Grade by Grade
Quadratic Relations of the Form y = ax² + bx + c · Quadratic Relations and Their Graphs

Collecting Data That Curves: Graphs and Curves of Best Fit

1.1About 70 minutes
Learning goals
  • I can collect data that can be represented by a quadratic relation, from an experiment or a secondary source.
  • I can graph two-variable data with suitable axes and scales.
  • I can draw a smooth curve of best fit, by hand and with technology.
  • I can use a curve of best fit to interpolate and extrapolate, and explain the limits of each.

Words to know

quadratic relation
A relation whose equation can be written y = ax² + bx + c with a ≠ 0. Its graph is a parabola.
parabola
The symmetric U-shaped (or upside-down U-shaped) curve that is the graph of a quadratic relation.
primary data
Data you collect yourself, for example in an experiment.
secondary data
Data collected by someone else, such as Statistics Canada or a textbook.
curve of best fit
A smooth curve that best follows the pattern of the points on a scatter plot.
interpolation
Estimating a value between known data points.
extrapolation
Estimating a value beyond the known data.
quadratic regression
Using technology to find the quadratic equation that best fits a set of data.

Not every pattern is a straight line

Roll a ball down a ramp and watch it. It does not move the same distance every second. It starts slowly and then speeds up. If you record how far it has rolled at equal time steps and graph the data, the points do not lie on a line. They bend upward in a smooth curve.

In this unit you will study the most important curve in high school math: the parabola. Relations whose graphs are parabolas are called quadratic relations. They describe falling and thrown objects, the area of shapes, the shape of satellite dishes and arch bridges, and the profit of a business. Today you will collect data that curves, graph it and draw a curve of best fit.

Remember from Grade 9

In Grade 9 math you sorted relations into linear (constant rate of change, constant first differences, a straight-line graph) and non-linear (curves). You saw that y = x² makes a U-shaped curve called a parabola, and in the data unit you drew lines of best fit and tested linear, quadratic and exponential regression models with technology. Grade 10 takes the quadratic relation and studies it in depth.

Where does quadratic data come from?

You can collect data in two ways:

  • From an experiment (primary data): you measure it yourself with tools such as a metre stick, a stopwatch or phone timer, a motion sensor (a scientific probe) or a graphing calculator.
  • From a secondary source: data someone else collected, from a textbook, a website or an agency such as Statistics Canada.

Some situations that often give quadratic data:

  • the distance a ball rolls down a ramp, measured at equal time steps
  • the height of a ball tossed straight up
  • the height of water as it drains out of a hole in the bottom of a bottle
  • the area of a rectangle when the perimeter stays the same
  • the stopping distance of a car at different speeds
booksstartt = 1 s24 cmt = 2 s97 cmt = 3 s218 cmd = distance rolled from the start (not to scale)
A ball released from rest rolls farther in each second than in the second before: 24 cm after 1 s, 97 cm after 2 s and 218 cm after 3 s. Equal time steps do not give equal distance steps.

Graphing the data

To graph two-variable data:

  1. Decide which variable is independent (you choose it or it goes up steadily, like time) and put it on the horizontal axis. The dependent variable goes on the vertical axis.
  2. Choose a scale for each axis that fits all the data and uses most of the grid.
  3. Label each axis with the variable and its unit, and give the graph a title.
  4. Plot each point carefully.

The curve of best fit

If the points follow a curved pattern, do not force a straight line through them. Draw a curve of best fit: one smooth curve that follows the pattern of the points.

  • It does not have to pass through every point. Measured data always has small errors.
  • About as many points should lie above the curve as below it.
  • Do not join the dots with straight segments. Nature does not move in zig-zags.
  • Only draw it where it makes sense. (A ramp has an end; time before the start makes no sense.)
t (s)d (cm)00.511.522.5304080120160200240curve of best fitd ≈ 24.1t² + 0.36t − 0.14
The ramp data from the class experiment. The points bend upward, so a curve fits them much better than a line would. The equation shown is the quadratic regression a graphing tool gives for these 7 points.
Worked example

Example 1: Is the ramp data quadratic?

A class released a ball from rest at the top of a long ramp and recorded the distance d it had rolled every half second.

t (s): 0, 0.5, 1.0, 1.5, 2.0, 2.5, 3.0

d (cm): 0, 6, 24, 55, 97, 152, 218

Graph the data, describe the pattern and decide whether a line or a curve is the better model.

