collect data, notice that it curves like a parabola, and only then write a model. In this lesson you will plan an investigation, collect data that can be modelled by a quadratic function, and graph it. In the next lesson, Finding a Quadratic Model for Data, you will find an equation for it.

Remember from Grades 9 and 10

In Grade 9 you made scatter plots, drew lines of best fit and tested linear, quadratic and exponential regression models. In Grade 10 you collected data that curves (a ball on a ramp, a draining bottle), drew curves of best fit, and used second differences to recognize quadratic relations: for equally spaced x-values, constant second differences mean quadratic.

Primary and secondary sources

Primary data is data you collect yourself, by measuring, timing, counting or recording a video.
Secondary data is data someone else collected: a table in a textbook or scientific article, a government website such as Statistics Canada, or sports statistics.

Both are useful. With primary data you control how it is measured. With secondary data you can study situations you can't set up yourself, but you must check the source and its units.

Situations that often give quadratic data

the distance travelled by an object that speeds up steadily (a ball rolling down a ramp, a dropped object)
the height of a thrown or kicked object over time
the height of water in a bottle draining through a small hole, over time (approximately)
area as a length changes (the area of a square or circle as its side or radius grows)
counting pairs (handshakes, games in a round robin, diagonals of a polygon)
revenue when a price change also changes the number sold

Tools for collecting

Rulers and metre sticks, stopwatches and phone timers, slow-motion or frame-by-frame phone video, a motion sensor connected to a graphing calculator, and spreadsheets or Desmos for recording and graphing.

A ball released from rest on a ramp. Marking its position every half second (from a phone video) shows the spaces between marks getting longer, a sign that distance is not a linear function of time.

A plan for a good investigation

State the question. "How does the distance a ball rolls depend on time?"
Name the variables. Independent (what you choose or control: time) and dependent (what you measure: distance). Keep everything else the same (same ball, same ramp angle).
Choose tools and equal steps for the independent variable (every 0.5 s). Equal steps make differences easy to use.
Collect at least 6 data points. Repeat trials and average if you can.
Record in a table with units.
Graph a scatter plot, independent variable on the horizontal axis. Do not join the dots; draw a smooth curve of best fit if the pattern curves.
Describe the pattern and note sources of error.

Worked example

Example 1: Ramp data

A student recorded a ball rolling down a ramp, using a phone video to read its position every 0.5 s. Graph the data and decide whether a quadratic model is reasonable.

t (s): 0, 0.5, 1.0, 1.5, 2.0, 2.5

d (cm): 0, 4, 15, 34, 61, 94

Plot t on the horizontal axis and d on the vertical axis. The points curve upward; a line would not fit.
First differences (t steps of 0.5): 4, 11, 19, 27, 33. They are not constant, so the relation is not linear.
Second differences: 7, 8, 8, 6. They are close to constant. Real measurements are never perfect, so "close to constant" is what we look for.
The pattern looks like half a parabola opening up, starting at the vertex (0, 0) (the ball starts from rest). A quadratic model is reasonable.
Sources of error: reading positions from a blurry video frame, the ball not starting at exactly t = 0, and friction.

The data curves upward with nearly constant second differences, so a quadratic model fits.

The ramp data as a scatter plot. A smooth curve of best fit passes very close to every point; it is not a dot-to-dot line.

Worked example

Example 2: Is it quadratic?

Which data sets are likely quadratic? (x = 0, 1, 2, 3, 4 in each)

(a) y = 5, 8, 11, 14, 17

(b) y = 2, 6, 18, 54, 162

(a) First differences 3, 3, 3, 3: constant. Linear, not quadratic.
(b) First differences 4, 12, 36, 108; second differences 8, 24, 72: not constant. Each y is 3 times the last, a constant ratio. This is exponential, a pattern you will study in the unit Exponential Functions.
(c) First differences −5, −3, −1, 1; second differences 2, 2, 2: constant. Quadratic, with a minimum near x = 3.

Only (c) is quadratic. (a) is linear and (b) is exponential.

Worked example

Example 3: Primary data you can draw

Draw polygons with n = 3, 4, 5, 6 and 7 sides and count their diagonals. Is the number of diagonals a quadratic function of n?

Counting (or using the fact that each vertex joins to n − 3 others, and each diagonal is counted twice): triangle 0, quadrilateral 2, pentagon 5, hexagon 9, heptagon 14.
First differences: 2, 3, 4, 5. Second differences: 1, 1, 1. Constant, so the relation is quadratic.
The model is D(n) = n(n − 3) ÷ 2. Check n = 6: 6 × 3 ÷ 2 = 9 ✓.
The domain is discrete: n = 3, 4, 5, …, so the graph is separate points.

