1. The quadratic coefficient of unary quadratic equation 5x2- 1 = 4x is ().
A.﹣ 1 B. 1
2. The opening direction of parabola y=3x2+2x is ().
A. Up B. Down C. Left D. Right
3. The root of the equation x2+x=0 is ()
a.x=﹣ 1 b . x = 0 c.x 1=0,x2=﹣ 1 d . x 1 = 0，x2= 1
4. As shown in the figure, it can be regarded as the isosceles right triangle rotated several times, so the degree of each rotation is ().
.45 caliber? B.50？ C.60？ D.72？
5. The following figure is an axisymmetric figure and a rotationally symmetric figure ().
A.①② B.①②③ C.②③④ D.①②③④
6. If the equation x2+8x+7=0 is solved by matching method, the correct formula is ().
A.(x﹣4)2=9 b .(x+4)2 = 9 c.(x﹣8)2= 16 d .(x+8)2 = 57
7. Given that the two elements of the equation x2+mx+3=0 are x 1, x2, x 1+x2=4, then the value of m is ().
B.﹣4
8. The vertex coordinate of parabola y = 2x2-8x-6 is ().
A.(﹣2,﹣ 14)b.(﹣2, 14(2 14)d.(2,﹣ 14)
9. As shown in the figure, it is known that the two diagonal lines AC and BD of the parallelogram ABCD intersect at the origin of the plane rectangular coordinate system, and the coordinate of point D is (3,2), then the coordinate of point B is ().
A.﹣2,﹣3 b.(﹣3,2 c.(3,﹣2 d.(﹣3,﹣2)
10. In the plane rectangular coordinate system, the number of times the parabola y = x2+2x-3 intersects the X axis is ().
A.0 B. 1 C.2 D.3
1 1. The number of a column arranged according to a certain rule is:? According to this rule, the seventh number in this column is ().
A.B. C. D。
12. The images of functions y=x2+bx+c and y=x are as shown in the figure, and the following conclusions are drawn:
①b2﹣4c>； 0; ②b+ c+ 1 = 0； ③3 b+ c+6 = 0； ④ When 1
The correct number is ()
A. 1
Fill in the blanks (this big question is ***6 small questions, each with 3 points, *** 18 points. Please fill in the answers on the answer sheet.
13. It is known that x= 1 is the root of the equation x2+mx+ 1=0, then m=.
14. The coordinates of the symmetrical point of point P (2 2,3) about the X axis are.
15. The known function y=2(x+ 1)2+ 1, when x >; Y increases with the increase of x.
16. As shown in the figure, on a rectangular site with a length of100m and a width of 80m, two roads with the same width and perpendicular to each other will be built, and the rest will be afforested. If the green area is 7644 square meters, how wide should the road be? Let the width of the road be x meters, then the equation can be listed as.
17. If the equation KX2-6x- 1 = 0 has two real roots, the range of k is.
18. for each nonzero natural number n, the parabola y = x2-X+ intersects with the x axis at two points, An and Bn, and the distance between these two points is expressed by an and Bn, then A 1B 1+A2B2+? The value of+a2013b2013+a2014b2014 is.
Third, answer the question (this big topic is ***8 small questions, ***66 points), please write the answer on the answer sheet.
19. Solve the equation: 9X2 ~ 1 = 0.
20. Solve the equation: x2-2x+ 1 = 25.
2 1. As shown in the figure, each small square in the grid paper is a square with a side length of 1 unit. After the plane rectangular coordinate system is established, the vertices of △ABC are all on the grid points, and the coordinates of point C are (4,-1).
(1) Take the origin o as the center of symmetry, draw △ABC and △ a1b1symmetrical about the origin o, and write the coordinates of C 1.
(2) Take the origin O as the rotation center, draw a picture and rotate △ABC clockwise by 90? Figure △A2B2C2. Write down the coordinates of C2.
22. It is known that parabola y = a (x- 1) 2 passes through point (2,2).
(1) Find the analytical formula corresponding to this parabola.
(2) When x takes what value, does the function have a maximum or minimum value?
23. As shown in the figure, point P is a point in the square ABCD, connecting AP, BP and CP, and rotating △PAB 90 clockwise around point B? To delta p? The location of CB. If AP=2 and BP=4, APB= 135？ , beg PP? And the length of the PC.
24. Planting Sydney has become an advantageous industry for increasing the income of township farmers in our county. This year, the Sydney planted by Wang Xiao's family got a bumper harvest. The sales of Sydney in Wang Xiaojia's two years are: the total sales in the first year 10000 yuan, and the total sales in the third year 12 1000 yuan.
(1) If the growth rate of total sales in the second year and the third year is the same, find the growth rate of total sales;
(2) According to the growth rate of total sales in Sydney in (1), what is the total sales of this farmer in the fourth year?
