A3.2 Determine a quadratic model for a data set

Ontario Grade 11 Mathematics

Quadratic Functions

Recognize a curved pattern, use a suitable model, and check how well it fits

Data can show how one quantity changes as another quantity changes. A straight-line model is useful when the change stays steady. A quadratic model can be useful when the change itself varies in a steady way. In this lesson, you will use patterns in a data table and a graphing tool to determine a quadratic model. You will also check what the model says and whether it suits the data.

What you will learn

Recognize when a data set may be modelled by a quadratic relation.
Use a table, graph, and quadratic regression to determine a model.
Interpret the model in the context of the data.
Check whether a quadratic model is a reasonable fit.

1. Prerequisite bridge: read a table and recognize change

A data set is a collection of related values. The input is the quantity you choose or measure first. The output is the quantity that changes in response. For example, time could be the input and height could be the output.

A function rule connects an input to an output. In a table, each input is paired with an output. In a graph, each pair is shown as a point. A model is an equation that represents the pattern in the data. A model may not pass through every measured point, especially if the measurements include small errors.

Before looking for a quadratic pattern, review first differences. These are the changes between consecutive output values when the inputs are equally spaced. If the first differences stay the same, the data follow a linear pattern. If the first differences change by a steady amount, the data may follow a quadratic pattern.

This pattern is a clue, not a guarantee. Real data can be uneven because of measurement limits or other influences. A graph and a regression model help you judge whether a quadratic model is reasonable.

\text{first difference}=\text{next output}-\text{current output}

Inputs and outputs are paired in a data set.
Equal first differences suggest a linear pattern.
Steadily changing first differences suggest a possible quadratic pattern.

2. What a quadratic model represents

A quadratic model is a rule with a squared input. Its graph is a parabola, a curve that opens upward or downward. The coefficient of the squared term affects which way it opens: a positive coefficient gives an upward opening, and a negative coefficient gives a downward opening.

The standard form of a quadratic model is

y=ax^2+bx+c

. Here,

x

is the input,

y

is the output, and

a

,

b

, and

c

are numbers chosen to represent the data. The value of

a

must not be zero, or the rule would be linear instead of quadratic.

A quadratic model can be determined from data in more than one way. For a small, evenly spaced table, second differences can help identify an exact quadratic pattern. For a larger or less exact data set, a graphing calculator or spreadsheet can perform quadratic regression. Quadratic regression finds a quadratic equation that fits the data as closely as possible overall.

When using technology, enter the input values and output values as paired lists. Choose a quadratic regression, not a linear regression. The technology returns model coefficients. Keep a sensible number of decimal places, and write the resulting equation with the variables named for the situation.

Always check the graph and the context. A model may fit the measured values well but behave poorly far beyond them. Use it mainly for inputs in or near the range of the data unless the situation supports a wider use.

y=ax^2+bx+c,\quad a\ne 0

A quadratic model has a squared input and a parabolic graph.
Use second differences for a clear pattern or quadratic regression for a data set.
A fitted equation is a model, not proof that the real relationship is exactly quadratic.

3. Connect the table, graph, and equation

Each representation gives different information. The table shows measured values and their changes. The graph shows the overall shape, including whether the data curve upward or downward. The equation gives a rule for estimating an output from an input.

A table with equal input steps is especially useful for checking differences. Find each first difference by subtracting one output from the next. Then find the differences between those first differences. These are second differences. Constant second differences support a quadratic pattern when inputs are equally spaced.

For a graphing tool, plot the points before fitting a model. Look for a curved pattern that could be represented by a parabola. Then perform quadratic regression and compare the curve with the points. If the curve misses many points or the points do not show a consistent curved trend, a quadratic model may not be suitable.

The equation can also help explain the situation. For example, a downward-opening model has a highest point, called its maximum. An upward-opening model has a lowest point, called its minimum. These features are useful only if they make sense for the real situation and occur within a relevant input range.

\text{second difference}=\text{next first difference}-\text{current first difference}

Tables reveal changes, graphs reveal shape, and equations support estimates.
Constant second differences support a quadratic pattern for equally spaced inputs.
Check the fitted curve against the plotted data and the real context.

4. Application: use the model carefully

Once you have a model, substitute an input to estimate the corresponding output. State what the estimate means and include appropriate units. If the model is used to estimate a value between measured inputs, that is interpolation. If it is used beyond the measured inputs, that is extrapolation. Extrapolation is less reliable because the data do not show whether the same pattern continues.

A model should match the purpose of the question. If the goal is to describe the data, explain the general pattern. If the goal is to estimate an output, use the equation and then check that the input is sensible. If the goal is to describe a maximum or minimum, check that the turning point is meaningful in the situation.

Avoid treating every data set as quadratic just because its graph bends. A quadratic model is suitable when the pattern, regression curve, and context support it. Report the model as an estimate and do not claim more accuracy than the data justify.

