A. Quadratic Functions · Quadratic Functions

Domain and Range in Real-World Contexts

A2.4 A2.3About 65 minutes

Learning goals

I can explain why a real-world model may need a restricted domain and range.
I can find the domain and range of a quadratic model of height, area or revenue.
I can tell discrete situations from continuous ones and graph them correctly.
I can use the zeros and the vertex to set restrictions.

Words to know

restriction
A limit on the domain or range so that a model only uses values that make sense in its context.

continuous
A function whose inputs can take every value in an interval; its graph is unbroken.

discrete
A function whose inputs are separate values (such as whole numbers); its graph is separate points.

whole numbers (W)
0, 1, 2, 3, …

model
A function used to describe a real situation.

A model with limits

You throw a snowball up from a 25 m high rooftop. Its height after t seconds is h(t) = −5t² + 20t + 25 metres. As a pure function, h has domain {t ∈ ℝ}. But plug in t = 7 and the model says h(7) = −80: the ball is 80 m underground, two seconds after it has already landed.

The formula does not know when the story starts and stops. You do. In real-world problems we restrict the domain and range to the values that make sense.

Where restrictions come from

Ask these questions about every model:

When does the story start and end? Time usually starts at t = 0. A thrown object's story ends when it lands (height 0).
Can the quantity be negative or zero? Lengths, widths and areas must be positive. Heights above the ground are at least 0.
Is there a physical limit? You only have 40 m of fence; the bus only has 48 seats; the wall is only 10 m long.
Is the quantity counted or measured? People, tickets and T-shirts are counted in whole numbers. Time, length and temperature are measured and can take any value in an interval.

A function whose inputs can take every value in an interval is continuous. Its graph is a smooth unbroken curve. A function whose inputs are only separate values (such as whole numbers) is discrete. Its graph is a set of separate dots.

Worked example

Example 1: A snowball from a rooftop

h(t) = −5t² + 20t + 25 gives the height in metres of a snowball t seconds after it is thrown from a roof. State the domain and range that make sense.

Start: the throw is at t = 0, so t ≥ 0.
End: the ball lands when h(t) = 0. −5t² + 20t + 25 = 0. Divide by −5: t² − 4t − 5 = 0, so (t − 5)(t + 1) = 0 and t = 5 or t = −1.
Reject t = −1 (before the throw). The ball lands at t = 5 s. D = {t ∈ ℝ | 0 ≤ t ≤ 5}.
Highest point: the axis of symmetry is halfway between the zeros: t = (5 + (−1)) ÷ 2 = 2. h(2) = −20 + 40 + 25 = 45.
The height runs from 0 (landing) up to 45, and every height in between happens. R = {h ∈ ℝ | 0 ≤ h ≤ 45}.

D = {t ∈ ℝ | 0 ≤ t ≤ 5}; R = {h ∈ ℝ | 0 ≤ h ≤ 45}.

Only the solid part of the parabola describes the snowball, from the throw at (0, 25) to landing at (5, 0). The dashed parts are still part of the function h(t) = −5t² + 20t + 25, but they have no meaning in this story.

Worked example

Example 2: Fencing a pen

Maya has 40 m of fencing to make a rectangular pen for her rabbits. If the width is w metres, the length is 20 − w metres, and the area is A(w) = w(20 − w). State the domain and range.

The width must be positive: w > 0.
The length must be positive too: 20 − w > 0, so w < 20.
D = {w ∈ ℝ | 0 < w < 20}. (w = 0 or w = 20 would make a "rectangle" with no area.)
A(w) = w(20 − w) has zeros 0 and 20, so its vertex is at w = 10: A(10) = 10 × 10 = 100. It opens down, so 100 is the maximum.
Areas between 0 and 100 all happen, but 0 itself does not (w can't be 0 or 20). R = {A ∈ ℝ | 0 < A ≤ 100}.

D = {w ∈ ℝ | 0 < w < 20}; R = {A ∈ ℝ | 0 < A ≤ 100}. The largest pen is a 10 m by 10 m square.

Worked example

Example 3: A discrete function

At a team party, every person shakes hands once with every other person. With n people, the number of handshakes is H(n) = n(n − 1) ÷ 2, or H(n) = 0.5n² − 0.5n. The team has 8 people. State the domain and range.

People are counted, so n is a whole number. With 8 people on the team, n can be 0, 1, 2, …, 8.
D = {0, 1, 2, 3, 4, 5, 6, 7, 8}.
Evaluate: H(0) = 0, H(1) = 0, H(2) = 1, H(3) = 3, H(4) = 6, H(5) = 10, H(6) = 15, H(7) = 21, H(8) = 28.
R = {0, 1, 3, 6, 10, 15, 21, 28}. The graph is a set of 9 dots, not a curve: there is no such thing as 2.5 people.

D = {0, 1, 2, …, 8}; R = {0, 1, 3, 6, 10, 15, 21, 28}.