25. The owner of a shopping mall records the sales of a newly listed commodity, and it is known that the purchase price of the commodity is 40 yuan per piece. Through the analysis of the records, it is found that when the sales unit price is between 40 yuan and 90 yuan (including 40 yuan and 90 yuan), the relationship between the monthly sales volume y (pieces) and the sales unit price x (yuan) can be approximately regarded as a linear function, and its image is as shown in the figure.
(1) Find the functional relationship between y and x 。
(2) Let the monthly profit of the shopkeeper be P (yuan), and find the functional relationship between P and X;
(3) If you want to make a profit of 2,400 yuan per month, what should the sales unit price be?
26. As shown in the figure, it is known that the intersection of the parabola y =-x2+bx+c and the X axis is A (4 4,0), and it intersects with the Y axis at point B (0,3).
(1) Find the function relation corresponding to this parabola;
(2) Is there a point M on the positive semi-axis of X axis, so that AM=BM? If it exists, find the coordinates of point m; If it does not exist, please explain why.
Reference answers and analysis of test questions
A, multiple-choice questions (this topic is entitled *** 12 small questions, 3 points for each small question, 36 points for each small question. Each small question gives four conclusions codenamed A, B, C and D, only one of which is correct. Please mark the selected answer sheet with 2B pencil).
1. The quadratic coefficient of unary quadratic equation 5x2- 1 = 4x is ().
A.﹣ 1 B. 1
General form of quadratic equation with one variable.
To determine the coefficients of quadratic term and constant term in analysis, we must first turn the equation into a general form.
Solution: 5x2 ~ 1 ~ 4x = 0,
5x2﹣4x﹣ 1=0，
The quadratic coefficient is 5.
Therefore, choose: d.
Comments on this topic mainly focus on the general form of quadratic equation in one variable: ax2+bx+c=0(a, B, C are constants, A? 0) pay special attention to a? 0. This is a knowledge point that is easily overlooked in the process of doing the problem. In general, ax2 is called quadratic term, bx is called linear term, and C is constant term. Among them, A, B and C are called quadratic coefficient, linear coefficient and constant term respectively.
2. The opening direction of parabola y=3x2+2x is ().
A. Up B. Down C. Left D. Right
Test the properties of central quadratic function.
It is enough to directly determine the opening direction of parabola by using the quadratic term coefficient.
Solution: ∫ parabola y=3x2+2x, a = 3>0,
? Parabolic opening is upward.
So choose: a.
This question examines the properties of quadratic function and determines that the opening direction of parabola is related to the coefficient of quadratic term.
3. The root of the equation x2+x=0 is ()
a.x=﹣ 1 b . x = 0 c.x 1=0,x2=﹣ 1 d . x 1 = 0，x2= 1
Solving quadratic equation by unary factorization.
Special calculation problems.
Analysis: factorize the left side of equation x(x+ 1)=0, and the equation can be reduced to two linear equations x=0 or x+ 1=0, and then solve the two linear equations.
Solution: x2+x=0,
? x(x+ 1)=0，
? X=0 or x+ 1=0,
? x 1=0,x2=﹣ 1.
So choose C.
Comment on this topic and solve quadratic equation ax2+bx+c=0(a? 0) Method: First, turn the equation into a general formula, and then factorize the left side of the equation, so as to turn the quadratic equation of one yuan into two linear equations of one yuan, and then solve the two linear equations of one yuan.
4. As shown in the figure, it can be regarded as the isosceles right triangle rotated several times, so the degree of each rotation is ().
.45 caliber? B.50？ C.60？ D.72？
The test center rotates symmetrically.
According to the analysis of the nature of rotation and combining a fillet is 360? Solve.
Solution: A fillet is 360 degrees, and the acute angle of an isosceles right triangle is 45 degrees.
? As shown in the figure, an isosceles right triangle rotates 45 degrees at a time and rotates 8 times.
? The angle of each rotation is 45 degrees? .
So choose: a.
This topic examines the essence of rotation: before and after the rotation change, the corresponding line segment and the corresponding angle are equal, and the size and shape of the figure remain unchanged.
5. The following figure is an axisymmetric figure and a rotationally symmetric figure ().
A.①② B.①②③ C.②③④ D.①②③④
The test center rotates symmetrically; Axisymmetric graph.
The analysis directly uses the definition of axisymmetric figure combined with the definition of rotationally symmetric figure to get the answer.
Solution: ① It is not an axisymmetric figure, but a rotationally symmetric figure, and the option is wrong;
② Axisymmetric graphics and rotationally symmetric graphics, with correct options;
③ It is an axisymmetric figure and a rotationally symmetric figure, and the options are correct;
④ It is an axisymmetric figure and a rotationally symmetric figure, and the option is correct.
So choose: C.
The review of this topic mainly examines the rotationally symmetric graphics and axisymmetric graphics, and correctly grasping the definition is the key to solving the problem.
6. If the equation x2+8x+7=0 is solved by matching method, the correct formula is ().
A.(x﹣4)2=9 b .(x+4)2 = 9 c.(x﹣8)2= 16 d .(x+8)2 = 57
Solving quadratic equation with a variable matching method.
