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Q6 · Collect quadratic data and draw a curve of best fit

Learn to collect quadratic data and draw a curve of best fit through clear examples and targeted practice.

Ontario Grade 10 Mathematics

Quadratic Relations

Recognizing U-shaped patterns in real data and sketching a smooth parabola through them

In Grade 9 you learned to draw a line of best fit through data that followed a straight-line pattern. Not all real-world data is linear. Drop a ball, launch a water stream, or roll a toy car off a ramp and the path curves. That curve is called a parabola, and data that follows this shape is called quadratic data. In this lesson you will collect and organize quadratic data, plot it, and sketch a smooth curve of best fit through the points — a parabola — so you can read off values and spot trends that the raw numbers alone do not reveal.

What you will learn

Explain what makes a set of collected data quadratic rather than linear.
Organize data in a table and plot the points on a scatter plot.
Identify the vertex and axis of symmetry of a parabolic data set by inspection.
Sketch a smooth curve of best fit through quadratic data and use it to make predictions.
Describe how the shape of the curve tells you whether the parabola opens upward or downward.

Reviewing What You Already Know: Scatter Plots and Lines of Best Fit

In Grade 9 (MTH1W) you organized pairs of numbers into a table, plotted them as points on a grid, and drew the straight line that came closest to all the points. That line let you estimate values between data points (interpolation) and beyond them (extrapolation).

The key ideas you already have are: every point has two coordinates written as

(x, y)

; the independent variable goes on the horizontal axis; the dependent variable goes on the vertical axis; and a best-fit model does not have to pass through every point — it captures the overall trend.

All of those ideas carry forward into this lesson. The only difference is that the trend is now a curve, not a straight line. Whenever the data bends and then bends back the other way, a line of best fit will miss many points. That is your signal that the pattern is quadratic.

A scatter plot shows the relationship between two variables as plotted points.
A line of best fit works only when data follows a roughly straight pattern.
When data rises and then falls (or falls and then rises), the pattern is curved, not linear.
The same interpolation and extrapolation ideas apply to a curve of best fit.

What Is Quadratic Data?

Data is called quadratic when the pattern of the points forms a U-shape (or an upside-down U-shape) on a scatter plot. This shape is a parabola. The turning point of the parabola — the lowest point of a U or the highest point of an upside-down U — is called the vertex.

A useful check is to look at the differences between consecutive

y

-values in your table. For linear data the differences are roughly equal (constant first differences). For quadratic data the first differences change, but the differences of those differences — called second differences — are roughly equal. Equal second differences are a strong signal that the data is quadratic.

The vertical line through the vertex is called the axis of symmetry. Because a parabola is symmetric, the

y

-values on either side of the vertex mirror each other. If you fold the parabola along the axis of symmetry, the two halves match. This symmetry is a powerful tool for checking your curve and for estimating missing values.

When the parabola opens upward (U-shape), the vertex is the minimum point. When it opens downward (arch shape), the vertex is the maximum point. Context always tells you which to expect: a ball thrown upward has a maximum height, while the distance a dropped object falls has a minimum of zero at the moment of release.

Quadratic data forms a parabola (U-shape or arch shape) on a scatter plot.
The vertex is the turning point: minimum for an upward parabola, maximum for a downward one.
Equal second differences in a table confirm that data is quadratic.
The axis of symmetry passes vertically through the vertex and splits the parabola into two mirror halves.
An upward parabola opens like a bowl; a downward parabola opens like an arch.

Collecting and Organizing Quadratic Data

Many classroom experiments produce quadratic data. A common one is recording the height of a ball at equal time intervals after it is tossed upward. Another is measuring the area of a square as its side length increases — but time-and-height experiments are especially clear because the data rises, peaks, and falls.

To organize your data, set up a table with the independent variable in the left column and the dependent variable in the right column. Label each column clearly and include units. Once you have your table, check for the U-shape trend before you plot anything — if the

y

-values go up and then come back down (or down then up), you already know the curve will be a parabola.

After plotting the points on a scatter plot, look at the overall shape. Do not connect the dots with straight line segments. Instead, you are going to draw one smooth, continuous curve that follows the general path of all the points. This is the curve of best fit.

Real data always has small measurement errors, so some points will be slightly above or below your curve. That is normal and expected. The goal is for the curve to pass as close as possible to all the points, with roughly the same number of points above the curve as below it.

Record data in a clearly labelled two-column table with units.
Check the $y$ -values for a rise-then-fall (or fall-then-rise) pattern before plotting.
Plot each $(x, y)$ pair as a point — do not connect dots with straight segments.
The curve of best fit is one smooth, continuous curve near all the points.
Small scatter above and below the curve is normal in real collected data.

