"When to Obtain Maximum Profit" Quadratic Function PPT Courseware

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"When to Obtain Maximum Profit" Quadratic Function PPT Courseware

Review questions

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.

Activity exploration 1

A store sells T-shirts. It is known that the unit price when purchased in batches is 2.5 yuan. According to market research, the sales volume and unit price satisfy the following relationship: within a period of time, when the unit price is 13.5 yuan, the sales volume is 500 pieces, and the unit price is 13.5 yuan per unit. If you reduce it by 1 yuan, you can sell 200 more pieces. Could you please help analyze, at what unit sales price can you make the most profit?

If the sales price is x yuan (x≤13.5 yuan), then

Sales volume can be expressed as: 500+200(13.5-x) pieces;

Sales can be expressed as: x[500+200(13.5-x)]yuan;

The profit obtained can be expressed as: (x-2.5)[500+200(13.5-x)]yuan;

When the sales unit price is 9.25 yuan, the maximum profit can be obtained, and the maximum profit is 9112.5 yuan.

Activity exploration 2

Remember the “how many orange trees to plant” question at the beginning of this chapter?

We have also used the list method to obtain a data. Now please verify whether your guess (how many more orange trees will be planted to maximize the total yield?) is correct.

Communicate with your peers about how you are doing.

The idea of ​​"quadratic function application"

Looking back at the process of solving the "maximum profit" and "maximum output" problems in this lesson, can you summarize the basic ideas for solving such problems?

1. Understand the problem;

2. Analyze the variables and constants in the problem and the relationship between them;

3. Express the relationship between them mathematically;

4. Do mathematical solutions;

5. Check the rationality and expansion of the test results, etc.

Class exercises

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 will be reduced by 20 units accordingly. When the selling price is increased by how much Yuan, the maximum profit can be obtained within half a month?

Solution: Assume that when the selling price increases by x yuan, the profit obtained within half a month is y yuan. Then

y=(x+30-20)(40-20x)

=-20x2+200x+4000

=-20(x-5)2+4500

∴When x=5, the maximum value of y is =4500

Answer: When the selling price increases by 5 yuan, the maximum profit can be made within half a month of 4,500 yuan.

Class Exercise 2

A travel agency organizes a group trip to other places. The minimum group size is 30 people, and the unit price per person is 800 yuan. Travel agencies offer discounts to groups of more than 30 people, that is, for each additional person in the tour group, the unit price per person will be reduced by 10 yuan. What is the maximum turnover that a travel agency can achieve when the number of people in a tour group is?

Solution: Suppose there are x people in a tour group, and the travel agency’s turnover is y yuan. Then

y=〔800-10(30-x)〕·x=-10x2+1100x

∴When x=55, the maximum value of y=30250

Answer: When there are 55 people in a tour group, the travel agency can earn a maximum profit of 30,250 yuan.

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Update Time: 2024-11-19

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