A2.3 Solve real-world problems involving quadratic functions

Ontario Grade 11 Mathematics

Characteristics of Functions

Using vertex form, factored form, and standard form to answer real questions about motion, area, and revenue

Quadratic functions appear in a surprising number of everyday situations — the height of a ball thrown into the air, the area of a rectangular garden with a fixed amount of fencing, or the profit a business earns as it adjusts its prices. In each case, the relationship between two quantities forms a smooth U-shaped curve called a parabola. Knowing how to read, build, and solve quadratic functions lets you answer practical questions such as "When does the ball hit the ground?" or "What price gives the most profit?" This lesson connects the algebra of quadratic functions to real decisions and real answers. Before diving in, you should be comfortable expanding and factoring simple quadratic expressions, and with completing the square to rewrite a quadratic in vertex form — both skills from Grade 10. Every new idea will be built on those foundations step by step.

What you will learn

Identify the key features of a quadratic function — vertex, zeros, and direction of opening — and explain what each feature means in context.
Choose the most useful algebraic form of a quadratic function (standard, vertex, or factored) for a given type of question.
Solve real-world problems that involve finding a maximum or minimum value, the zeros of a function, or the output at a specific input.
Interpret answers in the context of the problem, including deciding whether a negative or non-integer answer makes sense.

Prerequisite Bridge: Three Forms of a Quadratic Function

A quadratic function is any function that can be written as

f(x) = ax^2 + bx + c

where

a \neq 0

. The graph is a parabola. When

a > 0

the parabola opens upward, giving a minimum value. When

a < 0

it opens downward, giving a maximum value.

There are three algebraic forms, each useful for different questions. Standard form,

f(x) = ax^2 + bx + c

, is easiest to read the

y

-intercept from (

c

). Vertex form,

f(x) = a(x - h)^2 + k

, immediately gives the vertex

(h, k)

, which is where the maximum or minimum occurs. Factored form,

f(x) = a(x - r)(x - s)

, immediately gives the zeros

x = r

and

x = s

, which are where the function equals zero.

You choose the form that matches the question. Asking about a maximum height? Vertex form is your friend. Asking when something reaches zero? Factored form or the quadratic formula saves time. A quick review: the quadratic formula states that for

ax^2 + bx + c = 0

, the solutions are

x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}

.

x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}

Standard form: $f(x) = ax^2 + bx + c$ — read the $y$ -intercept directly as $c$ .
Vertex form: $f(x) = a(x-h)^2 + k$ — vertex is $(h, k)$ ; max or min value is $k$ .
Factored form: $f(x) = a(x-r)(x-s)$ — zeros are $x = r$ and $x = s$ .
Parabola opens upward when $a > 0$ (minimum) and downward when $a < 0$ (maximum).
Quadratic formula: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

Reading a Problem and Setting Up the Function

Most real-world quadratic problems give you information in words. Look for clues: a quantity that is squared or involves a product of two changing quantities is a strong hint that the relationship is quadratic. Common settings include projectile motion, area problems, and revenue problems.

Once you identify the function, decide what the question is asking. Questions like 'What is the maximum?' point you to the vertex. Questions like 'When does it return to the ground?' point you to the zeros. Questions like 'What is the height after 3 seconds?' ask you to evaluate the function at a given input.

Always define your variables carefully and state their units. After calculating a numerical answer, check whether it is realistic — a negative time or a height below ground level usually signals an error.

Identify variables and units before writing any equation.
Products of two changing quantities often produce quadratic relationships.
Match the question type to the right feature: vertex for max/min, zeros for when the output equals zero, evaluation for output at a specific input.
Always interpret your answer in the real-world context.

Choosing the Right Form

Goal	Algebraic Form
Find max/min	Vertex form
Find when object hits zero	Factored form
Find starting value	Standard form

Worked example

Example 1 — Projectile Motion

A student launches a model rocket from ground level. Its height in metres after

t

seconds is

h(t) = -5t^2 + 40t

. Find the maximum height and the time it lands.

Identify the vertex time
The maximum occurs at the vertex $t = -\frac{b}{2a}$ . Here $a = -5$ and $b = 40$ .
$t = -\frac{40}{2(-5)} = 4$
Calculate maximum height
Evaluate the function at the time found in the previous step.
$h(4) = -5(4)^2 + 40(4) = 80$
Find landing time
The rocket lands when $h(t) = 0$ . Factor out the common variable.
$-5t(t - 8) = 0$

Answer: Maximum height is 80 m at 4 s; it lands at 8 s.

Check: The vertex at 4 s is halfway between the zeros at 0 s and 8 s, confirming symmetry.

Worked example

Example 2 — Revenue Optimization

A product sells at CAD 5 for 200 units. Each CAD 1 increase causes 20 fewer sales. Find the price that maximizes revenue

R(x) = (5+x)(200-20x)

.

Expand the function
Multiply the binomials to reach standard form.
$R(x) = -20x^2 + 100x + 1000$
Locate vertex
Find the optimal price increase $x$ using the axis of symmetry.
$x = -\frac{100}{2(-20)} = 2.5$
Determine final price
The price is the original price plus the optimal increase.
$\text{Price} = 5 + 2.5 = 7.5$

Answer: The price that maximizes revenue is CAD 7.50.

Check: At

x=2.5

,

R(2.5) = 1125

. At

x=2

or

x=3

, revenue is 1120. The vertex is indeed the maximum.

Common mistakes and how to avoid them

Reporting the vertex

x

-coordinate as the maximum value.

Correction: The

x

-coordinate is the location; you must substitute this value into the function to find the maximum output value (

k

).

Keeping a negative time as a solution.

Correction: In most physical problems, time starts at zero, so discard any negative solutions.

Lesson summary

Quadratic functions can be expressed in standard, vertex, or factored form.
The vertex represents the maximum or minimum point of a parabola.
Zeros represent where the function crosses the horizontal axis.
Always define variables and include units in your final answer.
Verify your answers by testing nearby points or substituting back into the model.

Check your understanding

Question 1

For

h(t) = -4t^2 + 16t + 5

, what is the

t

-coordinate of the vertex?

-2
2
4
5

Show answer and explanation

2

Using

t = -b/2a

, we get

-16 / (2 * -4) = 2

.

Question 2

If a parabola opens upward, the vertex is:

A maximum
A minimum
A zero
A y-intercept

Show answer and explanation

A minimum

Upward opening parabolas look like a cup, so the vertex is the lowest point.

Key terms

Parabola: The curve formed by the graph of a quadratic function.
Vertex: The turning point of a parabola.
Zero: The input value where the output of the function is zero.

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About this lesson

Published by DoAssignment. This AI-assisted lesson follows Ontario Grade 11 Mathematics (MCR3U), expectation A2.3. It is a study resource, not an official curriculum publication.

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A2.3 · Solve real-world problems involving quadratic functions