Drawing the Curve of Best Fit

Follow these steps to draw a parabolic curve of best fit by hand. First, scan your scatter plot and estimate where the vertex is — this is the point that sits at the very top or very bottom of the data cloud. Mark that location lightly.

Next, look at whether the data is roughly symmetric. Imagine a vertical line through your estimated vertex. Do the points on the left look like a mirror image of the points on the right? If not, adjust your estimated vertex left or right until the symmetry is as close as possible.

Now draw a smooth U-shaped (or arch-shaped) curve that passes near the vertex and sweeps through the rest of the data points. Use a curved, fluid motion — not a series of short strokes. The curve should be widest at the ends and narrowest near the vertex.

When using graphing technology (such as Desmos or a graphing calculator), you can enter the data and ask it to display a quadratic regression curve. The technology calculates the best-fitting parabola automatically. You can then read coordinates directly from the displayed graph. In this course, you interpret the graph the technology gives you; you do not need to find the equation algebraically.

Estimate the vertex first, then use symmetry to check and adjust it.
Draw the curve in one smooth stroke — not connected straight segments.
Aim for roughly equal numbers of data points above and below the curve.
Graphing technology can generate the curve; you read and interpret that graph.
Always label your axes and include a title on any scatter plot you submit.

Reading the Curve to Make Predictions

Once the curve is drawn, you can use it exactly the way you used a line of best fit in Grade 9. To find the

y

-value for a given

x

-value, locate that

x

-value on the horizontal axis, move straight up to the curve, and then read across to the vertical axis. This is interpolation when the

x

-value is inside your data range, and extrapolation when it is outside.

Extrapolation with a curve of best fit requires extra care. A parabola continues to rise (or fall) steeply beyond the data you collected, so predictions far outside the data range may not be realistic. Always ask yourself whether the physical situation makes sense at the predicted value.

The vertex gives you the most important single prediction: the maximum or minimum value of the dependent variable and the

x

-value at which it occurs. You can read both coordinates of the vertex directly from the graph.

Use the curve to estimate $y$ for any $x$ by reading straight up to the curve, then across.
Interpolation (inside the data range) is more reliable than extrapolation (outside it).
Extrapolation with a parabola can give unrealistic values far from the data.
The vertex coordinates give the maximum or minimum value and when it occurs.
All predictions should be checked against whether the physical context makes sense.

First and Second Differences: Linear vs. Quadratic Data

x	y (linear)	1st diff (linear)	y (quadratic)	1st diff (quadratic)
0	3	—	1	—
1	6	3	4	3
2	9	3	11	7
3	12	3	22	11
4	15	3	37	15

Worked example

Example 1: Ball Toss — Finding the Vertex and Sketching the Curve

A student tosses a ball straight up and records its approximate height every half second using a motion sensor. The data is shown below. (a) Confirm the data is quadratic using second differences. (b) Identify the vertex. (c) Describe the curve of best fit, including whether it opens up or down and where the axis of symmetry is.

Time

t

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

h

(m): 1.2, 3.7, 5.0, 5.1, 3.8, 1.3

List the first differences
Subtract each height value from the next one. First differences show how much $h$ changes each half-second. Going left to right: $3.7-1.2=2.5$ , then $5.0-3.7=1.3$ , then $5.1-5.0=0.1$ , then $3.8-5.1=-1.3$ , then $1.3-3.8=-2.5$ .
$2.5,\ 1.3,\ 0.1,\ {-1.3},\ {-2.5}$
Calculate the second differences
Subtract each first difference from the one that follows it. This gives the second differences. The calculation goes: $1.3-2.5=-1.2$ , then $0.1-1.3=-1.2$ , then $-1.3-0.1=-1.4$ , then $-2.5-(-1.3)=-1.2$ .
$-1.2,\ -1.2,\ -1.4,\ -1.2$
Confirm the data is quadratic
The second differences are approximately $-1.2$ each time. The slight variation at the third value is due to rounding in the sensor readings. Because they are roughly constant and negative, the data is quadratic and the parabola opens downward.
Locate the vertex
Scan the $h$ column for the largest value, since this is a downward parabola with a maximum. The highest recorded height is $5.1$ m at $t = 1.5$ s. However, $h = 5.0$ at $t = 1.0$ s is very close, suggesting the true peak sits between $t = 1.0$ s and $t = 1.5$ s. Using the near-symmetry of the data, the vertex is approximately at $(1.25,\ 5.1)$ .
$\text{vertex} \approx (1.25,\ 5.1)$
State the axis of symmetry
The axis of symmetry is the vertical line that passes through the $t$ -coordinate of the vertex.
$t = 1.25$
Describe the curve of best fit
Plot all six points on a grid with $t$ on the horizontal axis (0 to 2.5 s) and $h$ on the vertical axis (0 to 6 m). The curve opens downward, peaks near the vertex at approximately $(1.25,\ 5.1)$ , and sweeps symmetrically down on both sides. Draw one smooth arch through all the points.

