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A1.2 · Interpret meaningful values on applied quadratic graphs

Learn to interpret meaningful values on applied quadratic graphs through clear examples and targeted practice.

Ontario Grade 11 Mathematics

Mathematical Models

Ontario Grade 11 MBF3C — A1.2

A curved graph can represent a real situation, such as the height of a ball over time or the profit from selling an item. Reading the curve is only the first step. You also need to connect its points to the situation and its units. A point on a graph is meaningful when it describes a possible value in that situation. For example, a graph may show a ball at a height of zero, but a negative time would not describe when the ball was in flight. In this lesson, you will practise reading important values and deciding what they mean in context.

What you will learn

Identify what the horizontal and vertical axes represent, including their units.
Interpret a vertex, intercepts, and other points in the context of an applied quadratic graph.
Decide which graph values make sense in the situation and explain why.
Use the graph's shape and labels to describe how a quantity changes.

1. Start with the axes and the situation

A graph has a horizontal axis and a vertical axis. The horizontal axis is usually the input, such as time or number of items. The vertical axis is usually the output, such as height or profit. Read the axis labels and units before interpreting the curve. A value of 4 has no clear meaning until you know whether it means 4 seconds, 4 metres, or 4 items.

A quadratic graph has a curved, U-shaped or upside-down U-shaped form. The turning point is called the vertex. It is the lowest point on a U-shaped graph or the highest point on an upside-down U-shaped graph. The vertex can describe a minimum or maximum value in an application.

The point where the graph crosses the vertical axis is the vertical intercept. It tells you the output when the input is zero. A horizontal intercept is a point where the output is zero. In an application, it may represent when a height reaches the ground or when a profit is zero. Interpret an intercept using the situation, not just its position on the graph.

Before reading specific values, ask: What does each axis measure? What units are used? What input values are included? What does the curve represent? These questions help prevent a mathematically readable value from being mistaken for a realistic one.

Read labels and units before reading points.
The vertex shows a maximum or minimum on the displayed curve.
An intercept describes a point where one coordinate is zero.

2. Connect graph features to real meaning

On a graph of height against time, a point gives the object's height at a particular time. If the vertex is a maximum, its coordinates give the time and height at the top of the object's path. A horizontal intercept may show when the object is at ground level. The vertical intercept gives its height at the starting time, if time zero is part of the model.

On a graph of profit against number of items sold, the vertex may show the greatest profit in the range shown. A horizontal intercept can show a break-even quantity: the number of items for which profit is zero. The vertical intercept may represent profit when no items are sold, if that value is included in the situation.

A graph may extend farther than the values that make sense in an application. For example, a time value cannot be negative when the clock begins at zero. A number of items sold is usually counted in whole items. A height below ground might not belong to a model that only describes the time an object is in the air. These restrictions describe the meaningful part of the graph, sometimes called the practical domain. The practical domain is the set of input values that fit the situation.

Do not assume that every visible part of a curve is useful. Check the story, the units, and the stated interval. If the graph shows only a selected time interval, do not claim it describes what happens outside that interval. If a value is estimated from a drawn graph, say it is approximate unless the graph gives an exact value.

Explain a coordinate using both quantities and their units.
Use the context to decide which inputs and outputs are realistic.
Distinguish a graph reading from a claim about values outside the shown interval.

3. Read points and describe change

A point is written as an ordered pair: the first coordinate is read from the horizontal axis, and the second from the vertical axis. In context, it is often clearer to say the input value first and the output value second, with units. For example, a point on a time-height graph might mean that at a certain time the object has a certain height.

You can also describe how the output changes as the input increases. On an upside-down U-shaped graph, the output rises toward the vertex and falls after it. On a U-shaped graph, it falls toward the vertex and rises after it. Use the graph and the situation to state what this means, such as a height increasing and then decreasing.

A scale matters. If vertical grid lines represent increments of 5 metres, a point halfway between two lines must be read using that scale. Read carefully rather than counting grid squares as if each one always represented one unit. When a graph is hand-drawn or an image is unclear, report a reasonable estimate and do not give more precision than the graph supports.

The same graph feature can have different meanings in different situations. A maximum height and a maximum profit are both maximum outputs, but their units and interpretations differ. Always name the quantity, not only the graph feature.

Read the horizontal coordinate first and the vertical coordinate second.
Describe rising and falling in terms of the quantities shown.
Use the marked scale and report estimates honestly.

4. Check whether an interpretation fits

A useful interpretation links a graph feature to the story in a complete statement. Include what is measured, the value, and the unit. For example, instead of saying, “The vertex is 8,” explain which coordinate is 8 and what it measures.

Check whether the reading is plausible. If an item-count graph gives a quantity between whole numbers, that may be a useful estimate for a graph, but the actual number of items sold must be a whole number. If an object is in flight, a negative time is not meaningful. If a graph's intercept is just beyond the interval shown, do not treat it as a directly read value.

Finally, check whether the graph supports the claim. A graph can show an approximate maximum, a trend, and values within its displayed range. It does not automatically prove that the same pattern continues forever. Accurate interpretation stays within the model and the information shown.

State the feature, its value, and its meaning with units.
Check realistic restrictions such as nonnegative time or whole-item counts.
Do not extend a graph's story beyond the information it provides.

Graph feature and possible meaning

Feature	What it shows	Example interpretation
Vertex	A maximum or minimum on the graph	The greatest height reached by a ball
Vertical intercept	Output when input is zero	Starting height at time zero
Horizontal intercept	Input value where output is zero	Time a ball reaches the ground
Other point	Output for a particular input	Profit at a stated number of tickets

Worked example

A ball's height over time

A graph of a ball's height has time in seconds on the horizontal axis and height in metres on the vertical axis. It is an upside-down U shape. The graph starts at the point with time 0 and height 1.5, reaches its vertex at time 2 and height 21, and crosses the horizontal axis at time 4. Interpret these values and state the meaningful time interval for the ball's flight.

