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A1.1 · Pose and solve application problems from quadratic tables and graphs

Learn to pose and solve application problems from quadratic tables and graphs through clear examples and targeted practice.

Ontario Grade 11 Mathematics

Quadratic Functions

Pose useful questions, read patterns, and interpret solutions in context

A quadratic relationship can describe a quantity that rises and then falls, or falls and then rises. For example, the height of a ball thrown upward may rise to a maximum and then fall. A table lists measured or modelled values. A graph shows the overall pattern and can help answer questions about values between listed entries. In this lesson, you will practise turning information from quadratic tables and graphs into questions about a real situation, then solving and interpreting those questions.

What you will learn

Recognize how a quadratic table and graph can describe a real situation.
Pose questions that can be answered using quadratic data.
Use a table, graph, and a simple quadratic model to answer an application question.
Explain what a mathematical answer means in the original situation.

1. Prerequisite bridge: connect quantities and values

An application problem describes a situation using quantities. A variable is a letter that stands for a quantity that can change. For example, time might be represented by

t

and height by

h

. The input is the quantity you choose or observe. The output is the quantity that depends on it.

A table pairs inputs with outputs. A graph places each pair as a point. The horizontal axis usually shows the input, and the vertical axis shows the output. Always check the labels and units before reading values.

A quadratic relationship has a curved graph called a parabola. It may open upward or downward. If it opens downward, its highest point is called the maximum. If it opens upward, its lowest point is called the minimum. That turning point is useful when the situation asks for a greatest or least value.

Read the labels and units before interpreting a table or graph.
A quadratic graph is a parabola.
The turning point gives a maximum or minimum, depending on the direction the parabola opens.

2. Pose a question from quadratic data

To pose a question means to write a question that the available information can answer. Begin by identifying what the input and output represent. Then notice a feature of the table or graph that matters in the situation.

A quadratic table often shows outputs increasing and then decreasing, or decreasing and then increasing. The outputs may not change by the same amount each time. Do not assume that equal changes in input produce equal changes in output.

Useful questions include: When does the output reach a certain value? What is the greatest or least output? At what input does that extreme happen? In an application, say what the answer means and include units where they make sense.

A table gives exact values at its listed inputs. A graph can also help estimate an output or input between listed values, if the graph is drawn to a suitable scale. Mark an estimate as approximate rather than claiming it is exact.

Connect every question to the quantities and units in the situation.
A table answers questions at listed inputs; a graph can support estimates between them.
Distinguish an exact listed value from an estimate read from a graph.

3. Read the table, graph, and model together

A table, graph, and equation can describe the same quadratic relationship in different ways. The table gives selected input-output pairs. The graph makes the overall shape, turning point, and crossings easier to see. An equation can calculate values that are not listed, when an appropriate model is available.

A graph's vertical coordinate is the output for a chosen horizontal coordinate. To answer a question such as “When is the height 16 metres?”, look across from that height to the curve, then down to the time axis. The graph may show two times because a rising-and-falling quantity can reach the same height twice.

In a model, the symbol for the output is written in terms of the input. For example, a height model may use

h

for height and

t

for time. Substitute a time to calculate a height. To find when a specified height occurs, set the model equal to that height and solve for the time. Keep only solutions that make sense in the situation, such as times during the object's flight.

The equation should fit the information and the question. Do not extend a model beyond the situation without a reason. A negative time before a launch, for example, may solve an equation but may not answer a question about the flight after launch.

Use the table for listed values and the graph for overall shape or estimates.
A given output may occur at two different inputs.
Check calculated answers against the graph, table, and real-world context.

4. A clear problem-solving routine

First, name the input and output and record their units. Next, describe what the table or graph shows. Then pose a question that can be answered from that information. Choose the representation that best answers it: read a table entry, estimate from a graph, or use a suitable equation.

After calculating, state the result as a sentence about the situation. Include units and explain whether the result is exact or approximate. Finally, check that the answer is within a sensible range. If the table and graph disagree with your calculation, revisit the values, scale, or arithmetic.

