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Q9 · Solve real problems by interpreting quadratic graphs

Learn to solve real problems by interpreting quadratic graphs through clear examples and targeted practice.

Ontario Grade 10 Mathematics

Quadratic Relations

MFM2P · Study Topic Q9

You have already sketched and plotted quadratic relationships in this course. Now the focus shifts: instead of drawing the graph, you are given one — or you generate one with technology — and your job is to read it carefully and use what you see to answer real questions. A ball thrown into the air, a profit that rises and then falls, a diver dropping off a platform: all of these situations produce a curve called a parabola. In this lesson you will learn exactly which points on that curve matter and why, so you can answer problem questions with confidence.

What you will learn

Identify and name key features of a quadratic graph (vertex, zeros, axis of symmetry, y-intercept) using correct vocabulary.
Connect each graph feature to its meaning in a real-world context.
Answer questions about a real situation by reading values directly from a quadratic graph.
Decide whether a value read from a graph is a maximum, a minimum, or a crossing point, and explain what that means in the problem.

Grade 9 Bridge: What Makes a Relationship Quadratic?

In Grade 9 you studied linear relationships, where the graph is a straight line and the equation looks like

y = mx + b

. A quadratic relationship is different: the highest power of the variable is

2

, so the equation looks like

y = ax^2 + bx + c

. The graph of a quadratic is a smooth, symmetrical curve called a parabola.

When

a > 0

the parabola opens upward, like a bowl, and has a lowest point. When

a < 0

the parabola opens downward, like an arch, and has a highest point. Most real-life quadratic situations — thrown objects, profit models, area problems — produce a downward-opening parabola because the quantity rises and then falls.

You do not need to work with the equation in this topic. You only need to read the graph. Think of the graph as a picture that stores all the answers; your job is to know which feature of the picture answers each question.

A quadratic graph is a parabola — a smooth, U-shaped or arch-shaped curve.
Downward-opening means a maximum (highest point); upward-opening means a minimum (lowest point).
Every answer in this topic comes from reading the graph, not from algebra.

Key Features of a Parabola and What They Mean

Four features appear on almost every quadratic graph problem. Learn each name and its meaning before moving on.

The vertex is the turning point of the parabola — the single point where the curve changes direction. If the parabola opens downward, the vertex is the highest point and gives the maximum value of the output. If it opens upward, the vertex is the lowest point and gives the minimum value. The vertex is written as an ordered pair

(x, y)

. The

x

-coordinate tells you when the maximum or minimum occurs; the

y

-coordinate tells you what that maximum or minimum value is.

The zeros (also called x-intercepts or roots) are the points where the parabola crosses the

x

-axis, meaning the output

y = 0

. In a real problem, zeros often mark the start and end of an event — for example, the moment a ball leaves the ground and the moment it lands. A parabola can have two zeros, one zero, or no zeros.

The axis of symmetry is an invisible vertical line through the vertex that cuts the parabola into two mirror-image halves. Its equation is

x =

(the

x

-coordinate of the vertex). This is useful when you need to find the vertex from the zeros: the axis of symmetry sits exactly halfway between the two zeros.

The y-intercept is where the parabola crosses the

y

-axis, at

x = 0

. It gives the starting value of the output — for instance, the height of an object at launch, or profit before any items are sold.

x_{\text{axis}} = \frac{x_1 + x_2}{2}

Vertex $(x, y)$ : the maximum or minimum point; read both coordinates.
Zeros: where $y = 0$ ; mark the beginning or end of the real event.
Axis of symmetry: vertical line $x =$ (vertex $x$ -value); halfway between the zeros.
y-intercept: the output value when $x = 0$ ; often the starting condition.
Every feature has a real-world meaning — always state what the numbers represent.

Connecting Graph Features to Problem Questions

Different problem questions point you to different features. Matching the question to the right feature is the core skill of this topic. The table in this lesson lists common question types and the feature that answers each one.

When a question asks 'What is the maximum height?' or 'What is the greatest profit?', look at the vertex. Read the

y

-coordinate of the vertex — that is your answer. Also note the

x

-coordinate: it tells you when that maximum occurs.

When a question asks 'When does the object hit the ground?' or 'How long until profit reaches zero?', look at the zeros. Identify the zero with the larger

x

-value (since time usually runs forward), and read its

x

-coordinate.

