Quadratic Relations of the Form y = ax² + bx + c · Quadratic Relations and Their Graphs

Using Parabola Features in Context: Maximum and Minimum Values

1.3 1.1About 70 minutes

Learning goals

I can explain what the vertex, intercepts and axis of symmetry mean in a real situation.
I can find a maximum or minimum value using the zeros and symmetry.
I can decide which parts of a quadratic model make sense in context.
I can answer a problem with a sentence and correct units.

Words to know

maximum height
The greatest height reached: the y-coordinate of the vertex of a parabola that opens down.

initial value
The value of the dependent variable when the independent variable is 0; on a graph, the vertical intercept.

break-even point
A point where profit is $0 (revenue equals cost): a zero of a profit model.

revenue
The money a business takes in: price × number sold.

constraint
A limit in a problem, such as a fixed amount of fencing or times that must be positive.

Highest, lowest, start and finish

When a basketball leaves your hands, three questions matter: how high does it go, when does it get there, and where does it land? Each answer is a key feature of a parabola. In this lesson you will use the vertex, the intercepts and symmetry to answer real questions, and you will think about which parts of a graph actually make sense in the real situation.

What each feature means in context

Suppose h is the height of a ball (in metres) t seconds after it is thrown.

The vertex tells you the maximum height (its y-coordinate) and when the ball reaches it (its x-coordinate).
The h-intercept (where t = 0) is the starting height.
A zero with t > 0 is the time the ball lands (height 0).
The axis of symmetry says the ball takes as long to go up as to come down (if it lands at the height it started from).

A ball thrown upward from the top of a 15 m platform: h = −5t² + 10t + 15. The solid part of the curve, from t = 0 to t = 3, is the real flight. The dashed part, before the throw, is not part of the situation.

Worked example

Example 1: A ball thrown from a platform

A ball is thrown upward from a 15 m platform. Its height in metres after t seconds is modelled by h = −5t² + 10t + 15. Use the graph above and the equation to find (a) the starting height, (b) the maximum height and when it happens, (c) when the ball hits the ground.

(a) Starting height: at t = 0, h = 15. The ball starts 15 m up.
(b) From the graph the vertex is (1, 20). Check: h = −5(1)² + 10(1) + 15 = −5 + 10 + 15 = 20. The maximum height is 20 m, reached 1 s after the throw.
(c) The graph meets the t-axis at t = 3 (and at t = −1). Check t = 3: −45 + 30 + 15 = 0. ✓
The zero t = −1 is a time before the ball was thrown, so it has no meaning here. Only t = 3 makes sense.
Notice the symmetry: the axis is t = (−1 + 3) ÷ 2 = 1, which is when the maximum happens.

(a) 15 m; (b) 20 m at t = 1 s; (c) at t = 3 s.

Does it make sense?

A quadratic model only describes a real situation for some values. Time cannot be negative before an event starts, lengths cannot be negative, and a ball stops when it hits the ground. Always ask which part of the parabola is meaningful before you answer.

Worked example

Example 2: The biggest dog run

Tasha has 40 m of fencing to make a rectangular dog run. If one side is w metres, the area is A = w(20 − w). Find the dimensions that give the largest area.

Why the formula: two widths and two lengths use 40 m, so w + length = 20 and the length is 20 − w. Area = width × length = w(20 − w).
Zeros: A = 0 when w = 0 or when 20 − w = 0, that is w = 20. (These are the "flat" rectangles with no area.)
Axis of symmetry: halfway between the zeros: w = (0 + 20) ÷ 2 = 10.
Maximum area: A = 10(20 − 10) = 10 × 10 = 100 m².
The length is 20 − 10 = 10 m, so the best shape is a 10 m by 10 m square. Only widths between 0 and 20 make sense.

A 10 m × 10 m square gives the largest area, 100 m².

Area of a rectangle with 40 m of fencing, A = w(20 − w). The points are symmetric about w = 10, where the area reaches its maximum of 100 m².

Worked example

Example 3: Break-even and maximum profit from a table

A school club sells snow-day hot chocolate kits. Its profit P (hundreds of dollars) depends on the price x (dollars): P = −2x² + 24x − 40.

x: 2, 3, 4, 5, 6, 7, 8, 9, 10

P: 0, 14, 24, 30, 32, 30, 24, 14, 0

Find the break-even prices, the best price and the maximum profit.

Break-even means profit = 0: the zeros. P = 0 at x = 2 and x = 10 (prices of $2 and $10).
The P-values are symmetric about x = 6 (30 and 30, 24 and 24, …), so the axis is x = 6. Check: (2 + 10) ÷ 2 = 6.
The vertex is (6, 32). Check: −2(36) + 24(6) − 40 = −72 + 144 − 40 = 32.
P is in hundreds of dollars, so 32 means $3 200.

