1. If the image of quadratic function y=ax2+bx+c has two common points with the x axis, then the unary quadratic equation ax2+bx+c=0 has two unequal real roots. Please solve the following problems according to your understanding of this sentence: If m, n(m
A.m
Test center: the intersection of parabola and x axis.
Analysis: Draw an image sketch of the function y = (x-a) (x-b) according to the meaning of the question, and solve it according to the increase or decrease of the quadratic function.
Solution: according to the meaning of the question, draw the image of the function y = (x-a) (x-b), as shown in the figure.
The function image is a parabola with an upward opening, and the abscissas of the two intersections of the X axis are A and b(a
Equation1(x-a) (x-b) = 0 is converted into (x-a) (x-b) =1,where two equations are parabola y = (x-b).
From the parabolic opening upward to the left of the symmetry axis, y decreases with the increase of x.
So choose a.
Comments: This topic examines the relationship between quadratic function and unary quadratic equation, and examines the mathematical thought of combining numbers with shapes. When solving a problem, draw a sketch of the function and draw a conclusion intuitively from the function image to avoid complicated calculation.
2. quadratic function y=ax2+bx+c(a, b, c are constants, a? Part of the corresponding values of x and y in 0) are as follows:
X ﹣ 1 0 1 3
y ﹣ 1 3 5 3
Draw the following conclusions:
( 1)AC & lt； 0;
(2) when x >; At 1, the value of y decreases with the increase of the value of x.
(3)3 is a root of equation ax2+(b﹣ 1)x+c=0;
④ When 10.
The correct number is ()
A.4 B. 3 C. 2 D. 1
Analysis: According to the table data, find out that the symmetry axis of quadratic function is a straight line x= 1.5, and then according to the properties of quadratic function, analyze and judge each term to get the solution.
Answer: From the data in the chart, it can be concluded that when x= 1, the value of y=5 is the largest, so the quadratic function y=ax2+bx+c opens downward, a; 0, so AC
∵ Quadratic function y=ax2+bx+c, with downward opening and axis of symmetry x= = 1.5. When x> is at 1.5, the value of y decreases with the increase of x value, so (2) is wrong;
When x = 3, y=3,? 9a+3b+c=3，∫c = 3，？ 9a+3b+3=3，？ 9a+3b=0，？ 3 is a root of the equation ax2+(b﹣ 1)x+c=0, so (3) is correct;
∵x=﹣ 1，ax2+bx+c=﹣ 1？ When x=﹣ 1, ax2+(b﹣ 1)x+c=0 and ax2+(b ﹣1) x+c = 0 ﹣ x = 3, the function has-
So choose B.
Comments: It is difficult to examine the properties of quadratic function, the relationship between quadratic function image and coefficient, the intersection of parabola and X axis, quadratic function and inequality. Mastering the properties of quadratic function images is the key to solve the problem.
3. Quadratic function y=ax2+bx+c(a? 0), the image passes through the point (-1, 0), and the symmetry axis is the straight line x=2. Draw the following conclusions:
①4a+b = 0； ②9a+c & gt； 3b； ③8a+7 b+2c & gt； 0; ④ When x >; When-1, the value of y increases with the increase of x value.
The correct conclusion is ()
A. 1 B. 2 C. 3 D.4
Analysis: According to the parabola's symmetry axis as a straight line X =-= 2, then 4a+b = 0；; Observing the function image, it is found that when x=﹣3 and the function value is less than 0, then 9a ﹣ 3b+c < 0, that is, 9a+c: 0; Because the symmetry axis is a straight line x=2, according to the properties of quadratic function, it is obtained that when x >: 2, Y decreases with the increase of X. 。
Answer: ∫ The symmetry axis of parabola is a straight line X =∫= 2,? B =-4a, that is, 4a+b=0, so ① is correct;
When x =-3 and y < 0,? 9a﹣3b+c<； 0, namely 9a+c
∵ The intersection of parabola and X axis is (-1, 0). a﹣b+c=0，
And b =-4a,? A+4a+c=0, that is, c =-5a,? 8a+7b+2c=8a﹣28a﹣ 10a=﹣30a，
∵ Parabolic opening down,? a & lt0,? 8a+7 b+2c & gt； 0, so ③ is correct;
The symmetry axis is a straight line x=2,
? -12, y decreases with the increase of x, so ④ is wrong, so B. 。
Comments: This question examines the relationship between quadratic function image and coefficient: quadratic function y=ax2+bx+c(a? 0), the quadratic coefficient a determines the opening direction and size of the parabola, when a >; 0, the parabola opens upwards; When a<0, the parabola opens downwards; Both linear coefficient b and quadratic coefficient a*** determine the position of the axis of symmetry. When A and B have the same sign (ab>0), the symmetry axis is on the left side of Y axis; When a and b have different numbers (that is, AB 0, parabola and x axis have two intersections; △ = B2-4ac = 0, and the parabola has 1 intersection with the X axis; △=b2﹣4ac<； When the value is 0, the parabola has no intersection with the x axis.
4, known quadratic function y=ax2+bx+c(a? 0) As the image is shown, the following statement appears:
①c = 0； ② The symmetry axis of parabola is a straight line x =-1; ③ when x= 1, y = 2a4am2+BM+a > 0(m？ ﹣ 1).
