Q2 Collect quadratic data and draw a curve of best fit

Ontario Grade 10 Mathematics

Quadratic Relations

Recognising Quadratic Patterns in Real Data and Sketching a Smooth Parabolic Curve

When you drop a ball, stretch a spring, or track the height of a launched object, the numbers you collect do not follow a straight line — they curve. In Grade 9 (MTH1W) you learned to recognise linear relationships and draw lines of best fit. In this lesson you will go one step further: you will learn how to collect data that follows a curved, U-shaped pattern called a quadratic relationship, check that the pattern really is quadratic, plot the points on a grid, and draw a smooth curve of best fit through them. No equations are needed yet — the focus is on the data and the shape of the curve.

What you will learn

Identify whether a table of values shows a quadratic (non-linear) pattern by checking first and second differences.
Plot collected data points accurately on a scatter plot.
Draw a smooth parabolic curve of best fit through or near the data points.
Describe what a curve of best fit tells you about the relationship between two quantities.

Bridge from Grade 9: Lines of Best Fit and Scatter Plots

In Grade 9 you collected data, plotted it as a scatter plot, and drew a line of best fit when the points looked roughly linear. A line of best fit is a straight line placed as close as possible to all the data points — it does not have to pass through every point, or even any point.

The key idea carried forward into Grade 10 is the same: real data is never perfectly neat, so we draw a model — a line or a curve — that captures the overall trend. The difference now is that the trend is curved, not straight, and the curve we use is a parabola.

A parabola is the U-shaped (or upside-down U-shaped) curve that you see when you graph a quadratic relation such as

y = x^2

or

y = -x^2 + 4

. Before drawing any curve, you need evidence that the data actually follows this shape. That evidence comes from a tool called finite differences.

A scatter plot shows individual data points on a grid without connecting them.
A curve of best fit is a smooth curve placed as close as possible to all the points.
A parabola is the curve produced by a quadratic relationship.
Real data is always slightly imperfect, so the curve may not touch every point.

Checking for a Quadratic Pattern: First and Second Differences

Before fitting a curve, you need to confirm the data is quadratic. You do this by calculating finite differences from your table of values. Always make sure the input values (the

x

-values) are equally spaced before you start — for example,

x = 0, 1, 2, 3, 4

(each step is

+1

).

First differences are found by subtracting each output value from the one directly below it in the table:

\Delta_1 = y_{\text{next}} - y_{\text{current}}

. If the first differences are all equal, the relationship is linear. If they are not equal, keep going.

Second differences are found by applying exactly the same subtraction process to the first differences:

\Delta_2 = \Delta_{1,\text{next}} - \Delta_{1,\text{current}}

. If the second differences are all equal (and not zero), the relationship is quadratic. This is the key test you will apply to every data set in this topic.

With real collected data, measurement errors mean the second differences will be approximately equal rather than perfectly equal. You are looking for a consistent pattern, not an exact match. If the second differences are roughly the same value, you have good evidence of a quadratic relationship.

\Delta_2 = \Delta_{1,\text{next}} - \Delta_{1,\text{current}}

The $x$ -values must be equally spaced before you calculate differences.
Equal first differences → linear relationship.
Approximately equal (constant) second differences → quadratic relationship.
Real data will show small variations in the differences due to measurement error.
Second differences being approximately equal is your evidence to draw a parabolic curve of best fit.

Setting Up and Collecting Quadratic Data

Quadratic relationships appear in many real situations: the distance a ball falls over time, the area of a square as its side length grows, the height of a thrown object. When collecting your own data, you choose one quantity to change (the independent variable, plotted on the horizontal axis) and you measure another quantity that responds to it (the dependent variable, plotted on the vertical axis).

Good data collection habits make your scatter plot more reliable. Use equally spaced input values whenever you can — this makes the difference test straightforward. Record at least five to eight data points so the curved shape is visible. Repeat each measurement two or three times and average the results to reduce the effect of errors.

Once you have your table, label both columns clearly with the variable name and its units — for example, time in seconds and height in centimetres. These labels will become the axis labels on your scatter plot.

Independent variable goes on the horizontal axis; dependent variable goes on the vertical axis.
Use equally spaced $x$ -values to make the difference test easy.
Collect at least five to eight data points to see the curve shape.
Average repeated measurements to reduce errors.
Label columns with variable names and units.

