A. Quadratic Functions · Quadratic Expressions and Equations

Representing Situations with Quadratic Expressions

A1.2About 70 minutes

Learning goals

I can define a variable and draw a diagram for a situation.
I can write a quadratic expression for an area, a revenue or another quantity, and simplify it.
I can test my expression with a value I can check by hand, and say which values make sense.

Words to know

variable
A letter that stands for an unknown or changing quantity. Always say what it represents and its units.

revenue
The money taken in from sales: price × number sold.

cross-section
The shape you see when you cut straight through a solid, such as the end view of a gutter.

model
An expression, equation or graph that describes a real situation.

restriction
A limit on the values of a variable that make sense in a situation, such as x ≥ 0 for a length.

From words to algebra

A quadratic expression is often the best way to describe a real quantity that depends on a choice you can make: how wide to make a frame, how much to raise a price, how far to fold a sheet of metal. In this lesson you will build those expressions yourself. Writing the expression is the hardest part of most word problems. Once you have it, expanding, factoring and solving (the rest of this unit) are routine.

A four-step recipe

Name the variable. Say exactly what x stands for, with units: "Let x be the width of the frame in centimetres."
Draw and label a diagram, or make a small table of a few cases.
Write each needed quantity in terms of x (lengths, number sold, price, and so on).
Combine them into one expression, then expand and simplify.

Then test your expression with a number you can check by hand.

A 30 cm by 20 cm photo in a frame of width x. Each outer side is the photo side plus x at both ends, so the outer dimensions are 30 + 2x and 20 + 2x.

Worked example

Example 1: The picture frame

A 30 cm by 20 cm photo has a frame of width x cm all around it. Write simplified expressions for (a) the total area of photo and frame and (b) the area of the frame alone. (c) Test your answer for x = 3.

Let x be the frame width in cm. From the diagram, the outer rectangle is (30 + 2x) by (20 + 2x). A common mistake is 30 + x: the frame is on both sides.
(a) Total area = (30 + 2x)(20 + 2x) = 600 + 60x + 40x + 4x² = 4x² + 100x + 600.
(b) Frame area = total area − photo area = 4x² + 100x + 600 − 600 = 4x² + 100x.
(c) If x = 3: the outer rectangle is 36 by 26, area 936; the frame is 936 − 600 = 336 cm². The expression gives 4(9) + 100(3) = 36 + 300 = 336 ✓.

(a) 4x² + 100x + 600 cm² (b) 4x² + 100x cm²

Worked example

Example 2: Raising a price

A canoe outfitter in Algonquin Park rents 120 canoes a day at $30 each. A survey suggests that for each $2 increase in price, 5 fewer canoes are rented. Write an expression for the daily revenue if the price is raised x times.

Let x be the number of $2 increases.
Make a small table: x = 0: price $30, 120 rented. x = 1: $32, 115 rented. x = 2: $34, 110 rented.
So price = 30 + 2x and number rented = 120 − 5x.
Revenue = price × number = (30 + 2x)(120 − 5x).
Expand: 3600 − 150x + 240x − 10x² = −10x² + 90x + 3600.
Test x = 3: price $36, 105 rented, revenue 36 × 105 = $3780. The expression: −90 + 270 + 3600 = 3780 ✓.

R = (30 + 2x)(120 − 5x) = −10x² + 90x + 3600 dollars

Revenue problems

Revenue = (price) × (number sold). Write each factor separately first, using a short table of x = 0, 1, 2. The negative x² term appears because one factor goes up while the other goes down, which is why revenue graphs are parabolas that open down (like the hoodie graph in the first lesson).

Folding up x cm on each side of a 30 cm sheet leaves a base of 30 − 2x. The cross-section is a rectangle with area x(30 − 2x).

Worked example

Example 3: A rain gutter

A 30 cm wide strip of aluminium is bent up x cm on each side to make a rain gutter with a rectangular cross-section. (a) Write the cross-sectional area as a simplified expression. (b) Make a table for x = 3, 6, 7.5, 9, 12. (c) Which fold gives the largest area?