The correct number is ()
A. 1 B. 2 C. 3 D. 4
Test site: the relationship between quadratic function image and coefficient.
Analysis: judge the relationship between C and 0 through the intersection of parabola and Y axis, and then infer the conclusion according to the intersection of symmetry axis and parabola and X axis.
Solution: Solution: The parabola intersects the Y axis at the origin, and c=0, so ① is correct;
The symmetry axis of this parabola is: straight line X =- 1, so ② is correct;
When x= 1 and y=2a+b+c,
The symmetry axis is a straight line x =- 1,
? ，b=2a，
∫c = 0，
? Y=4a, so ③ error;
The function value corresponding to x=m is y=am2+bm+c,
X =- 1 the corresponding function value is y = a-b+c, and the function takes the minimum value when x =- 1.
? a﹣b+c
∫b = 2a，
? am2+BM+a & gt； 0(m？ -1). So ④ is correct.
So choose: C.
Comments: This question examines the relationship between quadratic function images and coefficients. Quadratic function y=ax2+bx+c(a? 0) The sign of the coefficient is determined by the opening direction of the parabola, the axis of symmetry, the intersection point of the parabola and the Y axis, and the number of intersection points of the parabola and the X axis.
5. Given that point A (a-2b, 2-4ab) is on the parabola y=x2+4x+ 10, the coordinate of point A about the parabola axis symmetry point is ().
A.(﹣3,7)b .(﹣ 1,7)c .(﹣4, 10)d .(0， 10)
Test site: the coordinate characteristics of points on the quadratic function image; Coordinate and graphic changes-symmetry.
Analysis: substitute the coordinates of point A into the quadratic resolution function and arrange it according to the complete square formula, then calculate a and b according to the non-negative property formula, then calculate the coordinates of point A, then calculate the parabolic symmetry axis, and then calculate the solution according to the symmetry.
Solution: Point A (a-2b, 2-4ab) is on the parabola y=x2+4x+ 10.
? (a﹣2b)2+4？ (a﹣2b)+ 10=2﹣4ab，
a2﹣4ab+4b2+4a﹣8ab+ 10=2﹣4ab，
(a+2)2+4(b﹣ 1)2=0，
? a+2=0,b﹣ 1=0，
The solution is a =-2, b= 1,
? a﹣2b=﹣2﹣2？ 1=﹣4,
2﹣4ab=2﹣4？ (﹣2)? 1= 10,
? The coordinates of point A are (-4, 10).
The symmetry axis is a straight line x = =-2,
? The coordinate of the symmetry point of point A about the symmetry axis is (0, 10).
So choose D.
Comments: This topic examines the coordinate characteristics of points on the image of quadratic function, the symmetry of quadratic function, and the change-symmetry between coordinates and graphics. It is the key to solve the problem to substitute the coordinates of points into parabolic analytical expressions and arrange them into non-negative forms.
6 As shown in the figure, the vertex A of the rectangular ABCD is in the first quadrant, and the intersection of the diagonal lines of the axis AB∨x and AD∨y coincides with the origin O. In the process of changing the AB side from less than AD to greater than AD, if the perimeter of the rectangular ABCD remains unchanged, the inverse proportional function y= (k? The change of the value of k in 0) is ()
A. increasing B. decreasing C. increasing first and then decreasing D. decreasing first and then increasing.
Test site: the coordinate characteristics of points on the inverse proportional function image; The properties of rectangles.
Analysis: Let AB=2a and AD=2b in rectangular ABCD. Since the perimeter of the rectangular ABCD always remains the same, a+b is a constant. According to the coincidence degree between the intersection of the diagonal lines of the rectangle and the origin O and the geometric meaning of the proportional coefficient k of the inverse proportional function, it can be known that k= AB? AD=ab, and then when a+b is constant, when a=b, the greatest understanding of ab is that the value of K first increases and then decreases in the process of ab edge changing from less than AD to greater than AD.
Solution: Let AB=2a and AD= 2B in rectangular ABCD.
The perimeter of the rectangular ABCD always remains the same.
? 2(2a+2b)=4(a+b) is a fixed value,
? A+b is a fixed value.
The intersection of the diagonal lines of the rectangle coincides with the origin o.
? k= AB？ AD=ab，
When ∵a+b is a constant value, when a=b, ab is the largest.
? During the change of AB edge from less than AD to greater than AD, the value of K first increases and then decreases.
So choose C.
Comments: It is difficult to examine the nature of rectangle, the geometric meaning of the proportional coefficient k of inverse proportional function and the nature of inequality. According to the meaning of the question, k= AB? AD=ab is the key to solve the problem.
7, known function y = (x-m) (x-n) (where m
Morning+n <; 0b m+n & gt； 0 c . m-n & lt； 0d . m-n & gt； 0
Analysis: judging m
A: As can be seen from the figure, M
Therefore, the linear function y=mx+n passes through the second four quadrants and intersects the Y axis at point (0, 1).
The image of the inverse proportional function y= is located in the second quadrant,
Looking at all the options, only the C option graphics match. So I chose C.
Comments: This question examines quadratic function images, linear function images and inverse proportional function images. It is the key to solve the problem to observe the image of quadratic function and determine the values of m and n.