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A3.1 · Collect and graph data modelled by a quadratic function

Learn to collect and graph data modelled by a quadratic function through clear examples and targeted practice.

Ontario Grade 11 Mathematics

Quadratic Functions

Turning observations into a useful quadratic model

A quadratic relationship can describe situations where one quantity changes in a curved pattern as another quantity changes. For example, the height of a tossed object may rise, reach a maximum, and then fall. In this lesson, you will practise collecting or organizing paired measurements, graphing them, and using a quadratic model to represent the pattern. A model is a mathematical description of data; it is useful, but it does not make every measurement exact.

What you will learn

Plan and record data that may be represented by a quadratic function.
Organize paired data and graph it as a scatter plot.
Use the shape of the plotted points and a quadratic model to describe a relationship.
Check whether a quadratic model is a reasonable fit for the data.

1. Review: variables, tables, and graphs

A variable is a quantity that can change. In a data investigation, the input is the value you choose or record first. The output is the value that may depend on it. For a tossed object, time can be the input and height can be the output.

A data point is a pair of related values, such as a time and the height measured at that time. Write it as an ordered pair, with the input first. A table helps keep the pairs together. A graph shows the same pairs visually.

Before collecting data, decide what you will measure, choose units, and use a consistent method. Record enough input values to reveal the pattern. Measurements may vary slightly because tools and observations are not perfect.

Keep each input matched with its correct output.
Include units and use them consistently.
A table and a graph are two ways to show the same paired data.

2. Recognizing a possible quadratic pattern

A quadratic function has a squared input. Its graph is a parabola, a U-shaped or upside-down U-shaped curve. The turning point is where the curve changes direction. It is the lowest point on an upward-opening parabola or the highest point on a downward-opening parabola.

A quadratic model can be written in the form

y=ax^2+bx+c

, where

x

is the input,

y

is the output, and

a

b

, and

c

are numbers. The value of

a

affects the direction and width of the curve: if

a

is positive, it opens upward; if

a

is negative, it opens downward. You do not need to find these numbers by hand to make a useful model from measured data.

Data collected in real situations rarely land perfectly on a smooth curve. Instead, look for an overall curved pattern. A scatter plot displays the individual data points without joining them. If the points cluster around a parabolic shape, a quadratic model may be reasonable.

A graphing tool or spreadsheet can calculate a quadratic regression. Regression means finding a curve that fits the general pattern of the data. The resulting equation is a model, not a claim that every observed point lies exactly on the curve. Use the equation and graph together, and consider whether the curve makes sense for the situation.

y=ax^2+bx+c

A quadratic graph is a parabola with one turning point.
A scatter plot shows measured points; a regression curve summarizes their overall pattern.
Interpret the graph in context, including the units and the range of data collected.

3. From observations to a model

A careful investigation follows a clear sequence. First, identify the two quantities and decide which is the input. Next, collect or receive paired measurements and organize them in a table. Then graph the points, with the input on the horizontal axis and the output on the vertical axis.

Look at the plotted points before adding a model. Ask whether the pattern bends, whether it has a high or low region, and whether any point seems far from the overall pattern. If the shape is plausibly parabolic, use a graphing tool to fit a quadratic equation. Label the model and axes so another person can understand the result.

A model should only be used over a range supported by the data. For instance, measurements taken during the first few seconds of a toss do not automatically justify predictions far before or after that interval. Predictions beyond the collected range are extrapolations, meaning estimates outside the observed data. They may be unreliable.

The context also matters. A quadratic curve might fit measured heights over a short part of a toss, but the data should be collected in a safe and consistent way. If the points do not show a clear curved pattern, do not force a quadratic model onto them.

Plot the observations before choosing a model.
Use a quadratic regression only when the pattern supports it.
Treat predictions outside the measured input range with caution.

4. Reading and checking the graph

The graph communicates features of the relationship. The horizontal coordinate gives an input value; the vertical coordinate gives the corresponding output. The turning point suggests where the output reaches a local high or low value in the displayed pattern.

Check that the scale on each axis suits the data. A scale that is too wide can make important changes hard to see. A scale that is too narrow can make small differences look larger than they are. Plot points carefully and label each axis with the quantity and its unit.

After fitting a model, compare its curve with the scatter plot. A reasonable model follows the main pattern without ignoring large groups of points. It should also give sensible values for the situation. A close-looking curve is not automatically appropriate if it contradicts what the quantities mean.

When reporting the result, state what the variables represent, give the equation if useful, and describe what the graph shows. Avoid saying that the equation proves a cause. It represents an observed relationship in the data.

Choose a readable scale and label both axes.
Compare the model with the actual points.
Use the situation to judge whether the model's output is sensible.

