"When to Get the Maximum Profit" Quadratic Function PPT Courseware 5

"When to Get the Maximum Profit" Quadratic Function PPT Courseware 5

Endless aftertaste

1. The graph of the quadratic function y=a(x-h)²+k is a parabola, its axis of symmetry is the straight line x=h, and the vertex coordinates are (h, k).

2. The graph of the quadratic function y=ax²+bx+c is a parabola, its axis of symmetry is the straight line x=-b/2a, and the vertex coordinates are [-b/2a,4ac-b²/4a&# 093;. When a>0, the parabola opens upward, has a lowest point, and the function has a minimum value, which is 4ac-b²/4a; when a<0, the parabola opens downward, has a highest point, and the function has a maximum value, which is 4ac-b²/4a.

3. The symmetry axis of the quadratic function y=2(x-3)²+5 is the straight line x=3, and the vertex coordinate is (3, 5). When x=3, the minimum value of y is 5.

4. The symmetry axis of the quadratic function y=-3(x+4)²-1 is the straight line x=-4, and the vertex coordinates are (-4, -1). When x=-4, the function has a maximum value, which is -1.

5. The symmetry axis of the quadratic function y=2x²-8x+9 is the straight line x=2, and the vertex coordinate is (2, 1). When x=2, the function has a minimum value, which is 1.

Mr. Yang from a large shopping mall went to inspect the T-shirt department and learned the following: It is known that the unit price when purchased in batches is 20 yuan. According to market research, sales volume and sales unit price satisfy the following relationship: within a period of time, when the unit price is 35 yuan, the sales volume is 600 pieces, and every time the unit price decreases by 1 yuan, 200 more pieces can be sold. So Mr. Yang gave Manager Wang of the department a task to immediately formulate the most profitable sales plan. Did this stump Manager Wang? Can you help him solve this problem?

Manager Wang’s confusion: How to make more profits?

Manager Wang sells T-shirts, and the unit price when purchased was 20 yuan. Market research found that within a period of time, when the unit price was 35 yuan, the sales volume was 600 units; and for every unit price reduced by 1 yuan, 200 more units could be sold.

Manager Wang wants to know:

1. If the price drops and sales volume increases, will the total profit increase or decrease?

2. How much can you reduce the price to get the maximum profit?

Summarize :

Use functions to determine pricing problems:

Construct a quadratic function model: Convert the problem into a specific expression of a quadratic function.

Find the maximum (or minimum) value of a quadratic function

Discuss

There are 100 orange trees in an orchard, and each tree bears an average of 600 oranges. Now we are planning to plant more orange trees to increase production, but if we plant more trees, the distance between the trees and the sunlight received by each tree will It will decrease. According to empirical estimates, every time a tree is planted, each tree will bear 5 fewer oranges on average. How many more orange trees can be planted to maximize the total output of oranges?

Equivalent relationship: The total output of oranges = the output of each orange tree × the number of orange trees

y=(100+x)(600-5x) =-5x²+100x+60000=-5(x-10)²+60500

∵a<0 ∴ y has a maximum value

Summary:

General steps for finding the maximum and minimum values ​​of practical problems using the properties of quadratic functions:

Find the analytical expression of the function and the value range of the independent variable

Transform the formula, or use formulas to find its maximum or minimum value.

The value of the independent variable corresponding to the maximum or minimum value obtained by the check must be within the value range of the independent variable.

Practice in class

A store purchases a batch of daily necessities with a unit price of 20 yuan. If sold at a unit price of 30 yuan, 400 units can be sold within half a month. According to sales experience, increasing the unit price will lead to a reduction in sales volume, that is, every increase in the unit price of 1 Yuan, the sales volume is correspondingly reduced by 20 pieces. How to increase the selling price to obtain the maximum profit within half a month?

Solution: Assume that the sales unit price is x (x≥30) yuan and the sales profit is y yuan, then

y = (x-20) [400-20(x-30)]

= -20x2+140x-20000

∴When x=35, y has a maximum value of 4500.

35-30=5 (yuan)

Answer: When the sales unit price increases by 5 yuan, that is, when the unit price is 35 yuan, the maximum profit of 4,500 yuan can be obtained within half a month.

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