Name quantities and units before calculating.
Choose a method suited to the question.
Interpret and check the result in context.

Ball height data

Time, $t$ (s)	Height, $h$ (m)
0	1
1	16
2	21
3	16
4	1

Worked example

A ball’s height during a practice throw

A ball is thrown upward from a platform. The table gives its height

h

, in metres, after

t

seconds. A graph of the same relationship is a downward-opening parabola. Pose and answer two questions: What is the ball’s greatest height, and when is it 16 metres high?

Read the situation
Time is the input and height is the output. The table shows the height rising and then falling, so it is reasonable to ask for the greatest height. It also shows that a height of 16 metres occurs at two listed times.
Use the table
The largest listed height is 21 metres, at 2 seconds. The height is 16 metres at both 1 second and 3 seconds. These are exact table entries, not estimates.
Use a model to check the pattern
The graph’s turning point is at 2 seconds and 21 metres. A matching quadratic model in vertex form is shown here. In this form, the squared part is zero at the turning point, so the model gives a height of 21 metres there. It also gives the table’s height of 16 metres one second before and after that time.
$h=-5(t-2)^2+21$
Answer the application questions
Setting the model to 16 gives two times. Both fit the flight described by the table and graph. The ball reaches 16 metres while rising and again while falling.
$-5(t-2)^2+21=16$

Answer: The ball’s greatest height is 21 metres, reached after 2 seconds. It is 16 metres high after 1 second and after 3 seconds.

Check: The table directly confirms all three heights. The graph’s turning point should be at

(2,21)

, and the horizontal line at height 16 should meet the curve at times 1 and 3.

Common mistakes and how to avoid them

Reporting only one time when a height occurs twice.

Correction: Check both sides of the turning point. A downward-opening parabola can have the same height once while rising and once while falling.

Calling a value estimated from a graph exact.

Correction: Use “about” or “approximately” when reading between marked values. Table entries are exact as given.

Giving a number without saying what it measures.

Correction: Write a sentence that identifies the quantity and includes its unit, such as “The ball reaches 21 metres after 2 seconds.”

Keeping a calculated input that does not fit the situation.

Correction: Check whether the input is possible in context. For a question about the flight after the throw, use times during that flight.

Lesson summary

A quadratic table pairs inputs and outputs, while its graph shows the overall curved pattern.
Use the turning point to identify a maximum or minimum.
A question about a given output may have two answers.
Use the table for listed values, the graph for visual information or estimates, and a suitable equation for calculation.
State answers with meaning and units, then check them against the situation.

Check your understanding

Question 1

A quadratic table lists an output of 12 at inputs 2 and 6, with a maximum output between them. Which statement best describes the data?

The output reaches 12 at two different inputs.
The output must be 12 for every input from 2 to 6.
The maximum output is 12.
The graph cannot be quadratic because an output repeats.

Show answer and explanation

The output reaches 12 at two different inputs.

The table gives the same output at two different inputs. A quadratic graph can reach a given output on both sides of its maximum.

Question 2

A graph-based estimate gives a time of 4.3 seconds. The graph has no marked point at that time. How should you report it?

The time is exactly 4.3 seconds.
The time is approximately 4.3 seconds.
The time must be 4 seconds.
The graph provides no information about time.

Show answer and explanation

The time is approximately 4.3 seconds.

A value read between marked points is an estimate. Report it as approximate unless exact data or calculation supports greater precision.

Key terms

Input: The quantity chosen or observed, often shown on the horizontal axis.
Output: The quantity that depends on the input, often shown on the vertical axis.
Quadratic relationship: A relationship whose graph is a parabola.
Parabola: The curved graph of a quadratic relationship.
Turning point: The point where a parabola changes direction; it is a maximum or minimum.
Model: A mathematical representation, such as a table, graph, or equation, used to describe a situation.

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About this lesson

Published by DoAssignment. This AI-assisted lesson follows Ontario Grade 11 Mathematics (MCF3M), expectation A1.1. It is a study resource, not an official curriculum publication.

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