What this move does
In real problems, “the most”, “the largest” and “the cheapest” can very often be turned into the maximum or minimum of a quadratic function.
- Choose the variable: let be the quantity that changes and that you control, and write down its range.
- Write the other quantities in terms of : sales, the other side, time…
- Write the objective function: profit, area or height, as an expression in .
- Find the max or min within the range, then go back to the problem and check: is the answer sensible? Does it have to be a whole number?
Predict first
The shop above: at a price of yuan the profit is yuan; at yuan it is yuan, the same. So what price gives the highest profit?
The profit is a quadratic function of the price, and its graph is a parabola that opens downward. Prices of and yuan give the same profit, so these two points are symmetric about the axis, which lies exactly halfway: yuan. Then each item earns yuan, are sold, and the profit is yuan.
Work through one: setting a price
Example: cost 40 yuan; at 60 yuan, 300 sold; each 1 yuan more sells 10 fewer
Step 1 / 5- Choose the variable
Let the price go up by yuan, so the price is yuan. Sales cannot be negative, so . (A price cut is fine too: just let be negative.)
A different scene: a fence
One problem, several solutions
Using meters of fence against a long enough wall, you fence off a rectangle. What is the largest possible area?
Complete the square on the quadratic
Let the sides perpendicular to the wall be meters (); the side parallel to the wall is meters.
The area is largest at : square meters. The side parallel to the wall is then meters.
CostThe standard method; still works when the range is restricted.
Fixed sum, largest product
The sum of and is always . When two numbers have a fixed sum, their product is largest when they are equal.
So , , and .
CostVery quick; only works when there are no extra restrictions.
When it fails
Break it
Counterexample lab: remove one condition
Conclusion: The largest area is the vertex's y-coordinate ← still true?
With a long enough wall, can be anything between and . The vertex is in range, and the largest area is .
Now the wall is only meters long. The side parallel to the wall, , cannot be longer than , so , and the vertex falls into the red no-go zone. Drag and find the largest area within the allowed range.
On the area decreases as increases, so the maximum is at the boundary: at , . The pricing problem works the same way. Suppose the price may not exceed yuan:
One more scene: throwing upward
A ball is thrown straight up at meters per second. After seconds its height above the ground is about (meters).
- : the ball reaches its highest point, meters, at seconds.
- Set : or , so it lands at seconds. Going up and coming down take seconds each: that is the symmetry of the parabola.
- Note that the horizontal axis of the right-hand graph is time. The curve is not the path of the ball.
Practice
An item costs yuan. At a price of yuan, are sold per day; for every yuan increase, fewer are sold per day. At what price (in yuan) is the daily profit largest?
A ball is thrown straight up; after seconds its height is (meters). How high, in meters, can the ball get?
Using meters of fence against a long enough wall, you fence off a rectangle and add one more fence across the middle, perpendicular to the wall, to split it into two pens. What is the largest possible total area, in square meters?
Using meters of fence against a wall, you fence off a rectangle (three sides of fence). The wall is only meters long. What is the largest possible area, in square meters?
A travel agency runs a tour: with people, each person pays yuan; for every extra person, the price for everyone drops by yuan. How many people in the group give the highest total income?