Example 1. (1)y and x are proportional functions, when y=5. Find the analytic expression of this proportional function.

(2) The image of a known linear function passes through two points A (- 1, 2) and B (3 3,5), and the analytical expression of this linear function is obtained.

Solution: (1) Let the analytical formula of the proportional function be

Substitute into the above formula, y=5.

Get it, solve it, get it.

The analytic formula of the positive proportional function of ∴ is

(2) Let the analytical formula of the linear function be

∵ This image passes through two points, A (- 1, 2) and B(3, -5), and the coordinates of these two points must be satisfied. Substitute y=2 and x=3 into the above formula respectively, and get

solve

∴ The analytical formula of this linear function is

Comments: (1) cannot divide the score exactly. (2) There are several undetermined coefficients in the set analytic formula, and several equations need to be listed according to the known conditions.

Example 2. When the tractor started working, there was 20 liters of oil in the tank. If the fuel consumption is 5 liters per hour, find the functional relationship between the remaining fuel quantity Q (liter) in the fuel tank and the working time T (hour), point out the value range of the independent variable X, and draw an image.

Analysis: The tractor consumes 5 liters per hour and 5 liters per t hour. 20 liters MINUS 5 liters is the remaining fuel.

Solution:

The image is shown in the figure below.

Comments: Pay attention to the value range of function independent variables. The image depends on the range of independent variables. It is a line segment, not a straight line.

Example 3. Given that the image of a linear function passes through the point p (-2,0) and the area of the triangle intersecting with the two coordinate axes is 3, find the analytical expression of this linear function.

Analysis: As can be seen from the figure, the intersection of the image with point P as a linear function and the Y axis may be on the positive half axis of the Y axis or on the negative half axis of the Y axis, so it is necessary to study it in two cases, which is the mathematical thinking method of classified discussion.

Solution: Find the resolution function first.

∫ The coordinate of point p is (-2,0).

∴|OP|=2

Let the function image intersect the Y axis at point B (0, m).

According to the meaning of the question, S δ POB = 3.

∴

∴|m|=3

∴

∴ The image of the linear function intersects the Y axis at b1(0 (0,3) or B2 (0 0,3).

Substitute the coordinates of p (-2,0) and b1(0 (0,3) or p (-2,0) and B2 (0 0,3) into y=kx+b, and you get it.

solve

∴ The analytical formula for finding a linear function is

Comments: (1) This topic adopts the mathematical thinking method of classified discussion. Speaking of the problem of making a straight line between a fixed point and the intersection of two coordinate axes, we must consider the direction and which direction to do it. We can think intuitively with graphics to prevent losing a straight line. (2) When it comes to area, choose half of the product of two right-angled sides of a right triangle, and the result must be positive.

Synthetic test

First, multiple-choice questions:

1. If the image of the proportional function y=kx passes through one or three quadrants, the value range of k is ().

A.B. C. D。

2. The candle is 20cm long and burns 5cm every hour after lighting. The functional relationship between the residual height y(cm) and the combustion time x (hours) is graphically represented as ().

3. (Beijing) The quadrant that the image of a linear function does not pass through is ()

A. first quadrant B. second quadrant C. third quadrant D. fourth quadrant

4. The area of triangle surrounded by straight line, X axis and Y axis is ()

A.3 B. 6 C. D

5. The approximate image of (Hainan Province) linear function is ()

Second, fill in the blanks:

1. If the image of a linear function y=kx+b passes through (0, 1) and (-1, 3), then the analytical expression of this function is _ _ _ _ _ _ _ _.

2. If the image of the proportional function y=kx passes through the point (1, 2), the analytical formula of the function is _ _ _ _ _ _ _ _ _.

Third,

The intersection point between the image of the linear function and the Y axis is (0, -3), and the area of the triangle surrounded by the coordinate axes is 6. Find the analytical expression of this linear function.

IV. (Wuhu Curriculum Reform Experimental Zone)

The functional relationship between the mechanical efficiency η of a diesel locomotive measured before the trial operation of the Qinghai-Tibet Railway and the altitude h (,unit km) is shown in the figure.

(1) Please write the functional relationship between locomotive mechanical efficiency η and altitude h(km) according to the image;

(2) What is the mechanical efficiency of locomotive running at an altitude of 3km?

Verb (abbreviation of verb) (Lishui City, Zhejiang Province)

As shown in the figure, the plane rectangular coordinate system of badminton match scene is established. In the figure, the height OD of the net is 1.55 m and the length OA=OB=6.7 (m). Badminton players take off at point C, which is 5 m away from the net, and the ball flies straight from point E at the top of the net, with a DE of 0.05 m, which just lands at point B of the opponent's court.

(1) An analytical formula for finding the straight line of badminton flight path;

(2) What is the height FC of the hitting point of the badminton racket from the ground in this straight spike? (The result is accurate to 0. 1 m)

Comprehensive test answer

First, multiple-choice questions:

1.B 2。 B 3。 D 4。 A 5。 B

Second, fill in the blanks:

1.2.

Analysis: The analytical formula y=kx+b of a linear function has two undetermined coefficients, and two equations need to be established by using two conditions. One condition in the topic is obvious, that is, the vertical coordinate of the intersection of the image and the Y axis is -3, and the other condition is hidden and needs to be determined from "the area enclosed by the coordinate axis is 6".

Solution: Let the analytical expression of the linear function be,

The ordinate of the intersection of the functional image and the Y axis is -3,

∴

The analytical formula of this function is.

Find the intersection of this function image and the x axis, that is, solve the equations:

get

That is, the coordinates of the intersection point are (,0).

Since the area of the right triangle enclosed by the image of the linear function and the two coordinate axes is 6, from the triangle area formula, we can get

∴

∴

∴ The analytical formula of this linear function is

Fourth, the solution: (1) As can be seen from the image, the functional relationship with H is a linear function.

set up

∫ This function image passes (0,40%) and (5,20%).

Get a solution

∴

(2) When h=3km,

When the locomotive runs at an altitude of 3km, the mechanical efficiency of the locomotive is 28%.

Solution of verb (abbreviation of verb): (1) According to the meaning of the question, let the straight line BF be y = kx+b.

∫OD = 1.55，DE=0.05

∴

That is, the coordinate of point E is (0, 1.6).

OA = OB = 6.7

∴ The coordinate of point B is (-6.7,0).

Because the straight line passes through point E (0, 1.6) and point B (-6.7,0), it is obtained that

Solve, that is,

(2) If the coordinate of point f is (5,), then when x=5,

Then FC=2.8.

In this straight spike, the hitting point of the badminton racket is 2.8 meters above the ground. If not, I still have it.