  1. Time is the independent variable, so t goes on the horizontal axis and d on the vertical axis.
  2. Plot the 7 points (see the graph above). They curve upward: the graph gets steeper as time goes on.
  3. Check the first differences (the change in d for each half second): 6 − 0 = 6, 24 − 6 = 18, 55 − 24 = 31, 97 − 55 = 42, 152 − 97 = 55, 218 − 152 = 66. They are far from constant, so the relation is not linear.
  4. The first differences grow by about the same amount each time: 18 − 6 = 12, 31 − 18 = 13, 42 − 31 = 11, 55 − 42 = 13, 66 − 55 = 11. These second differences are all close to 12. In the next lesson you will see that roughly constant second differences are the sign of a quadratic relation.

The data is non-linear. A smooth upward-bending curve (part of a parabola) is the best model, so draw a curve of best fit, not a line.

Worked example

Example 2: Using the curve to estimate

Use the ramp data and its curve of best fit to estimate (a) the distance after 1.75 s and (b) the time when the ball has rolled 2.4 m.

  1. (a) 1.75 s is between measured times, so this is interpolation. On the graph, go up from t = 1.75 to the curve, then across: about 74 cm. (Check with the regression equation: 24.1(1.75)² + 0.36(1.75) − 0.14 ≈ 74.)
  2. (b) 2.4 m = 240 cm is beyond the last data point, so this is extrapolation. Extend the curve gently past t = 3: it reaches 240 cm a little after 3 s, at about 3.15 s.
  3. Extrapolation is less reliable than interpolation. Here it only works if the ramp is longer than 2.4 m and the ball keeps speeding up in the same way.

(a) about 74 cm; (b) about 3.1 s to 3.2 s, if the ramp is long enough.

Worked example

Example 3: Letting technology fit the curve

Use a graphing tool such as Desmos (free online) or a graphing calculator to find a quadratic curve of best fit for the ramp data.

  1. In Desmos, add a table and type the t-values in the x₁ column and the d-values in the y₁ column. The points appear on the grid.
  2. On a new line type y₁ ~ ax₁² + bx₁ + c (the ~ symbol means "fit this model"). Desmos finds the values of a, b and c that fit best.
  3. For this data it gives a ≈ 24.1, b ≈ 0.36 and c ≈ −0.14, so the model is d ≈ 24.1t² + 0.36t − 0.14. It also reports R² very close to 1, which means a very good fit.
  4. Since b and c are tiny, the model is close to d = 24t². Try it: at t = 2, 24(2)² = 96, and the measured value was 97.

A quadratic regression model is d ≈ 24.1t² + 0.36t − 0.14, very close to d = 24t².

Which kind of curve?

When the points curve, look at the shape. If the curve rises and then falls (or falls and then rises) like a U or an upside-down U, or if it looks like one side of a U, a quadratic model is a good first choice. Later in this unit you will compare it with an exponential curve such as y = 2ˣ, which never turns around.

th0408012005101520Not like this: dot to dotth0408012005101520Like this: one smooth curvet = time (s), h = height of water (cm)
Water draining from a hole near the bottom of a bottle. The height drops quickly at first and then more slowly. On the left the dots are joined with straight pieces (wrong). On the right one smooth curve passes close to all the points (right).
Just for fun

Why did the ball refuse to roll down the ramp at a steady speed?

It had a lot of acceleration in its life and didn't want to fall into a linear routine.

Hands-on activity

The draining bottle experiment

You need: an empty plastic water bottle (500 mL to 2 L) with straight sides, a nail or thick skewer, a ruler and tape, a phone timer, a sink or a basin, paper and pencil (graph paper if you have it).

Grown-up step: an adult makes one small hole in the side of the bottle about 2 cm above the bottom with the nail or skewer. Sharp points can slip, so the adult does this part and pushes away from their hand.

  1. Tape the ruler to the outside of the bottle so that 0 cm is level with the hole.
  2. Cover the hole with your finger, fill the bottle with water and stand it in the sink.
  3. Start the timer as you remove your finger. Every 10 seconds, read the height of the water above the hole. A helper reading the timer out loud makes this easier.
  4. Stop when the water stops flowing. Put the results in a table: time (s) and height (cm).
  5. Graph height against time. Draw a smooth curve of best fit.
  6. Work out the first differences of the heights. Are they constant? Are they getting smaller?

