A1.1 Build tables and graphs for applied quadratic relations

Ontario Grade 11 Mathematics

Mathematical Models

Use practical situations to organize quadratic values and show their patterns visually.

A table and a graph can show how two quantities change together. For example, a table can show the height of a moving object at different times, while a graph can make its rise and fall easier to see. In this lesson, you will build tables and graphs for applied quadratic relations. An applied relation connects quantities in a real situation, such as time and height or a garden’s width and area. You will use a provided rule, calculate values, and display the resulting pattern. You will not need to solve a quadratic equation.

What you will learn

Recognize a quadratic relation in a practical situation when its rule is provided.
Build a table of input and output values for a practical quadratic relation.
Plot the table values to make a graph with suitable labels and scales.
Use the table and graph to describe a pattern in the situation.

1. Bridge from inputs and outputs

A relation connects an input to an output. The input is the value you choose or measure. The output is the value that depends on it. In a table, each row pairs one input with its matching output.

For example, if time is the input and height is the output, each row pairs one time with the height at that time. Before building a table, check what each variable means and what units it uses. Units such as seconds, metres, and square metres help keep the situation clear.

A quadratic relation is one whose rule includes a squared variable. A simple example is

y=x^2

. Its outputs do not increase by the same amount each time the input increases by one. For an applied quadratic relation, the rule is usually given to you. Substitute useful input values into the rule to find the outputs.

y=x^2

Identify the input and output before calculating.
Keep the units attached to the quantities they describe.
A squared variable is a clue that a relation is quadratic.

2. Build a useful table

To build a table, first choose input values that make sense in the situation. The allowed inputs are called the domain. Here, domain simply means the input values that are reasonable for the model. For time after an object is launched, negative time would not fit the situation.

Next, substitute each input into the rule and calculate the output. Use parentheses around a negative input when squaring it. For example, squaring

-2

means multiplying

-2

by

-2

, which gives

4

. Do not confuse that with subtracting

2

after squaring.

Choose enough input values to show the pattern. Evenly spaced inputs often make a table easier to read. Include values on both sides of a central value when the situation permits; this can help reveal whether the outputs rise and then fall. Check calculations before moving to the graph.

Choose input values that fit the situation and its units.
Substitute each input into the rule and calculate carefully.
A table is a set of input-output pairs, not just a list of outputs.

3. Turn the table into a graph

A graph displays the same input-output pairs visually. The horizontal axis shows the input, and the vertical axis shows the output. Label each axis with the quantity and its unit. For example, write time in seconds on the horizontal axis and height in metres on the vertical axis.

Choose a scale that fits all the table values and uses the graph space well. A scale is the amount represented by each step or interval on an axis. The intervals should be equal. Plot each pair as a point: move to the input along the horizontal axis, then to the matching output along the vertical axis.

For an applied quadratic relation, the plotted points often follow a curved pattern. Connect the points with a smooth curve when that suits the situation; do not join them with sharp corners. A smooth curve helps show how the output changes between the selected inputs. The graph is still based on the table and the practical model, so show only the part that makes sense for the situation.

Read the table and graph together. They show the same information in different ways: the table gives exact selected values, while the graph makes the overall pattern easier to see. A graph can help you describe where an output increases, decreases, or reaches a high or low value within the displayed situation.

Put the input on the horizontal axis and the output on the vertical axis.
Label both axes with the quantity and unit.
Plot table pairs accurately and use a suitable, even scale.

4. Use the display to describe an application

A table and graph are useful only when they remain connected to the original situation. A point on a graph is not just two numbers: it represents an input and its matching output, with units. When describing a pattern, say what the quantities mean. For example, instead of saying “the output goes up,” say “the height rises as time increases over this part of the model.”

A practical model may apply only over a limited range. Use the stated situation to decide which inputs are sensible, and avoid treating a calculated value outside that range as a real prediction. A neat curve does not make every possible input meaningful.

When checking your work, compare the table and graph. Each plotted point should match a table row. The highest or lowest plotted point should also agree with the nearby table values. If not, revisit the substitution, arithmetic, or plotting.

Describe changes using the quantities and units in the situation.
Use the practical context to decide which inputs are reasonable.
Check that every plotted point matches its table pair.

Garden width and area values

Width, w (m)	Area, A (m²)
0	0
2	20
4	32
6	36
8	32
10	20
12	0

Worked example

A ball’s height over time

A simple model for a ball’s height is

h=-0.5t^2+3t+1

, where

t

is time in seconds and

h

is height in metres. Build a table for whole-number times from 0 to 6 seconds, then describe how to graph and interpret the values.

Choose the inputs
The situation describes time after the ball begins moving. Use the whole-number times from 0 to 6, as requested. These are the inputs, measured in seconds.
$t=0,1,2,3,4,5,6$
Calculate the heights
Substitute each time into the rule. For instance, at 2 seconds, the squared term is $2^2=4$ , so the height is $-0.5(4)+3(2)+1=5$ metres. Repeating this substitution gives each table output.
$h=-0.5t^2+3t+1$
Organize the pairs
Keep time and height in separate columns and include the units in the headings. These pairs are the exact points to plot.
$(0,1),(1,3.5),(2,5),(3,5.5),(4,5),(5,3.5),(6,1)$
Graph and describe
Label the horizontal axis time in seconds and the vertical axis height in metres. Plot the pairs and draw a smooth curve through them. The heights rise to 5.5 metres at 3 seconds, then fall over the listed times.
$(3,5.5)$

Answer: The table has times 0, 1, 2, 3, 4, 5, and 6 seconds and matching heights 1, 3.5, 5, 5.5, 5, 3.5, and 1 metres. The graph rises to the listed maximum height of 5.5 metres at 3 seconds and then falls.