When a question asks 'What is the height at launch?' or 'What is profit before any sales?', look at the

y

-intercept. Read the

y

-coordinate where the curve crosses the vertical axis.

Some questions ask you to estimate an output for a given input, or find an input for a given output. For these, trace along the graph: move vertically from the

x

-axis to the curve to read

y

, or move horizontally from the

y

-axis to the curve to read

x

. Always check whether your reading makes sense in the real context.

Maximum or minimum value → read the vertex's $y$ -coordinate.
When does the max/min occur → read the vertex's $x$ -coordinate.
When does the event start or end → read the zero(s).
Starting value (at $x = 0$ ) → read the $y$ -intercept.
Output for a given input → trace vertically from the $x$ -axis to the curve.

Domain and Reasonableness in Context

A parabola extends infinitely in both directions on a pure math graph, but real situations have limits. The meaningful part of the graph is called the domain in context — the set of

x

-values that make sense for the problem.

For example, if

x

represents time in seconds after a ball is thrown, then negative

x

-values do not make physical sense. The meaningful domain starts at

x = 0

(launch) and ends at the zero where the ball lands. Always restrict your graph reading to the part that fits the real scenario.

Similarly, check whether your answers are reasonable. A height of

-3

m before the ball lands is a sign that you have read the graph in a region outside the real context. A maximum of

45

m for a ball thrown by hand is likely unreasonable — re-examine your reading.

Only read from the section of the graph that makes sense in the real problem.
Time and distance are usually non-negative, so start from $x = 0$ .
Always check that your answer is physically or contextually reasonable.

Using Technology to Generate the Graph

In MFM2P you are allowed — and encouraged — to use graphing technology (such as Desmos or a graphing calculator) when a quadratic equation is given. Enter the equation, view the graph, and then read the features directly from the screen.

Most graphing tools have a trace or point feature that lets you click on the vertex or intercepts and read the coordinates precisely. Use these tools to get accurate readings rather than guessing from a sketch.

Whether the graph is given to you on paper or generated by technology, the reading skills are identical: locate the feature you need, read both coordinates carefully, and connect the numbers to the real-world meaning.

Enter the quadratic equation into technology to produce a precise graph.
Use the trace or intersection feature to read exact coordinates.
Graph-reading skills are the same regardless of how the graph was produced.

Matching Problem Questions to Graph Features

Problem Question	Feature to Read	What to Record
What is the maximum (or minimum) value?	Vertex	$y$ -coordinate of the vertex
When does the max/min occur?	Vertex	$x$ -coordinate of the vertex
When does the event end (or start)?	Zeros (x-intercepts)	$x$ -coordinate(s) where $y = 0$
What is the starting value?	y-intercept	$y$ -coordinate where $x = 0$
What is the output for a given input?	Trace the curve	Read $y$ directly above the given $x$ -value

Worked example

Example 1 — Ball Launched from a Roof

A ball is launched upward from the edge of a flat roof. The graph of its height

h

(in metres) above the ground versus time

t

(in seconds) is a downward-opening parabola. From the graph, the vertex is at

(3, 49)

, the

y

-intercept is at

(0, 4)

, and the parabola crosses the

t

-axis at

t = -0.08

and

t = 6.08

(approximately). Answer each question using the graph. (a) What is the maximum height of the ball, and when does it occur? (b) What was the height of the ball at the moment it was launched? (c) How long is the ball in the air after launch?

Identify the feature needed for part (a)
The question asks for the maximum height and when it occurs. The maximum of a downward-opening parabola is at the vertex. The vertex is $(3, 49)$ , so the $t$ -coordinate is $3$ and the $h$ -coordinate is $49$ .
$(t, h) = (3, 49)$
State the answer to part (a) in context
The $h$ -coordinate of the vertex is $49$ , so the maximum height is $49$ m. The $t$ -coordinate is $3$ , so this maximum occurs $3$ seconds after launch.
Identify the feature needed for part (b)
The height at launch means the height when time equals zero — that is the $y$ -intercept. Reading the graph, the $y$ -intercept is at $(0, 4)$ , so the launch height is $4$ m above the ground.
$(t, h) = (0, 4)$
State the answer to part (b) in context
The ball started at a height of $4$ m above the ground. This makes sense: the roof is $4$ m above ground level.
Identify the feature needed for part (c)
The ball is in the air from launch ( $t = 0$ ) until it hits the ground ( $h = 0$ ). The ground corresponds to $h = 0$ , which is a zero of the graph. The graph shows two zeros: one at approximately $t = -0.08$ (before launch, outside the real context) and one at approximately $t = 6.08$ . Because time starts at launch ( $t = 0$ ), only the zero at $t \approx 6.08$ is meaningful.
$t \approx 6.08$
State the answer to part (c) in context
The ball is in the air for approximately $6.08$ seconds after it is launched. The negative zero is outside the real domain and is ignored.