What to look for: the height drops quickly at first and then more and more slowly, so the graph is a curve that bends and flattens out near the bottom, like the right-hand graph above. The first differences are negative and shrink toward 0; the second differences are roughly the same positive number. Your numbers will differ from the example because bottles and holes differ.

Your turn

1

A student records the height of a plant every week: 2 cm, 4 cm, 6 cm, 8 cm, 10 cm. Would you model this data with a line or with a curve? Why?

2

Name the independent and dependent variables in the draining bottle experiment, and say which axis each goes on.

3

Give two rules for drawing a good curve of best fit.

4

Using the ramp data (d = 0, 6, 24, 55, 97, 152, 218 cm at t = 0, 0.5, …, 3.0 s), estimate the distance at t = 2.25 s. Is this interpolation or extrapolation?

5

Water drains from a bottle: h = 20, 15.6, 11.7, 8.3, 5.6, 3.4, 1.7 cm at t = 0, 20, 40, …, 120 s. Describe the shape of the graph in words.

6

Name one primary and one secondary source of data you could use to study how a car's stopping distance depends on its speed.

More practice: warm-up

1

What is the name of the U-shaped graph of a quadratic relation?

2

True or false: a curve of best fit must pass through every data point.

3

What do we call estimating a value between known data points?

4

What do we call estimating a value beyond the known data?

5

Which table is linear? (A) y = 3, 6, 9, 12 (B) y = 1, 4, 9, 16 (x = 1, 2, 3, 4 in both)

More practice: on your own

1

A rectangle has a perimeter of 20 cm. Its width w can be 1, 2, 3, 4, 5, 6, 7, 8 or 9 cm. Find the area for each width. Describe the shape of the graph of area against width.

2

For the rectangle data in the last question, find the first and second differences. What do you notice?

3

A curve of best fit for a set of data has 7 points above it and 1 point below it. What is wrong, and how would you fix it?

4

A ball is dropped from a bridge. A motion sensor records the distance it has fallen: 0 m, 1.2 m, 4.9 m, 11.0 m, 19.6 m at t = 0, 0.5, 1.0, 1.5, 2.0 s. Find the first differences and explain why the graph is a curve.

5

Why is extrapolating the ramp model to t = 10 s probably a bad idea?

Challenge

1

The curriculum's sample experiment: a can is placed 30 cm, 40 cm, 50 cm and 60 cm up a ramp, and the time to roll to the bottom is recorded. Here, distance is the independent variable. Sketch roughly how time depends on distance. Is the graph of distance against time (time on the horizontal axis) a parabola?

2

Using d = 24t² for the ramp, how far does the ball roll during the 4th second (from t = 3 s to t = 4 s)? Compare it with the distance during the 1st second.

3

A data set has points (1, 5), (2, 8), (3, 9), (4, 8), (5, 5). Without technology, describe a curve of best fit and predict where it is highest.

Check your understanding

Quiz

1.

Which situation is most likely to give quadratic data?

2.

A good curve of best fit…

3.

Using a model to predict a value beyond the measured data is called…

4.

For the ramp data, the first differences were 6, 18, 31, 42, 55, 66. What does this tell you?

5.

In Desmos, which line asks for a quadratic curve of best fit for a table with columns x₁ and y₁?

6.

Which is a secondary source of data?

For parents and teachers

Big idea: Expectation 1.1 asks students to collect data that can be represented by a quadratic relation (from experiments or secondary sources), graph it and draw a curve of best fit, with or without technology. The key skills are choosing axes and scales, drawing a smooth curve (not dot-to-dot) and telling interpolation from extrapolation.

How to help: Do the draining-bottle experiment together: one person reads the timer, the other reads the height. Make the hole yourself and keep the nail or skewer away from younger children. If you have a computer, try the free Desmos graphing calculator together: type the data into a table and fit y₁ ~ ax₁² + bx₁ + c.

Common difficulties:

  • Joining the dots. Remind your teen that measured data has small errors; the curve shows the trend.
  • Swapping the axes. Ask: "Which variable did you control or does it go up steadily?" That one goes across.
  • Trusting extrapolation too much. Ask: "Would the situation still be the same that far out?"
  • Squashed graphs. Encourage scales that use most of the grid.