Check: The values at 2 and 4 seconds are both 5 metres, and the values at 1 and 5 seconds are both 3.5 metres. This matching pattern supports the table and graph.

Worked example

Area of a rectangular garden

A rectangular garden has 24 metres of fencing around its perimeter. If its width is

w

metres, its length is

12-w

metres, so its area is

A=w(12-w)

. Build a table for widths from 0 to 12 metres in steps of 2, and describe the graph.

Choose the inputs
Use the requested widths. In this model, a width from 0 to 12 metres gives a non-negative length from 12 to 0 metres. The endpoints represent limiting cases with zero area.
$w=0,2,4,6,8,10,12$
Find each area
Substitute each width into the area rule. At a width of 4 metres, the length is $12-4=8$ metres, so the area is $4(8)=32$ square metres. Apply the same calculation to each width.
$A=w(12-w)$
Record the pairs
List the width in metres and its matching area in square metres. The area unit is square metres because area measures a two-dimensional surface.
$(0,0),(2,20),(4,32),(6,36),(8,32),(10,20),(12,0)$
Graph and interpret
Put width on the horizontal axis and area on the vertical axis. Plot the pairs and connect them with a smooth curve. The displayed area increases to 36 square metres at a width of 6 metres, then decreases across the remaining listed widths.
$(6,36)$

Answer: The areas for widths 0, 2, 4, 6, 8, 10, and 12 metres are 0, 20, 32, 36, 32, 20, and 0 square metres. The graph rises to 36 square metres at 6 metres wide and then falls.

Check: Widths 4 and 8 metres both give an area of 32 square metres. Their matching areas appear as equal-height points on opposite sides of the 6-metre input.

Common mistakes and how to avoid them

Putting the output on the horizontal axis and the input on the vertical axis.

Correction: Use the input on the horizontal axis and the output on the vertical axis, unless the question specifically asks for a different arrangement.

Using an uneven scale or forgetting units.

Correction: Choose equal intervals on each axis, and label each axis with its quantity and unit.

Plotting an input and output as separate points rather than as a pair.

Correction: Each table row makes one point: its input is the horizontal coordinate and its output is the vertical coordinate.

Including values that do not make sense in the application.

Correction: Use the situation to choose the allowed input range, and interpret the graph only where the model applies.

Lesson summary

An applied quadratic relation connects quantities in a practical situation and includes a squared variable in its rule.
Choose sensible inputs, substitute them into the rule, and record the matching outputs in a table.
Graph the table pairs with the input on the horizontal axis and the output on the vertical axis.
Use labels, units, an even scale, and the situation’s input range to make the graph meaningful.

Check your understanding

Question 1

A model gives area

A=x(10-x)

. What area matches an input of

x=2

?

16
8
20
4

Show answer and explanation

16

Substitute 2 for

x

:

A=2(10-2)=2(8)=16

.

Question 2

A table records time in seconds and height in metres. Which axis arrangement is appropriate?

Time on the horizontal axis and height on the vertical axis
Height on the horizontal axis and time on the vertical axis
Both quantities on the horizontal axis
Either quantity may be omitted

Show answer and explanation

Time on the horizontal axis and height on the vertical axis

Time is the input and height is the output, so time goes on the horizontal axis and height goes on the vertical axis.

Question 3

For

A=w(12-w)

, which two listed widths have equal areas?

2 m and 10 m
2 m and 6 m
4 m and 6 m
6 m and 12 m

Show answer and explanation

2 m and 10 m

At 2 metres, the area is

2(10)=20

square metres. At 10 metres, it is

10(2)=20

square metres.

Key terms

Input: A value chosen or measured that is used in a relation.
Output: The value that matches an input in a relation.
Quadratic relation: A relation whose rule includes a variable squared.
Domain: The input values that are allowed or make sense for the situation.
Scale: The amount represented by each equal step on a graph axis.

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About this lesson

Published by DoAssignment. This AI-assisted lesson follows Ontario Grade 11 Mathematics (MBF3C), expectation A1.1. It is a study resource, not an official curriculum publication.

Official curriculum reference

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A1.1 · Build tables and graphs for applied quadratic relations

Ontario Grade 11 Mathematics

Use practical situations to organize quadratic values and show their patterns visually.

What you will learn

1. Bridge from inputs and outputs

2. Build a useful table

3. Turn the table into a graph

4. Use the display to describe an application

Garden width and area values

A ball’s height over time

Area of a rectangular garden

Common mistakes and how to avoid them

Lesson summary

Check your understanding

Question 1

Question 2

Question 3

Key terms

Continue through MBF3C

About this lesson