Example 1. An athlete jumped up and shot 4 meters away from the basket. The trajectory of the ball is a parabola. When the horizontal distance of the ball is 2.5 meters, it reaches the maximum height of 3.5 meters and then falls into the basket accurately. It is known that the distance from the center of the iron ring to the ground is 3.05 meters.
(1) Establish the rectangular coordinate system as shown in the figure and find the parabolic analytical formula;
(2) The athlete's height is1.8m. In this jump shot, the ball was shot 0.25m above his head.
Q: How high did he jump off the ground when the ball was released?
Simple solution:
(1) Since the vertex of the parabola is (0,3.5), its analytical formula can be set to y=ax2+3.5. Because the parabola passes through (1.5,3.05), a=-0.2 is obtained. The analytical formula of parabola is y=-0.2x2+3.5.
(2) When x=-2.5, y=2.25. When the ball is released, its height from the ground is 2.25- 1.8-0.25=0.20 (m).
Comments: It is not uncommon to use the trajectory of the ball when pitching, the trajectory of the human body when diving and the quadratic function designed by parabolic bridge opening. Solving such problems is generally divided into the following four steps:
(1) Establish an appropriate rectangular coordinate system (as given in the title, no reconstruction is needed);
(2) According to the given conditions, find out the known points on the parabola and write the coordinates;
(3) Using the coordinates of known points, the analytical formula of parabola is obtained. ① When the coordinates of three points are known, the general formula y=ax2+bx+c can be used to find its analytical formula; (2) When the coordinates of the vertex are known as (k, h) and the coordinates of another point, the analytical formula can be obtained from the vertex y=a(x-k)2+h; (3) When the coordinates of the two intersections of parabola and X axis are known as (x 1, 0) and (x2, 0) respectively, an analytical formula can be obtained from the dichotomy formula y=a(x-x 1)(x-x2);
(4) The coordinates of points related to the problem are obtained by using parabolic analytical formula, so that the problem can be solved.
Example 2: A shopping mall bought a batch of daily necessities at a unit price of 16 yuan. Through experiments, it is found that if you sell at the price of each piece in 20 yuan, you can sell 360 pieces per month, and if you sell at the price of each piece in 25 yuan, you can sell 2 10 pieces per month. Assume that the number of pieces sold per month y (pieces) is a linear function of the price x (yuan/piece).
(1) Try to find the relationship between y and x;
(2) Without considering factors such as overstock, what is the selling price with the largest monthly profit? What is the maximum monthly profit?
Solution: (1) Let y=kx+b according to the meaning of the question, then there is
So y =-30x+960 (16 ≤ x ≤ 32).
(2) monthly profit P=(-30x+960)(x- 16)
=30(-x+32)(x- 16)
= 30(+48-5 12)
=-30 + 1920.
So when x=24, p has a maximum value, and the maximum value is 1920.
A: When the price is 24 yuan, you can get the maximum profit every month, and the maximum profit is 1920 yuan.
Note: Mathematical application problems come from practice and are used in practice. In today's social market economy environment, we should master some knowledge about commodity prices and profits. The total profit is equal to the total income minus the total cost, and then a quadratic function is used to find the maximum value.
Example 3: In the physical education examination, a tall boy shot put in grade three, and the known shot put route is part of a quadratic function image. As shown in the figure, if the coordinate of point A in this male student's hand is (0,2), then the coordinate of point B at the highest point of the shot put path is (6,5).
(1) Find the analytic expression of this quadratic function;
(2) How far did the male students push the shot put? (accurate to 0.0 1 m,)
Solution: (1) Let the analytic formula of quadratic function be
, the vertex coordinates are (6,5)
A (0 0,2) is on a parabola.
(2) When,
(Never mind, give up)
(meter)
A: The raw shot put 13.75 meters.
Example 4. A shopping mall buys a kind of clothing at the price of each piece in 42 yuan. According to the trial sale, the daily sales volume (pieces) of this kind of clothing can be regarded as a linear function with the sales price (yuan/piece) of each piece:
1. Write the functional relationship between the daily sales profit of this kind of clothing sold in the shopping mall and the sales price of each piece (daily sales profit refers to the difference between the sales price and the purchase price of the clothing sold);
2. Through the formula of the obtained functional relationship, it is pointed out that if the shopping mall wants to get the maximum sales profit every day, what is the most suitable sales price of each piece; What is the maximum sales profit?
Analysis: The profit of a shopping mall is determined by multiplying the profit of each commodity by the number of daily sales.
In this problem, if the profit of each garment is () and the number of pieces sold is (+204), then we can get a functional relationship between and, which is a quadratic function.
The maximum profit required for sales is the maximum required for this quadratic function.
Solution: (1) From the meaning of the question, the functional relationship between the sales profit and the sales price of each piece is
= (-42) (-3+204), which means =-3 2+8568.
(2) Formula =-3 (-55) 2+507.
When the sales price of each piece is 55 yuan, the maximum profit can be obtained, and the maximum daily sales profit is 507 yuan.
Example 5: When a diver performs platform diving training of10m, the movement route of his body (as a point) in the air is a parabola passing through the origin O in the coordinate system as shown in the figure (the data marked in the figure are known conditions). When jumping a specified action, under normal circumstances, the highest point in the air is meters away from the water surface, and the distance from the water entry point to the pool edge is 4 meters. Before the athlete leaves the water for 5 meters,
(1) Find the analytical expression of this parabola;
(2) In a trial jump, the athlete's movement route in the air is measured as a parabola in (1), and the horizontal distance from the pool edge is meters when the athlete adjusts his entry posture in the air. Will there be any mistakes in this diving?