Plotting the Scatter Plot and Drawing a Curve of Best Fit

Once you have confirmed the quadratic pattern and labelled your table, you are ready to plot. Choose a scale for each axis so that all your data points fit on the grid with some space around them. Mark the scale in equal intervals and write units on each axis.

Plot each ordered pair

(x, y)

as a small dot or cross. Do not connect the dots with straight line segments — that would produce a jagged shape, not a smooth curve. When all points are plotted, step back and look at the overall shape.

To draw the curve of best fit, use a pencil and sketch a single smooth, continuous curve that follows the general curved trend of the points. The curve should be as close as possible to all the points — some points will sit exactly on the curve, some slightly above, and some slightly below. The curve should not be forced through every single point; instead, it balances the points above and below it, just as a line of best fit does. Aim for a symmetrical U-shape (or inverted U-shape) unless the data clearly shows an asymmetric portion of a parabola.

Finally, label your graph with a title that describes what the data shows, for example: 'Height of a Bouncing Ball vs. Time'.

Choose a scale so all points fit on the grid with room to spare.
Plot each point as a dot or cross — do not connect points with line segments.
Draw one smooth, continuous curve; do not sketch a jagged or angular path.
The curve balances points above and below it — it does not have to touch every point.
Title the graph and label both axes with names and units.

Interpreting the Curve of Best Fit

A curve of best fit is a model — it represents the trend in your data, not the exact measurements. You can use it to make reasonable estimates. For example, if your curve shows height versus time for a bouncing ball, you can read off the approximate height at a time you did not measure by finding that time on the horizontal axis and reading across to the curve.

You can also describe the relationship in words: 'As the time increases from 0 to 1.5 seconds, the height decreases rapidly, reaching a minimum near

t = 1.5

s, and then increases again.' This kind of description — identifying whether the parabola opens up or down, where the vertex appears to be, and whether the data covers a full parabola or just one arm — is exactly what is expected at the Grade 10 level.

Remember that your curve is only reliable within the range of your data. Extending the curve far beyond the last data point is risky, because you do not know whether the quadratic pattern continues outside the range you measured.

The curve of best fit is a model of the trend, not a perfect record of every measurement.
You can estimate values within your data range by reading from the curve.
Describe the curve: does it open up or down? Where does the vertex appear to be?
Avoid extending the curve far beyond your data range.

First and Second Differences: What They Tell You

Pattern in first differences	Pattern in second differences	Type of relationship	Best-fit model
All equal	All zero	Linear	Straight line
Not equal	All approximately equal (non-zero)	Quadratic	Parabolic curve
Not equal	Not equal	Neither linear nor quadratic	Cannot use a line or parabola

Worked example

Example 1: Checking Differences and Identifying a Quadratic Data Set

A student rolls a toy car down a ramp and records the total distance it has travelled at each second. The table below shows the results. Determine whether the data is linear or quadratic by calculating first and second differences.

| Time

t

(s) | Distance

d

(cm) |
|---|---|
| 0 | 0 |
| 1 | 6 |
| 2 | 24 |
| 3 | 54 |
| 4 | 96 |

List the output values in order
Write the $d$ -values in a column in the same order as the table: 0, 6, 24, 54, 96. Check that the $t$ -values are equally spaced — each step is $+1$ second, so the difference test is valid.
Calculate the first differences
Subtract each $d$ -value from the one directly below it, working down the column. This gives the first differences $\Delta_1$ .
$6-0=6, \quad 24-6=18, \quad 54-24=30, \quad 96-54=42$
Check whether first differences are equal
The first differences are 6, 18, 30, 42. These are not equal, so the relationship is not linear.
Calculate the second differences
Subtract each first difference from the one directly below it. This gives the second differences $\Delta_2$ .
$18-6=12, \quad 30-18=12, \quad 42-30=12$
Interpret the second differences
All three second differences equal 12. Because the second differences are constant and non-zero, the data follows a quadratic pattern. A parabolic curve of best fit is appropriate for this data set.
$\Delta_2 = 12$

Answer: The second differences are all equal to

12

, confirming the data is quadratic. A smooth parabolic curve of best fit should be drawn through the scatter plot of these points.

Check: Verify: first differences 6, 18, 30, 42 increase by

12

each time. Second differences:

18-6=12

,

30-18=12

,

42-30=12

. All equal. Confirmed.