Read the axes
The horizontal coordinate measures time in seconds, and the vertical coordinate measures height in metres. This tells us how to describe each coordinate pair.
Interpret the starting point
At time zero, the ball is at a height of 1.5 metres. This is the starting height shown by the graph, because zero seconds is the start of the recorded flight.
$(0,1.5)$
Interpret the vertex
The curve opens downward, so its vertex is the highest point on this graph. The ball reaches a maximum height of 21 metres at 2 seconds.
$(2,21)$
Interpret the horizontal intercept
At the horizontal intercept, the height is zero. The graph shows the ball at ground level at 4 seconds, so this is the end of the flight described by the graph.
$(4,0)$
Choose meaningful times
The flight starts at time zero and reaches the ground at 4 seconds. Negative times do not describe this flight, and times after landing are outside the flight interval being modelled.
$0\leq t\leq 4$

Answer: The ball starts 1.5 metres above the ground. It reaches a maximum height of 21 metres after 2 seconds and is at ground level after 4 seconds. The meaningful flight interval is from 0 to 4 seconds, including both endpoints.

Check: The maximum occurs halfway between the start and ground-contact times on the stated graph, and the height at ground contact is zero, as an intercept should show.

Worked example

Profit from selling tickets

A graph shows profit in dollars on the vertical axis and tickets sold on the horizontal axis. The displayed curve is an upside-down U shape. Its vertex is at 30 tickets and CAD 600. It crosses the horizontal axis at 10 and 50 tickets. Interpret the important values, and decide whether every input from 10 to 50 is a possible ticket count.

Interpret the vertex
Because the graph opens downward, the vertex gives the greatest profit on the displayed curve. The graph indicates a maximum profit of CAD 600 when 30 tickets are sold.
$(30,600)$
Interpret the intercepts
At each horizontal intercept, profit is zero. The graph therefore indicates break-even quantities of 10 tickets and 50 tickets. Break-even means that the profit shown is neither positive nor negative.
$(10,0),\ (50,0)$
Check which counts are possible
The graph's horizontal values from 10 to 50 describe where the curve lies between the two break-even points. But tickets are counted in whole numbers, so a value such as 12.5 tickets is not a possible sale count. The graph can show the overall pattern, while actual ticket counts must be whole numbers. n∈\{10,11,\ldots,50\}
Keep the interpretation within the model
The graph gives information for its displayed range. It supports the stated maximum and break-even readings, but it does not tell us what happens for ticket counts outside the range shown.

Answer: The graph shows a maximum profit of CAD 600 at 30 tickets and break-even values at 10 and 50 tickets. Actual ticket counts within that range are whole numbers, not fractional values.

Check: The vertical coordinate at each horizontal intercept is zero profit. Since 30 lies between 10 and 50, the stated vertex is between the two break-even quantities, consistent with the described curve.

Common mistakes and how to avoid them

Saying that a vertex has a value without naming which quantity it measures.

Correction: Give both coordinates and interpret each with its axis label and units.

Treating every part of the curve as realistic.

Correction: Use the situation to restrict the graph to meaningful inputs and outputs.

Reading a horizontal intercept as a time or quantity when it is actually an output value.

Correction: At a horizontal intercept, the vertical output is zero; the horizontal coordinate gives the input where that happens.

Reporting a very precise value from a rough graph.

Correction: Use an approximate value when the graph only supports an estimate.

Lesson summary

Read the axis labels, units, and scale before interpreting a graph.
Use the vertex to identify a maximum or minimum in context.
Use intercepts to interpret starting values or points where the output is zero.
Restrict your interpretation to values that fit the situation and the displayed graph.

Check your understanding

Question 1

A height-time graph has its vertex at (3, 18), with time in seconds and height in metres. What does the vertex mean if the graph opens downward?

The object reaches a maximum height of 18 metres after 3 seconds.
The object reaches a maximum height of 3 metres after 18 seconds.
The object is at ground level after 3 seconds.
The object starts at a height of 18 metres.

Show answer and explanation

The object reaches a maximum height of 18 metres after 3 seconds.

The horizontal coordinate is time and the vertical coordinate is height. A downward-opening graph has a maximum at its vertex.

Question 2

A profit graph crosses the horizontal axis at 12 items sold. What does that point tell you?

Profit is zero when 12 items are sold.
Twelve dollars are earned for each item.
Profit is 12 dollars when no items are sold.
The greatest profit is made at 12 items.

Show answer and explanation

Profit is zero when 12 items are sold.

At a horizontal intercept, the vertical output is zero. Here the output is profit, so the graph shows zero profit at 12 items.

Question 3

A graph describes an object's height from time zero until it lands at 5 seconds. Which time is outside the meaningful flight interval?

2 seconds
0 seconds
5 seconds
Negative 1 second

Show answer and explanation

Negative 1 second

The described flight runs from time zero through 5 seconds. A negative time is not part of this flight interval.

Key terms

Vertex: The turning point of a quadratic graph; it is the highest or lowest point on the curve.
Intercept: A point where a graph meets an axis. A horizontal intercept has output zero; a vertical intercept has input zero.
Practical domain: The input values that make sense for the situation being modelled.
Break-even: A situation in which the profit is zero.

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About this lesson

Published by DoAssignment. This AI-assisted lesson follows Ontario Grade 11 Mathematics (MBF3C), expectation A1.2. It is a study resource, not an official curriculum publication.

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