And explain the reason through calculation.
Analysis: (1) In a given rectangular coordinate system, to determine the analytical formula of parabola, it is necessary to determine the coordinates of three points on the parabola, such as the jumping point O (0 0,0) and the water entry point (2,-10), and the vertical point of the highest point is marked as.
(2) After finding the analytical formula of parabola, it is necessary to judge whether diving will be wrong, that is, whether the athlete is 5 meters above the water when the horizontal distance from the swimming pool is meters.
Solution: (1) In a given rectangular coordinate system, let the highest point be A, the water entry point be B, and the analytical formula of parabola be.
From the meaning of the question, we know that O (0 0,0), B(2,-10), and the ordinate of vertex A is.
Solve or
The symmetry axis of parabola is on the right side of the axis.
And ∵ parabolic opening is downward, ∴ A < 0, B > 0.
∴ The analytical formula of parabola is
(2) When the horizontal distance between athletes in the air and the pool edge is 100 meters,
In the end,
At this time, the height of the athletes from the water is
So this diving will go wrong.
Example 6. A clothing dealer's A stock purchase price is 400 yuan's A brand clothing 1200 sets. During normal sales, each set of 600 yuan can buy 1000 sets per month, which is just sold out within one year. B brand clothes are popular in the market now. The purchase price of this brand clothing is 200 yuan, and the price is 500 yuan. You can buy 1 20 sets (two sets of clothing) every month. At present, there is an opportunity to enter the B brand. If we miss this opportunity, it is estimated that this kind of clothing will be gone within one year. However, the dealer has no liquidity at hand and only transfers A brand clothing at a low price. After consultation with dealer b, an agreement was reached. The transfer price (RMB/set) has the following relationship with the transfer quantity (set):
Transfer quantity (set)12001001000 900 800 700 600 500 400 300 200100.
Price (RMB/set) 240 250 260 270 280 290 300 365 438+00 320 330 340 350
Scheme 1: neither transfer A brand clothing nor distribute B brand clothing;
Option 2: All the clothes of brand A are transferred, and after the clothes of brand B are purchased with the transferred funds, the clothes of brand B are distributed;
Scheme 3: Transfer a part of brand A clothing, and distribute brand B clothing and brand A clothing at the same time after purchasing brand B clothing with the transferred funds.
Q:
(1) How much profit does dealer A get from option 1 and option 2 in a year?
(2) Which scheme does dealer A choose to make the most profit in one year? If Option 3 is selected, what is the number of clothes of a certain brand that he transferred to Dealer B (accurate to 100 sets)? How much does he earn a year at this time?
Solution: The purchase cost of dealer A = = 480,000 yuan.
① If the scheme 1 is selected, the profit will be 1200 600-480000=240000 yuan.
If you choose the second option, the transfer fee is1200 240 = 288,000 yuan, and you can buy a B brand clothing suit. If you only short within one year, you can make a profit 1440 500-480000=240000 yuan.
② If X sets of A-brand clothes are transferred, the transfer price is RMB per set, and you can buy B-brand clothes and get RMB after selling all B-brand clothes. At this time, there are (1200-x) sets of A-brand clothes left, and after all the A-brand clothes are sold, they will get RMB x=600( 1200-x), so X = * * profits.
Third, exercise questions:
1. A shopping mall buys a commodity at the price of 30 yuan. In the process of trial sale, it is found that the daily sales volume (pieces) of the commodity and the sales price (yuan) of each piece meet the linear function relationship:
(1) Write the functional relationship between the daily sales profit and the sales price of each commodity in the mall.
(2) If the mall wants to get the maximum sales profit every day, what is the most suitable price for each commodity? What is the maximum sales profit?
2. As shown in the figure, one side is close to the school wall, and the other three sides are surrounded by a 40-meter-long fence to form a 40-square-meter rectangular garden.
(1) Find the functional relationship between: and, and find the value when the meter is 2;
(2) Let the side length of the rectangle meet the relationship that the rectangle becomes a golden rectangle, and find the length and width of the golden rectangle.
Exercise 1 answer:
When the price is 42 yuan, the maximum sales profit is 432 yuan.
Exercise 2 Answer: (1)
When,
(2) If ①
(2)
From the solutions of ① and ②,
Twenty of them are irrelevant, so they are discarded.
When the rectangle becomes a golden rectangle, the width is and the length is.
A circular fountain will be built somewhere. A flower-shaped column OA is installed in the center of the fountain perpendicular to the water surface, and O is right in the center of the water surface. The nozzle placed at the top of the A-pillar sprays water outwards, and the water flows downwards along a parabolic path, with the same shape in all directions. On any plane passing through OA, the parabola shape is as shown in the figure, and the rectangular coordinate system is established as shown in the figure. The relationship between the height of the water jet and the horizontal distance is as follows.
Please answer the following questions:
What is the height of 1.OA column?
2. What is the maximum height of sprayed water from the horizontal plane?
3. If other factors are not considered, how many meters should the radius of the pool be at least, so that the sprayed water does not fall outside the pool?
Exercise 3 Answer:
(1) The height of OA is meters.
(2) When, that is, the maximum height of the water flow from the horizontal plane is meters.
(3)
Among them, it is irrelevant.
Answer: the radius of the pool should be at least 2.5 meters, so that the sprayed water will not fall outside the pool.