Worked example

Example 2: Plotting a Scatter Plot and Drawing a Curve of Best Fit

A student stands on a bridge and drops a water balloon. She measures the balloon's height above the ground (in metres) every half-second. Her results are shown below. Plot the data on a scatter plot, verify that the data is approximately quadratic using second differences, and describe how to draw the curve of best fit.

| Time

t

(s) | Height

h

(m) |
|---|---|
| 0.0 | 20.0 |
| 0.5 | 18.8 |
| 1.0 | 15.1 |
| 1.5 | 9.0 |
| 2.0 | 0.4 |

Check that input values are equally spaced
Each time value increases by $0.5$ s, so the spacing is equal and the difference test is valid.
Calculate the first differences of height
Subtract each $h$ -value from the one directly below it, working down the table.
$18.8-20.0=-1.2, \quad 15.1-18.8=-3.7, \quad 9.0-15.1=-6.1, \quad 0.4-9.0=-8.6$
Calculate the second differences
Subtract each first difference from the one directly below it.
$-3.7-(-1.2)=-2.5, \quad -6.1-(-3.7)=-2.4, \quad -8.6-(-6.1)=-2.5$
Interpret the second differences
The second differences are $-2.5,\,-2.4,\,-2.5$ . These are approximately equal — the small variation in the middle value is due to measurement rounding. This is strong evidence of a quadratic relationship. Because the second differences are negative, the parabola opens downward.
$\Delta_2 \approx -2.5$
Set up the scatter plot
Draw a horizontal axis labelled 'Time $t$ (s)' with values from $0$ to $2.0$ in steps of $0.5$ . Draw a vertical axis labelled 'Height $h$ (m)' with values from $0$ to $21$ in steps of $3$ . Plot each ordered pair: $(0.0,\,20.0)$ , $(0.5,\,18.8)$ , $(1.0,\,15.1)$ , $(1.5,\,9.0)$ , $(2.0,\,0.4)$ .
Draw the curve of best fit
Use a pencil to sketch a single smooth curve that starts near the top-left of the grid and sweeps downward following the general shape of the plotted points. Because the parabola opens downward and the data shows only the descending arm, the curve should arc smoothly toward the horizontal axis without any sharp corners or zigzags. Adjust the curve so it passes as close as possible to all five points. Add the graph title 'Height of Falling Balloon vs. Time'.

Answer: The second differences are approximately

-2.5

, confirming a quadratic (downward-opening) relationship. The scatter plot shows five points sweeping from top-left to bottom-right in a curved arc. A smooth downward-opening parabolic curve of best fit is drawn through these points.

Check: Re-check second differences:

(-3.7)-(-1.2)=-2.5

;

(-6.1)-(-3.7)=-2.4

;

(-8.6)-(-6.1)=-2.5

. The values

-2.5, -2.4, -2.5

are approximately equal, confirming the quadratic pattern. The slight difference of

0.1

in the middle is within expected rounding variation for real data.

Common mistakes and how to avoid them

Calculating differences without first checking that the

x

-values are equally spaced.

Correction: Always verify that the gap between consecutive

x

-values is the same before applying the difference test. If the spacing is unequal, the test does not work.

Connecting plotted data points with straight line segments instead of a smooth curve.

Correction: Use a single continuous, smooth pencil curve. Straight segments between points produce a jagged graph that is not a parabola and misrepresents the quadratic trend.

Expecting second differences to be perfectly equal when using real collected data.

Correction: Real measurements always contain small errors. Look for second differences that are approximately equal, not identical. A small variation (such as

-2.4

versus

-2.5

) is normal and still supports a quadratic conclusion.

Forcing the curve of best fit through every single data point.

Correction: A curve of best fit models the overall trend. It should be as close as possible to all the points but does not need to pass through each one. Forcing it through every point produces a wiggly curve that is not a parabola.

Forgetting to label axes with variable names and units, or omitting a graph title.

Correction: Always label both axes with the variable name and its units (e.g., 'Height

h

(m)') and give the graph a descriptive title. Without these, the graph cannot be interpreted by someone who has not seen the original data.

Lesson summary

To check whether data is quadratic, first confirm that the $x$ -values are equally spaced, then calculate first differences and then second differences from the $y$ -values.
If the second differences are approximately equal and non-zero, the data follows a quadratic pattern and a parabolic curve of best fit is appropriate.
A scatter plot displays data as individual points on a labelled grid; you do not connect the points with line segments.
A curve of best fit is a single smooth parabolic curve drawn as close as possible to all the points; it does not have to pass through every point.
Describe the curve by stating whether it opens up or down and where the vertex approximately appears, based on the shape of the data.
The curve of best fit is a model of the trend within your data range; use it cautiously and avoid extending it far beyond the data you collected.