Answer: Second differences are approximately

-1.2

, confirming quadratic data. The parabola opens downward with vertex approximately at

(1.25,\ 5.1)

and axis of symmetry

t = 1.25

Check: Check symmetry:

h

t = 0

1.2

m and

h

t = 2.5

s is

1.3

m — these are nearly equal, as expected for points at the same distance from the axis of symmetry on a parabola. The data and conclusion are consistent.

Worked example

Example 2: Using the Curve of Best Fit to Make Predictions

A student rolls a toy car off a ramp onto a flat floor and measures how far the car travels (in cm) for different ramp heights (in cm). The data is plotted on a scatter plot and a smooth parabolic curve of best fit is drawn. The curve passes through the following approximate points read from the graph:

Ramp height

x

(cm): 0, 5, 10, 15, 20, 25
Distance

d

(cm): 0, 18, 40, 64, 90, 118

(a) Does the curve open upward or downward? How do you know?
(b) Use the curve of best fit to estimate the distance the car travels when the ramp height is 12 cm.
(c) Estimate the ramp height needed for the car to travel 50 cm.

Determine the direction the parabola opens
Look at the $d$ values as $x$ increases from 0 to 25. The distances keep growing without ever turning back down. There is no maximum in the data — the curve rises continuously. This means the parabola opens upward, with its vertex at or near the origin $(0, 0)$ .
Check second differences to confirm quadratic shape
Find the first differences of $d$ : $18-0=18$ , $40-18=22$ , $64-40=24$ , $90-64=26$ , $118-90=28$ . Now find the second differences: $22-18=4$ , $24-22=2$ , $26-24=2$ , $28-26=2$ . These are roughly constant, confirming the data is quadratic.
$4,\ 2,\ 2,\ 2$
Estimate d when x = 12 cm (interpolation)
On the scatter plot, locate $x = 12$ on the horizontal axis. Move straight up until you hit the curve of best fit, then read across to the vertical axis. The point $x = 12$ sits $\frac{2}{5}$ of the way from $x = 10$ to $x = 15$ . A linear estimate between those two known points gives a starting approximation.
$40 + \frac{2}{5}(64 - 40) = 40 + 9.6 = 49.6 \approx 50$
Refine the interpolation estimate
Because the curve bends (it is not a straight line between any two points), reading directly from the smooth curve is more accurate than the linear step above. A carefully drawn parabolic curve of best fit gives approximately $d \approx 48$ cm at $x = 12$ cm. The curve's rate of increase is still growing in this region, so the true curve value is slightly below the linear estimate.
$d \approx 48 \text{ cm}$
Estimate x when d = 50 cm (reading the curve in reverse)
Locate $d = 50$ cm on the vertical axis. Draw a horizontal line across until it meets the curve, then drop straight down to the horizontal axis. The curve reaches $d = 50$ between $x = 10$ and $x = 15$ , very close to $x = 12$ cm. Reading directly from the smooth curve gives the estimate below.
$x \approx 12 \text{ cm}$

Answer: (a) The parabola opens upward —

d

keeps increasing with no maximum in the data range. (b) When the ramp height is 12 cm, the car travels approximately 48 cm. (c) The car travels 50 cm when the ramp height is approximately 12 cm.

Check: Cross-check:

d \approx 48

cm at

x = 12

cm and

d = 50

cm at

x \approx 12

cm are consistent with each other — the distances are very close, so the ramp heights should be very close too. Both estimates fall between the known data points, so interpolation is appropriate and reliable.

Common mistakes and how to avoid them

Connecting data points with straight line segments instead of drawing one smooth curve.

Correction: A curve of best fit is a single, continuous, smooth stroke that follows the overall parabolic shape of the data — never a series of ruler-drawn segments between consecutive points.

Assuming the curve must pass through every data point.

Correction: A best-fit curve models the trend, not every individual measurement. Real data has small errors, so some points will sit above or below the curve. Aim for roughly equal numbers of points on each side.

Forgetting to check whether the parabola opens up or down before sketching.