Yes: D(n) = n(n − 3) ÷ 2, with second differences of 1.

Just for fun

Legend says Galileo dropped balls from the Leaning Tower of Pisa, but historians are not sure he ever did. What he certainly did was roll balls down long ramps, and he found that the distance grows with the square of the time: quadratic data, about 400 years ago.

Hands-on activity

Roll, record, graph

You need: a long, smooth board, shelf or stiff piece of cardboard (about 1.5 to 2 m if possible), a stack of books, a ball (a tennis ball, golf ball or marble), masking tape, a measuring tape or metre stick, a phone that records video, and graph paper. Keep marbles away from small children.

Rest one end of the board on two or three books to make a gentle ramp. Put a tape mark near the top as the start line.
Lay the measuring tape along the ramp, or put a tape mark every 10 cm.
Start the video, then release the ball from rest at the start line (do not push it).
Play the video back slowly. Every 0.25 s or 0.5 s (use the video's time display), record how far the ball has rolled.
Record t and d in a table with units. Repeat the trial twice and average the distances for each time.
Make a scatter plot and draw a smooth curve of best fit. Compute first and second differences.
Write 3 sentences: Is the data quadratic? What tells you? What were your sources of error?

Grown-up option: an adult can poke a small hole near the bottom of a plastic bottle (with a nail or skewer) for the draining-bottle experiment: measure the water height every 10 s as it drains into a sink.

What to look for: a table with units and equal time steps; a scatter plot curving upward from (0, 0); second differences that are roughly equal; and honest notes on error (blurry frames, the start time, the ball wobbling).

Your turn

Give two examples of primary data and two examples of secondary data for a quadratic investigation.

Is the data quadratic? x = 0, 1, 2, 3, 4 and y = 1, 3, 9, 19, 33.

Is the data quadratic? x = 0, 1, 2, 3, 4 and y = 2, 6, 18, 54, 162.

Why should the x-values be equally spaced when you use differences?

A tossed ball's height from a video: t = 0, 0.2, 0.4, 0.6, 0.8, 1.0 s and h = 1.0, 2.2, 3.0, 3.4, 3.4, 3.0 m. Describe the pattern. Is it quadratic?

Name two sources of error when timing a rolling ball with a hand stopwatch.

More practice: warm-up

What is the difference between primary and secondary data?

What type of graph shows the relationship between two measured variables?

In the ramp experiment, which variable is independent?

Find the second differences of 2, 5, 10, 17 (x steps of 1).

Should a curve of best fit join the dots?

More practice: on your own

Use the curve d ≈ 15t² for the ramp data to estimate the distance at t = 1.75 s.

Water draining from a bottle: t = 0, 20, 40, 60, 80, 100 s and h = 20, 15.2, 11.0, 7.4, 4.6, 2.4 cm. Is a quadratic model reasonable?

Extend the diagonal data: how many diagonals does an octagon (n = 8) have? Check with D(n) = n(n − 3) ÷ 2.

Is the data quadratic? x = 1, 2, 3, 4, 5 and y = 3, 7, 13, 21, 31.

Real data rarely has exactly constant second differences. Why, and what should you look for instead?

Challenge

A student measured a tossed ball at t = 0, 1, 3 and 4 s. Why can't she use second differences directly? What could she do instead?

Show that D(n) = n(n − 3) ÷ 2 gives second differences of exactly 1.

Design your own investigation that should give quadratic data. Name the question, the variables, the tools, and one source of error.

Check your understanding

Quiz

Which is primary data?

Why should the independent variable go up in equal steps?

What suggests that measured data is quadratic?

A scatter plot rises, levels off and falls symmetrically. The best model is…

Which tool is best for recording a fast-moving ball's position many times per second?

The data 3, 6, 12, 24, 48 (x steps of 1) is best described as…

For parents and teachers

Big idea: expectation A3.1 asks students to collect data that can be modelled as a quadratic function, through investigation with and without technology, from primary sources using a variety of tools, or from secondary sources, and to graph the data.

How to help: the ramp activity takes about 30 minutes and works well as a family project. A phone video played back frame by frame is the key tool. Help your teen keep conditions the same between trials.

Common difficulties:

Joining the points dot-to-dot instead of drawing a smooth curve of best fit.
Expecting real data to have exactly constant second differences.
Using differences when the x-values are not equally spaced.
Forgetting units in the table and on the axes.