Answer: (a) Maximum height:

49

m, reached at

t = 3

s. (b) Launch height:

4

m above the ground. (c) Time in air: approximately

6.08

Check: Check (a): The vertex is the highest point on a downward parabola — confirmed by the graph. Check (b): At

t = 0

the curve crosses the

h

-axis at

4

— confirmed. Check (c): The positive zero is at

t \approx 6.08

; the ball lands then. Symmetry check — the axis of symmetry is

t = 3

, which is halfway between

-0.08

and

6.08

(-0.08 + 6.08) \div 2 = 6 \div 2 = 3

✓

Worked example

Example 2 — Fundraiser Ticket Profit

A school club sells tickets to a fundraiser. The profit

P

(in CAD) depends on the ticket price

p

(in CAD). A graphing tool produces the parabola for this relationship. From the graph, profit equals zero at

p = 2

and

p = 22

, and the vertex is at

(12, 720)

. Use these graph features to answer the following. (a) What ticket price gives the greatest profit, and what is that profit? (b) Between which two ticket prices is the club actually making a profit (i.e.,

P > 0

)? (c) Estimate the profit when the ticket price is CAD 17.

Find the axis of symmetry from the two zeros
The zeros occur at $p = 2$ and $p = 22$ . The axis of symmetry sits exactly halfway between them. Add the two zero values and divide by $2$ to find the midpoint.
$p_{\text{axis}} = \frac{2 + 22}{2} = 12$
Confirm the vertex location
The vertex lies on the axis of symmetry, so its $p$ -coordinate must equal $12$ . The problem states the vertex is at $(12, 720)$ , which matches. The $P$ -coordinate of the vertex is $720$ .
$(p, P) = (12, 720)$
Answer part (a)
The vertex is the highest point of the downward-opening parabola, so it gives the maximum profit. A ticket price of CAD 12 produces the greatest profit of CAD 720.
Answer part (b) using the zeros
Profit is positive ( $P > 0$ ) wherever the parabola sits above the $p$ -axis. The parabola is above the axis between its two zeros. The zeros are at $p = 2$ and $p = 22$ , so the club earns a positive profit for ticket prices strictly between those two values.
$2 < p < 22$
Estimate profit at a ticket price of CAD 17 using graph symmetry
The axis of symmetry is at $p = 12$ . The value $p = 17$ is $5$ units to the right of the axis. By symmetry, the point $5$ units to the left of the axis is $p = 7$ , and both points share the same profit value. Reading the graph at either $p = 7$ or $p = 17$ gives approximately $P = 495$ . This value is below the vertex ( $720$ ) and well above zero, which is consistent with the curve's shape between the vertex and the zero at $p = 22$ .
$P \approx 495$

Answer: (a) Greatest profit: CAD 720 at a ticket price of CAD 12. (b) Profit is positive for ticket prices between CAD 2 and CAD 22. (c) Profit at a ticket price of CAD 17 is approximately CAD 495.

Check: Check axis of symmetry:

(2 + 22) \div 2 = 12

✓ matches the vertex

p

-coordinate. Check symmetry for part (c):

p = 7

and

p = 17

are each

5

units from the axis, so they share the same profit — graph readings at both should agree. Reasonableness: CAD 495 is between

0

(zero at

p = 22

) and

720

(vertex at

p = 12

), which is consistent with the curve's shape ✓

Common mistakes and how to avoid them

Reading the vertex as just one number (e.g., only the maximum value) and forgetting to also report when it occurs.

Correction: The vertex is an ordered pair

(x, y)

. Always read and report both coordinates, then connect each to its real-world meaning.

Using a zero that falls outside the real domain (e.g., a negative time value) as the answer.

Correction: Check whether each zero makes sense in the context. Negative time usually has no meaning; use only the zero inside the valid domain.

Confusing the y-intercept with the vertex because both feel like 'important starting points'.

Correction: The y-intercept is always at

x = 0

(a fixed location on the vertical axis). The vertex is the turning point, which is rarely at

x = 0

in real problems.