(a) The two sides are x cm high and the base is 30 − 2x cm wide. A = x(30 − 2x) = 30x − 2x² cm².
(b) x = 3: 3(24) = 72. x = 6: 6(18) = 108. x = 7.5: 7.5(15) = 112.5. x = 9: 9(12) = 108. x = 12: 12(6) = 72.
(c) The table is symmetric about x = 7.5, so the largest area is 112.5 cm² with a fold of 7.5 cm. (You will meet ways to find maximums directly in the Quadratic Functions unit.)

A = 30x − 2x² cm²; maximum 112.5 cm² when x = 7.5 cm

The gutter area A = 30x − 2x². The model only makes sense for 0 < x < 15 (the base would vanish at x = 15). The best fold is in the middle of that interval.

Just for fun

Algebra is the only place where you can say "Let x be the frame width" and the frame width just has to go along with it.

Hands-on activity

Model a real frame or border

You need: a measuring tape or ruler, and any rectangular object with a border around it: a framed picture, a window with its frame, a placemat with a border, or a book cover on a table.

Measure the inner rectangle (the picture or glass). Call its dimensions L and W.
Let x be the border width. Write the total area (L + 2x)(W + 2x) and the border area, both expanded and simplified, using your real L and W.
Measure the real border width and substitute it into your border-area expression.
Check by measuring the outer rectangle and subtracting the inner area.

What to look for: both methods should agree (within measuring error of a few square centimetres). If your border is not the same width on every side, say so and explain how that changes the model.

Your turn

A square has side length x + 3. Write its area as a simplified expression.

A rectangle is 2x + 1 long and x − 2 wide. Write its area in standard form.

A garden 8 m by 5 m has a path of width x m all around it. Write the area of the path.

Write the product of two consecutive even integers if the smaller is n.

A triangle has base 2x + 6 and height x. Write its area in simplified form.

A theatre sells 300 tickets at $12. For each $1 increase, 20 fewer tickets sell. Write the revenue after x increases, in standard form.

Four squares of side x are cut from the corners of a 10 cm by 10 cm card. Write the remaining area.

More practice: warm-up

Write the area of a square with side 2x.

Write the area of a rectangle x by x + 5.

Write the perimeter of the same rectangle. Is it quadratic?

Write "a number times 4 more than the number" as an expression.

A circle has radius r + 1. Write its area, expanded.

More practice: on your own

A 15 cm by 10 cm photo has a frame of width x. Write the frame's area.

A shop rents 50 bikes a day at $40. Each $5 increase means 2 fewer rentals. Write the revenue after x increases in standard form.

A farmer has 60 m of fence for a rectangular pen against a barn (no fence on the barn side). The two sides touching the barn are x m long. Write the area, and find it when x = 15.

A 12 m by 9 m lawn is enlarged by adding x m to both its length and its width. Write the increase in area.

Write the sum of the squares of two consecutive integers, n and n + 1, in simplified form.

Challenge

For the 30 cm by 20 cm photo, the frame's area must equal the photo's area. Write this as an equation in standard form, and check that x = 5 cm works.

A 40 cm sheet is folded into a gutter with sides x cm high. Write the area, find its zeros, and use symmetry to find the best x.

In the canoe problem, R = (30 + 2x)(120 − 5x). For which values of x does the model make sense if the outfitter only raises prices? Explain.

Check your understanding

Quiz

A 10 cm by 6 cm card has a border of width x all around it. The outer dimensions are:

Revenue equals:

Tickets cost $8 and 200 are sold. Each $1 increase loses 10 sales. After x increases, revenue is:

Which quantity is quadratic in x for a rectangle x by x + 3?

A 20 cm sheet is folded up x cm on each side. The base of the gutter is:

What is the first step of the four-step recipe?

For parents and teachers

Big idea (A1.2, representing situations): students turn a description into a quadratic expression: area of a frame or border, revenue after price changes, cross-sections. This is the step most students find hardest in word problems.

How to help: ask your teen to say out loud what x stands for, and to test their expression with one easy number (like x = 1 or x = 3) worked out by hand from the picture.

Common difficulties:

Adding x instead of 2x for a border that goes all the way around.
In revenue problems, mixing up which factor increases and which decreases. A small table for x = 0, 1, 2 makes the pattern obvious.
Forgetting to subtract the inner area when only the border is wanted.
Not thinking about which x-values make sense (negative widths, negative numbers sold).