Special calculation problems.
The constant term of analytical equation is shifted to the right, and 16 is added to both sides to judge the formula.
Solution: Equation x2+8x+7=0,
Deformation: x2+8x =-7,
Formula: x2+8x+ 16=9, that is, (x+4)2=9,
So choose B.
The key to solve this problem is to understand the quadratic equation of one variable and master the complete square formula skillfully.
7. Given that the two elements of the equation x2+mx+3=0 are x 1, x2, x 1+x2=4, then the value of m is ().
B.﹣4
Test the relationship between root and coefficient.
The analysis shows that the two roots of the equation x2+mx+3=0 are x 1, x2 and x 1+x2=4. According to the relationship between root and coefficient, we can get-m = 4, and then get the answer.
Solution: The two roots of the equation x2+mx+3=0 are x 1, x2,
? x 1+x2=﹣m，
∫x 1+x2 = 4，
? ﹣m=4，
Solution: m =-4.
So choose B.
Comment on this question and examine the relationship between roots and coefficients. Note that if the coefficient of the quadratic term is 1, the following relationship is commonly used: x 1, x2 is the two roots of the equation x2+px+q=0, x 1+x2=﹣p, x1x2 = q. 。
8. The vertex coordinate of parabola y = 2x2-8x-6 is ().
A.(﹣2,﹣ 14)b.(﹣2, 14(2 14)d.(2,﹣ 14)
Test the properties of central quadratic function.
This paper analyzes the general formula of the known parabola analytical formula, and transforms it into the vertex by matching method to get the vertex coordinates.
Solution: ∫y = 2 x2 ~ 8x ~ 6 = 2(x ~ 2)2 ~ 14,
? The coordinate of the vertex is (2,-14).
Therefore, choose: d.
The properties of quadratic function are reviewed. It is a common method to find the vertex coordinates and symmetry axis of parabola by collocation method.
9. As shown in the figure, it is known that the two diagonal lines AC and BD of the parallelogram ABCD intersect at the origin of the plane rectangular coordinate system, and the coordinate of point D is (3,2), then the coordinate of point B is ().
A.﹣2,﹣3 b.(﹣3,2 c.(3,﹣2 d.(﹣3,﹣2)
The nature of the parallelogram in the examination center; Coordinates and graphic properties.
By analyzing the properties of parallelogram, it is concluded that B and D are symmetrical about the origin O, and the coordinates of point B can be obtained.
Solution: ∵ Quadrilateral ABCD is a parallelogram, O is the intersection of angle lines AC and BD,
? B and d are symmetrical about the origin o,
∫ The coordinate of point d is (3,2),
? The coordinates of point B are (-3,-2);
Therefore, choose: d.
This topic reviews the nature of parallelogram, the nature of coordinates and graphics, and the coordinate characteristics of points symmetrical about the origin; It is the key to solve the problem to master the properties of parallelogram and get the coordinates of point B from the coordinate characteristics of points symmetrical about the origin.
10. In the plane rectangular coordinate system, the number of times the parabola y = x2+2x-3 intersects the X axis is ().
A.0 B. 1 C.2 D.3
The intersection of parabola and x axis of test center.
Let y=0 through analysis, and get the unary quadratic equation x2+2x-3 = 0 about x, and then judge the number of solutions of the equation according to △.
Solution: Let y=0 to get: x2+2x-3 = 0,
∵△=b2﹣4ac=22﹣4？ 1? (﹣3)=4+ 12= 16>； 0,
? A parabola has two intersections with the x-axis.
So choose: C.
This topic mainly investigates the intersection of parabola and X axis, and the key to solving the problem is to transform the function problem into the equation problem.
1 1. The serial number of a column arranged according to a certain rule is:? According to this rule, the seventh number in this column is ().
A.B. C. D。
Regularity of test sites: diversity of numbers.
General types of topics.
Analysis By observing and analyzing the data, we can know that the numerator is a constant value 1, and the changing law of denominator is: the denominator of odd term is n2+ 1, and the denominator of even term is N2- 1.
Solution: molecular law: molecules are constant1;
Denominator Law: 1 The denominator of this number is: 12+ 1=2.
The denominator of the second number is 22- 1 = 3,
The denominator of the third number is: 32+ 1= 10,
The denominator of the fourth number is 42 ~ 1 = 15.
The denominator of the fifth number is: 52+ 1=26,
The denominator of the sixth number is: 62- 1 = 35,
The denominator of the seventh number is 72+ 1=50.
?
The denominator of odd terms is n2+ 1.
The denominator of even terms is N2- 1,
So the seventh number is.
So choose D.
It is a basic ability to observe, analyze, summarize, find out the rules and apply the found rules to solve problems. The key to this problem is to find out the changing law of denominator by analyzing denominator. The denominator of odd items is n2+ 1, and the denominator of even items is N2- 1.