Check your understanding

Question 1

A student records the following

y

-values for equally spaced

x

-values: 2, 5, 10, 17, 26. What are the second differences?

3, 5, 7, 9
2, 2, 2
3, 3, 3
2, 2, 2, 2

Show answer and explanation

2, 2, 2

First differences:

5-2=3

,

10-5=5

,

17-10=7

,

26-17=9

. Second differences:

5-3=2

,

7-5=2

,

9-7=2

. All second differences equal

2

, confirming a quadratic pattern. Option A lists the first differences, not the second. Options C and D have the wrong values.

Question 2

You have plotted a scatter plot and the second differences of the data are approximately equal. What should you draw as the best-fit model?

A straight line through as many points as possible
A smooth parabolic curve positioned as close as possible to all the points
A series of straight line segments connecting every consecutive pair of points
A horizontal line at the average $y$ -value

Show answer and explanation

A smooth parabolic curve positioned as close as possible to all the points

Approximately equal second differences indicate a quadratic relationship, so the correct model is a smooth parabolic curve of best fit. A straight line suits linear data, connected segments produce a jagged shape that is not a parabola, and a horizontal line ignores the trend entirely.

Question 3

Which of the following is the correct first step before applying the finite difference test to a table of values?

Calculate the average of all the $y$ -values.
Check that the $x$ -values are equally spaced.
Plot the points on a scatter plot.
Draw a curve through the first and last data points.

Show answer and explanation

Check that the

x

-values are equally spaced.

The difference test is only valid when the

x

-values are equally spaced. If the gaps between

x

-values differ, the test will give misleading results. Plotting and averaging are useful steps, but they come after confirming equal spacing.

Question 4

A scatter plot shows data that curves downward in a smooth arc. The second differences are all approximately

-4

. What does the negative sign tell you about the parabola?

The parabola opens upward (U-shape).
The data is linear with a negative slope.
The parabola opens downward (inverted U-shape).
There is no quadratic pattern because the value is negative.

Show answer and explanation

The parabola opens downward (inverted U-shape).

When the constant second difference is negative, the parabola opens downward, producing an inverted U-shape. A positive constant second difference would indicate an upward-opening parabola. A negative value does not mean the relationship is linear or non-quadratic.

Key terms

Scatter plot: A graph that displays collected data as individual points on a coordinate grid, without connecting the points.
Curve of best fit: A smooth curve drawn through or near the data points on a scatter plot to model the overall trend in the data.
Parabola: The U-shaped (or inverted U-shaped) curve that represents a quadratic relationship on a graph.
First differences: The values obtained by subtracting each output value from the next output value in a table where the input values are equally spaced.
Second differences: The values obtained by applying the same subtraction process to the first differences. Approximately equal second differences indicate a quadratic relationship.
Independent variable: The variable you choose or control in an experiment; plotted on the horizontal axis.
Dependent variable: The variable that responds to changes in the independent variable; plotted on the vertical axis.
Quadratic relationship: A relationship between two variables in which the second differences of equally spaced data are approximately equal and non-zero, producing a parabolic graph.

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About this lesson

Published by DoAssignment. This AI-assisted lesson follows Ontario Grade 10 Mathematics (MPM2D), study topic Q2. It is a study resource, not an official curriculum publication.

Official curriculum reference

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Q2 · Collect quadratic data and draw a curve of best fit

Ontario Grade 10 Mathematics

Recognising Quadratic Patterns in Real Data and Sketching a Smooth Parabolic Curve

What you will learn

Bridge from Grade 9: Lines of Best Fit and Scatter Plots

Checking for a Quadratic Pattern: First and Second Differences

Setting Up and Collecting Quadratic Data

Plotting the Scatter Plot and Drawing a Curve of Best Fit

Interpreting the Curve of Best Fit

First and Second Differences: What They Tell You

Example 1: Checking Differences and Identifying a Quadratic Data Set

Example 2: Plotting a Scatter Plot and Drawing a Curve of Best Fit

Common mistakes and how to avoid them

Lesson summary

Check your understanding

Question 1

Question 2

Question 3

Question 4

Key terms

Continue through MPM2D

About this lesson