Substitute an input to estimate an output and keep the context and units.
Interpolation is within the data range; extrapolation is beyond it.
A useful model must fit both the data pattern and the situation.

Ball height measurements

Time, $t$ (s)	Height, $h$ (m)	First difference
0	5	—
1	11	6
2	13	2
3	11	-2

Worked example

Determine a model from a small data set

A student records the height of a ball at equal time intervals. The table gives time

t

in seconds and height

h

in metres. Determine a quadratic model for the data and estimate the height at

t=2.5

seconds.

Inspect the differences
The times increase by equal steps of one second, so compare consecutive height changes. The first differences are $6$ , $2$ , and $-2$ . Their changes are both $-4$ , so the constant second difference supports a quadratic pattern.
$6,\ 2,\ -2;\quad -4,\ -4$
Fit a quadratic rule
For this exact pattern, use a quadratic equation in standard form. The constant second difference is twice the coefficient of $t^2$ , so the squared-term coefficient is $-2$ . The value at $t=0$ gives the constant term, $5$ . Using the value at $t=1$ determines the remaining coefficient as $8$ .
$h=-2t^2+8t+5$
Estimate at the requested time
Substitute $2.5$ for $t$ . This input is between measured times, so the estimate is an interpolation. The model gives a height of $12.5$ metres.
$h=-2(2.5)^2+8(2.5)+5=12.5$
Check the result
The model reproduces each table value, so it fits these data exactly. In real measurements, small differences may occur, so quadratic regression and a graph check would be appropriate.

Answer: A quadratic model is

h=-2t^2+8t+5

. At

2.5

seconds, the estimated height is

12.5

metres.

Check: Substitution gives the recorded values

5

,

11

,

13

, and

11

metres at times

0

,

1

,

2

, and

3

seconds.

Common mistakes and how to avoid them

Calling changing first differences a quadratic pattern without checking their changes.

Correction: For equally spaced inputs, find the second differences. Constant second differences support a quadratic pattern.

Using a linear regression when the plotted data curve.

Correction: Plot the points first, then choose quadratic regression when a parabolic pattern is supported.

Reporting an estimate without units or context.

Correction: State what the output represents and include the units from the data.

Assuming a model is reliable far beyond the measured inputs.

Correction: Use caution with extrapolation because the data do not confirm that the pattern continues.

Lesson summary

A quadratic model represents data with a squared input and a parabolic graph.
For equally spaced inputs, constant second differences support a quadratic pattern.
Quadratic regression can determine a model from a larger or imperfect data set.
Check the model against the graph, data range, and context before using it.

Check your understanding

Question 1

A data set has equal input steps and first differences of

3

,

1

, and

-1

. What do the differences suggest?

A possible quadratic pattern, since the second differences are constant.
A linear pattern, since the first differences are constant.
No pattern, since some first differences are negative.
A quadratic pattern only if all output values are positive.

Show answer and explanation

A possible quadratic pattern, since the second differences are constant.

The second differences are

-2

and

-2

. Constant second differences support a quadratic pattern for equally spaced inputs.

Question 2

A quadratic regression model is used to estimate an output for an input beyond all measured inputs. What is this called?

Interpolation
Extrapolation
A first difference
A measured value

Show answer and explanation

Extrapolation

An estimate beyond the measured input range is extrapolation, so it should be treated with caution.

Question 3

Which feature should you check after fitting a quadratic regression?

Whether the curve follows the overall pattern of the plotted data.
Whether every measured point has the same output.
Whether the model works for every possible input.
Whether the data have no measurement units.

Show answer and explanation

Whether the curve follows the overall pattern of the plotted data.

The plotted curve should represent the overall pattern. A model is not guaranteed to fit every point or remain suitable for all inputs.

Key terms

Data set: A collection of related values, often recorded as input-output pairs.
Model: An equation or rule that represents a pattern in data.
First difference: The change between consecutive output values.
Second difference: The change between consecutive first differences.
Quadratic regression: A technology method that finds a quadratic equation fitting a set of data as closely as possible overall.
Interpolation: Estimating an output for an input within the measured input range.
Extrapolation: Estimating an output for an input beyond the measured input range.

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About this lesson

Published by DoAssignment. This AI-assisted lesson follows Ontario Grade 11 Mathematics (MCF3M), expectation A3.2. It is a study resource, not an official curriculum publication.

Official curriculum reference

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A3.2 · Determine a quadratic model for a data set

Ontario Grade 11 Mathematics

Recognize a curved pattern, use a suitable model, and check how well it fits

What you will learn

1. Prerequisite bridge: read a table and recognize change

2. What a quadratic model represents

3. Connect the table, graph, and equation

4. Application: use the model carefully

Ball height measurements

Determine a model from a small data set

Common mistakes and how to avoid them

Lesson summary

Check your understanding

Question 1

Question 2

Question 3

Key terms

Continue through MCF3M

About this lesson