Ball-height observations

Time, $t$ (s)	Height, $h$ (m)
0.0	1.0
0.5	4.7
1.0	5.9
1.5	4.7
2.0	1.0

Worked example

Modelling the height of a tossed ball

A student records the height of a ball at several times after it is tossed. The values are rounded measurements in metres. Organize the data, describe the graph, and give a quadratic model that represents the pattern.

Organize the paired measurements
Use time as the input and height as the output. Keeping each pair in one row prevents the measurements from being mixed up.
Graph the observations
Put time in seconds on the horizontal axis and height in metres on the vertical axis. The points rise at first and then fall, so the plot has a downward-curving pattern.
Fit and interpret a model
A graphing tool gives the displayed quadratic model for these rounded data. Its negative squared-term coefficient matches the downward opening. The graph reaches its highest point near the middle of the recorded interval. The model is appropriate for describing the observed part of the toss, not for claiming exact measurements at every time.
$h=-4.9t^2+9.8t+1.0$
Check a model value
At two seconds, substitute that time into the model. The predicted height is one metre, which agrees with the recorded value at that time.
$h(2)=-4.9(2)^2+9.8(2)+1.0=1.0$

Answer: The data show a rise-and-fall pattern that is reasonably represented by a downward-opening quadratic. The model is

h=-4.9t^2+9.8t+1.0

, where

t

is time in seconds and

h

is height in metres. The data and model should be used only for the recorded interval from zero to two seconds.

Check: At

t=0

, the model gives

h=1.0

, matching the initial recorded height. At

t=2

, it also gives

h=1.0

. These checks agree with the table.

Common mistakes and how to avoid them

Putting the output on the horizontal axis and the input on the vertical axis without a reason.

Correction: For this kind of relationship, put the input on the horizontal axis and the output on the vertical axis. Label both axes with units.

Connecting every measured point with straight line segments and calling the result a quadratic graph.

Correction: A scatter plot shows the measured points. A quadratic model is a smooth curve fitted to the overall pattern, not a set of straight segments.

Assuming a quadratic equation must pass exactly through every measurement.

Correction: Real measurements can vary. Judge whether the curve follows the overall pattern and whether it makes sense in context.

Using the model far outside the range of collected inputs as if its predictions were certain.

Correction: Limit conclusions to the observed range unless there is a good reason to trust the model beyond it.

Lesson summary

Plan what to measure, use consistent units, and keep each input paired with its output.
Organize the data in a table and graph the points as a scatter plot.
A parabolic pattern may be represented by a quadratic model such as $y=ax^2+bx+c$ .
Use graphing technology to fit a curve when appropriate, then check it against the data and the context.

Check your understanding

Question 1

A table records the time and height of a tossed object. Which graph setup is most appropriate?

Time on the horizontal axis and height on the vertical axis, with units labelled.
Height on the horizontal axis and time on the vertical axis, with no labels.
Time and height on the same axis so the points can be joined.
A bar graph with one bar for each measured time.

Show answer and explanation

Time on the horizontal axis and height on the vertical axis, with units labelled.

Time is the input and height is the output. A labelled scatter plot keeps the paired measurements visible.

Question 2

A scatter plot rises, reaches a high region, and then falls. What is a reasonable next step?

Fit a quadratic model with graphing technology and compare the curve with the data.
Assume every point must lie exactly on a straight line.
Delete points that do not match the curve you expected.
Use the model to make certain predictions at any input value.

Show answer and explanation

Fit a quadratic model with graphing technology and compare the curve with the data.

A rise-and-fall pattern may be quadratic. Fit a model and check whether it represents the points and makes sense in context.

Question 3

Measurements were collected only from zero to four seconds. Which statement about a model prediction at ten seconds is best?

It is outside the observed range and may be unreliable.
It is guaranteed accurate because the equation is quadratic.
It must equal the measurement at four seconds.
It is a measured value because it comes from the equation.

Show answer and explanation

It is outside the observed range and may be unreliable.

A prediction beyond the recorded input values is an extrapolation. The collected data alone do not show that it will be reliable.

Key terms

Input: The quantity chosen or recorded first in a relationship; it is usually placed on the horizontal axis.
Output: The quantity paired with an input; it is usually placed on the vertical axis.
Scatter plot: A graph that displays paired data as separate points.
Quadratic function: A function that includes a squared input and has a parabolic graph.
Regression: A method used by technology to find a curve that represents the overall pattern in data.
Extrapolation: A prediction made for an input outside the range of the collected data.

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About this lesson

Published by DoAssignment. This AI-assisted lesson follows Ontario Grade 11 Mathematics (MCF3M), expectation A3.1. It is a study resource, not an official curriculum publication.

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