Describing the axis of symmetry as a point rather than a vertical line.

Correction: The axis of symmetry is a line, written as

x =

(some value). It passes through the vertex but is not the vertex itself.

Forgetting to state units or context when giving an answer (e.g., writing '49' instead of '49 metres').

Correction: Every answer from a real-world graph must include units and a brief statement of meaning, such as 'the ball reaches a maximum height of 49 m'.

Lesson summary

A parabola's vertex gives the maximum value (downward-opening) or minimum value (upward-opening); read both coordinates and state their real-world meaning.
The zeros of the graph show where the output equals zero — often marking the start or end of a real event such as landing or zero profit.
The y-intercept shows the output at $x = 0$ , which is usually the starting condition of the problem.
The axis of symmetry is the vertical line $x =$ (vertex $x$ -value) and lies exactly halfway between the two zeros.
Restrict your graph reading to the domain that makes sense in the problem; ignore features outside the real context.
Always attach units and a plain-language interpretation to every numerical answer you read from the graph.

Check your understanding

Question 1

A graph shows the height of a ball over time. The vertex of the parabola is at

(4, 36)

. What does the number

36

represent?

The time when the ball is highest
The maximum height of the ball in metres
The time when the ball hits the ground
The height from which the ball was launched

Show answer and explanation

The maximum height of the ball in metres

The vertex is

(4, 36)

. The first coordinate,

4

, is the time (in seconds) when the maximum occurs. The second coordinate,

36

, is the output value — the maximum height in metres. So

36

represents the maximum height.

Question 2

A parabola crosses the x-axis at

x = 1

and

x = 9

. What is the x-coordinate of the vertex?

$x = 4$
$x = 5$
$x = 8$
$x = 10$

Show answer and explanation

x = 5

The vertex lies on the axis of symmetry, which is exactly halfway between the two zeros. Calculate

(1 + 9) \div 2 = 10 \div 2 = 5

. So the vertex is at

x = 5

Question 3

A profit graph (profit in CAD vs. number of items sold) is a downward-opening parabola. The graph crosses the horizontal axis at

n = 0

and

n = 80

. What does the zero at

n = 80

tell you?

The club makes its greatest profit when 80 items are sold.
The club breaks even (profit equals zero) when 80 items are sold.
The club's profit is at its lowest when 80 items are sold.
The club sells 80 items before making any profit.

Show answer and explanation

The club breaks even (profit equals zero) when 80 items are sold.

A zero of the graph is where the output equals zero. Here the output is profit, so profit equals zero (the club neither gains nor loses money) when

n = 80

items are sold. This is the break-even point, not the maximum profit.

Question 4

You are reading a quadratic graph that shows height versus time for a diver. The y-intercept of the graph is at

(0, 10)

. What does this tell you?

The diver reaches a maximum height of 10 m.
The diver is in the air for 10 seconds.
The diver's starting height above the water is 10 m.
The diver hits the water after 10 seconds.

Show answer and explanation

The diver's starting height above the water is 10 m.

The y-intercept occurs at

t = 0

, the starting moment. The output value is

10

, so the diver's height at

t = 0

10

m. This is the height of the platform above the water — the starting height, not the maximum or the landing time.

Key terms

Parabola: The smooth, symmetrical U-shaped (or arch-shaped) curve that is the graph of a quadratic relationship.
Vertex: The turning point of a parabola — the highest point if the parabola opens downward, or the lowest point if it opens upward. Written as an ordered pair $(x, y)$ .
Zero (x-intercept): A point where the parabola crosses the x-axis; the output value is zero at this point. Also called a root or x-intercept.
Axis of symmetry: The vertical line that passes through the vertex and divides the parabola into two mirror-image halves. Written as $x =$ (some value).
y-intercept: The point where the parabola crosses the y-axis, occurring when the input $x = 0$ . It usually represents the starting value in a real problem.
Domain in context: The set of input values that make sense for a real situation — for example, only non-negative time values for a problem about a moving object.
Maximum value: The greatest output value on the graph, found at the vertex of a downward-opening parabola.
Minimum value: The smallest output value on the graph, found at the vertex of an upward-opening parabola.

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Published by DoAssignment. This AI-assisted lesson follows Ontario Grade 10 Mathematics (MFM2P), study topic Q9. It is a study resource, not an official curriculum publication.

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