Correction: Look at the data: if

y

-values rise then fall, the parabola opens downward and has a maximum. If

y

-values fall then rise, it opens upward and has a minimum. The direction must match the data.

Placing the vertex at the data point with the highest (or lowest)

y

-value without considering symmetry.

Correction: The true vertex may sit between two data points. Use the symmetry of the parabola — the

y

-values on either side of the vertex should mirror each other — to position the vertex more accurately.

Using first differences alone to decide whether data is quadratic.

Correction: Constant first differences indicate linear data. You must check second differences (differences of the first differences). Roughly constant second differences confirm the data is quadratic.

Lesson summary

Quadratic data forms a parabola (U-shape or arch) on a scatter plot; it is confirmed when the second differences in a table are roughly constant.
The vertex is the turning point of the parabola — a maximum for a downward-opening curve and a minimum for an upward-opening curve.
The axis of symmetry is the vertical line through the vertex; the two halves of the parabola mirror each other across this line.
To draw a curve of best fit, estimate the vertex, use symmetry to check it, then sketch one smooth continuous curve near all the data points.
The curve of best fit is used for interpolation (inside the data range) and extrapolation (outside it); interpolation is more reliable.
Always read predictions directly from the graph by locating the input on one axis, moving to the curve, and reading across to the other axis.

Check your understanding

Question 1

A set of data has these second differences: 3.9, 4.1, 3.8, 4.0. What does this tell you?

The data is linear because the differences are small.
The data is quadratic because the second differences are roughly constant.
The data is quadratic because the first differences are constant.
The data has no pattern because the second differences are not perfectly equal.

Show answer and explanation

The data is quadratic because the second differences are roughly constant.

Roughly constant second differences are the signature of quadratic data. The small variation (3.8 to 4.1) is normal rounding in real measurements. Constant first differences would indicate linear data, not constant second differences.

Question 2

A scatter plot of height vs. time for a tossed ball shows points that rise then fall. Where is the vertex of the curve of best fit?

At the left-most data point, where time equals zero.
At the right-most data point, where the ball hits the ground.
At the highest point of the curve, representing the maximum height.
At the point where the curve crosses the horizontal axis.

Show answer and explanation

At the highest point of the curve, representing the maximum height.

Because the ball rises then falls, the parabola opens downward and its vertex is the maximum point — the highest point on the curve. That represents the greatest height the ball reaches.

Question 3

You have drawn a parabolic curve of best fit on a scatter plot. How do you estimate the

y

-value when

x = 8

Calculate the average of all the $y$ -values in the table.
Find $x = 8$ on the horizontal axis, move straight up to the curve, then read across to the vertical axis.
Draw a straight line between the two nearest data points and use its midpoint.
Use the vertex coordinates and count squares outward from it.

Show answer and explanation

Find

x = 8

on the horizontal axis, move straight up to the curve, then read across to the vertical axis.

To read a value from any best-fit curve, locate the input (

x = 8

) on the horizontal axis, move vertically to where the curve is, then move horizontally to the vertical axis to read the output. This is the standard graph-reading method for interpolation.

Question 4

A student's curve of best fit passes through all but one data point, which sits clearly above the curve. The student says the curve is wrong. Is the student correct?

Yes — a correct curve of best fit must pass through every data point.
Yes — the one point above the curve means the curve is not symmetric.
No — a curve of best fit models the overall trend; a few points above or below the curve are expected.
No — only the vertex point must lie exactly on the curve.

Show answer and explanation

No — a curve of best fit models the overall trend; a few points above or below the curve are expected.

A best-fit curve captures the overall pattern of the data, not every individual measurement. Real data contains small errors, so it is normal and expected for some points to sit slightly above or below the curve of best fit.

Key terms

Quadratic data: Data whose scatter plot forms a parabola (U-shape or arch), confirmed by roughly constant second differences in a table.
Parabola: A smooth, symmetric U-shaped (or arch-shaped) curve that is the graphical shape of quadratic relationships.
Vertex: The turning point of a parabola — the highest point if the parabola opens downward, or the lowest point if it opens upward.
Axis of symmetry: The vertical line that passes through the vertex and divides the parabola into two mirror-image halves.
Curve of best fit: A single smooth curve drawn as close as possible to all the data points on a scatter plot, representing the overall trend of the data.
First differences: The differences between consecutive y-values in a table of values; constant first differences indicate a linear relationship.
Second differences: The differences between consecutive first differences; roughly constant second differences indicate a quadratic relationship.
Interpolation: Using a best-fit curve to estimate a value within the range of collected data.

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