H(n) = n(n − 1) ÷ 2 is quadratic, but in this context n is a whole number of people, so the graph is separate dots. The dotted curve only shows the pattern the dots follow.

A checklist for every context

Name the variables and their units.
Find where the story starts and stops (often a zero of the function).
Decide if any quantity must be positive.
Check for physical limits.
Decide whether the variable is counted (discrete) or measured (continuous).
Find the maximum or minimum (the vertex) if it is inside the domain.

Just for fun

Plug t = −2 into the snowball model h(t) = −5t² + 20t + 25 and it says the snowball was 35 m underground two seconds before you threw it. That is why we restrict the domain: snowballs do not have secret tunnels.

Hands-on activity

The string-rectangle investigation

You need: a piece of string or yarn about 70 cm long, tape, a ruler, a sheet of paper, and a calculator. (Scissors to cut the string: take care.)

Tie or tape the string into a loop with a perimeter of 60 cm (measure carefully).
Stretch the loop around four fingers (yours and a helper's), or tape it to a table, to make a rectangle. Make widths of 5, 10, 15, 20 and 25 cm. For each, measure the length (it should be 30 − w) and compute the area.
Record w and A(w) = w(30 − w) in a table and plot the points.
Answer: what is the smallest width you could make? The largest? Write the domain in set notation.
Which width gives the largest area? Write the range.

What to look for: D = {w ∈ ℝ | 0 < w < 30}, because both the width and the length 30 − w must be positive. The largest area is a 15 cm by 15 cm square: A = 225 cm². R = {A ∈ ℝ | 0 < A ≤ 225}. Measured areas will be close to the formula, not exact.

Your turn

A stone is dropped from an 80 m cliff. Its height is h(t) = 80 − 5t² metres after t seconds. State the domain and range in this context.

A rectangle has a perimeter of 30 m, so A(w) = w(15 − w). State the domain and range.

Discrete or continuous? (a) the number of concert tickets sold (b) the temperature during a day (c) the cost of n pizzas

A school club can order at most 200 hoodies. P(n) is its profit for selling n hoodies. State the domain.

A ball is thrown so that h(t) = −5t² + 10t + 15 metres. State the domain and range.

For A(w) = w(15 − w), why can't w = 20 be in the domain?

More practice: warm-up

A model starts when a ball is thrown. Can t be negative?

Discrete or continuous: the number of students on a field trip?

Can the width of a real rectangle be 0?

h(t) = 20 − 5t². When does the object land?

A ball's height model is h(t) = −4.9t² + 8t + 1.5. What is h(0) and what does it mean?

More practice: on your own

A diver's height above the water is h(t) = −4.9t² + 4.9t + 10 metres. State the domain and range. Use the quadratic formula (from Quadratic Expressions and Equations) and round to two decimal places.

A pen is built against a barn using 50 m of fencing on three sides: A(x) = x(50 − 2x), where x is the side perpendicular to the barn. State the domain and range.

Use H(n) = n(n − 1) ÷ 2. How many handshakes are there in a class of 30? Explain why n = 30.5 is not in the domain.

A theatre sells 300 tickets at $20. For each $1 increase, it sells 10 fewer. Revenue is R(x) = (20 + x)(300 − 10x), where x is the number of $1 increases. State a sensible domain and the maximum revenue.

Explain why the range of a height function for a ball thrown from the ground and landing on the ground always starts at 0.

Challenge

A garden against a house is fenced on three sides with 24 m of fence. The side parallel to the house is 24 − 2x, where x is each perpendicular side, but the house wall is only 10 m long. Find the domain and range of A(x) = x(24 − 2x).

Describe a real situation that the function f(x) = x(10 − x) could model, and give its domain and range in that context.

A ball is thrown from 1.2 m and caught at the same height. Its height is h(t) = −4.9t² + 14.7t + 1.2 metres. State the domain and range. Why doesn't the range start at 0?

Check your understanding

Quiz

h(t) = −5t² + 20t + 25 models a thrown ball that lands at t = 5. What is the domain in context?

Which quantity is discrete?

In A(w) = w(20 − w) for a rectangle, why must 0 < w < 20?

What is the range of A(w) = w(10 − w) for a rectangle?

A ball thrown from the ground reaches 30 m and lands at t = 4 s. What is the range?

The graph of a discrete function looks like…

For parents and teachers

Big idea: expectation A2.4 asks students to explain restrictions on the domain and range of a quadratic function in contexts from real-world applications.

How to help: for any formula your teen meets, ask "When does this story start and stop? Can this quantity be negative? Is it counted or measured?" Encourage full sentences with units: "The ball is in the air from 0 s to 5 s."

Common difficulties:

Giving the mathematical domain {x ∈ ℝ} instead of the context domain.
Keeping a negative root (a negative time) as a landing time.
Using ≤ when an endpoint can't happen (a width of 0), or < when it can (the moment of landing).
Drawing a smooth curve for a discrete situation (people, tickets).
Forgetting the vertex when finding the range.