Break-even at $2 and $10; the best price is $6, giving a maximum profit of $3 200.

Just for fun

A funny-but-true fact: on the Moon, gravity is about one sixth as strong as on Earth, so a ball thrown the same way follows a much wider, taller parabola. In 1971, astronaut Alan Shepard hit golf balls on the Moon.

Hands-on activity

The 40 cm string rectangle

You need: a piece of string tied into a loop whose total length is exactly 40 cm (tie it a little long, measure, and retie until it is right), four push pins or four fingers, a ruler, and a piece of cardboard or a cork board.

Stretch the loop into a rectangle around four pins. Make the width 2 cm, measure the length (it should be about 18 cm) and record the area.
Repeat for widths 4, 6, 8, 10, 12, 14, 16 and 18 cm.
Make a table of width and area and graph it. Draw a smooth curve of best fit.
Find the vertex. What shape gives the largest area? What are the zeros, and what rectangles do they represent?

What to look for: areas of about 36, 64, 84, 96, 100, 96, 84, 64 and 36 cm² (A = w(20 − w)). The graph is a parabola opening down with vertex (10, 100): a 10 cm by 10 cm square encloses the most area. The zeros 0 and 20 are "rectangles" squashed flat with no area. Small measuring errors are normal.

Your turn

A diver's height above the water is h = −5t² + 5t + 10 (h in metres, t in seconds). What is the height of the diving board?

For the diver, h = 0 at t = −1 and t = 2. When does the diver enter the water? Why do you reject the other value?

Find the diver's maximum height and when it happens.

With 60 m of fencing, a rectangle with width w has area A = w(30 − w). Find the maximum area.

A table of a cannonball's height gives h = 0 m at x = 0 m and x = 80 m (x = horizontal distance). How far away is it when it is highest?

Revenue R (thousands of dollars) for a concert depends on ticket price p: R = −p² + 50p. Find the price that maximizes revenue and the maximum revenue.

More practice: warm-up

In a height-time model, what does the vertex tell you?

In a height-time model, what does the h-intercept tell you?

Why might one zero of a height-time model be rejected?

What shape of rectangle gives the most area for a fixed perimeter?

A ball's height graph has zeros t = 0 and t = 4. When is it highest?

More practice: on your own

A rocket's height is h = −5t² + 40t (metres, seconds). Find the time in the air, the time of the maximum and the maximum height.

Use the rocket in the last question. At what other time is the height the same as at t = 1 s?

A farmer uses 100 m of fencing for three sides of a rectangular pen against a long barn wall. If each side touching the wall is x m, the area is A = x(100 − 2x). Find the zeros, then the maximum area.

A theatre's profit (dollars) is P = −10x² + 600x − 5 000, where x is the ticket price. The break-even prices are $10 and $50. Find the best price and the maximum profit.

A soccer ball's path is modelled with height in metres and horizontal distance in metres. The ball is kicked from the ground and lands 30 m away, reaching a height of 9 m. Name three points on the parabola.

Challenge

A ball is thrown from a 2 m height and lands at a height of 0 m. Explain why the time to go up to the maximum is not the same as the time to come back down to the ground.

For h = −5t² + 20t + 1, find the maximum height without a table. (Hint: find two times with the same height as t = 0.)

A rectangular poster must have a perimeter of 2.4 m. Write the area A as a relation of the width w, find the zeros, and find the largest possible area.

Check your understanding

Quiz

h = −5t² + 30t models a ball's height. What does the vertex represent?

A model of a diver's height has zeros t = −0.8 and t = 2.5. When does the diver enter the water?

A rectangle has a perimeter of 36 cm. Which dimensions give the maximum area?

A rocket's height is modelled by h = −5t² + 50t. What is its maximum height?

In a profit model, what do the zeros represent?

A ball reaches the same height at t = 0.5 s and t = 2.3 s. When is it highest?

For parents and teachers

Big idea: Expectation 1.3 includes describing key features with the right terms, and the strand's overall goals include solving problems with quadratic relations. This lesson connects each feature to its real meaning (maximum height, starting value, break-even) and asks students to decide which parts of a graph make sense.

How to help: The string rectangle is a quick, convincing experiment. When you see a ball thrown or a fountain, ask "Where would the vertex be? What does it mean?" Ask your teen to always finish with a sentence and units ("The ball reaches 20 m after 1 s").

Common difficulties:

Keeping a negative time or length as an answer. Ask: "Does that make sense here?"
Reporting only one coordinate of the vertex when the question asks for both the maximum and when it happens.
Forgetting units (hundreds of dollars, metres, seconds).
Using symmetry for points at different heights (see the first Challenge question).