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In the rectangular coordinate system, there are two intersections between the parabola y = ax ^ 2 + BX + C and the X axis, and the distance from the intersection to the origin is greater than 1 Then the minimum value of ABC is (), and a + B + C = ()
The problem is wrong, the minimum value can be infinitesimal, such as a (- 30000,0) B (- 10000,0) vertex (- 2000010000)
If it is the absolute value of ABC, it can also be wireless close to zero, the vertex (0,0.000000000 1) can be
It is known that the intersection of the parabola y = ax & sup2; + 2x + C and X axis is on the right side of the origin, and the point m (a, c) is in the () quadrant. Why?
1. It is known that the intersection of the parabola y = ax & sup2; + 2x + C and X axis is on the right side of the origin, and the point m (a, c) is in the () quadrant. Why?
2. No matter what real number the independent variable x takes, the function value of quadratic function y = 2x & sup2; - 6x + m is always positive, then the value range of M is (). Why?
3. The vertex of the parabola y = x & sup2; - 2x + m is on the x-axis. Why do we find its vertex coordinates and symmetry equation?
Because the intersection of the parabola and the x-axis is on the right side of the origin, the axis of symmetry x = - B / 2a is also on the right side of the origin
So - B / 2A 〉 0, B = 2
That is - 1 / a ＞ 0
So a 〈 0
So the parabola opens down
The intersection point with X axis should be greater than 0 on the right side of the origin
So when x = 0, the intersection of the parabola and the y-axis, y = C, is on the negative half axis of the y-axis
So C 〈 0
So m (a, c) is in the third quadrant
If we know that the intersection points of the parabola y = AX2 + 2x + C and X axis are on the right side of the origin, then the point m (a, c) is on the second side___ Quadrant
Let X1 and X2 be the roots of the equation AX2 + 2x + C = 0, then the relationship between the roots and the coefficients shows that X1 + x2 = - BA = - 2A, x1x2 = Ca, the intersection points of ∵ function and X axis are on the right side of the origin, ∵ X1 + X2 > 0, x1x2 > 0, ∵ a < 0, C < 0, and the point m (a, c) is in the third quadrant
As shown in the figure, the parabola passes through three points a (4,0), B (1,0), C (0, - 2). (1) find out the analytical formula of the parabola; (2) P
Y = AX2 (quadratic) + BX + C, substituting into three points, C = - 2, a = - 1 / 2, B = 5 / 2
Y = (- 1 / 2) X2 (2nd power) + (5 / 2) X-2
And then there's no graph, P doesn't know
RELATED INFORMATIONS
- 1. As shown in the figure, the parabola passes through three points a (4,0), B (1,0), C (0, - 2). (1) find the analytical formula of the parabola (2) There is a point on the parabola above the line AC (3) P is a moving point on the parabola, passing through P as the PM ⊥ X axis, and the perpendicular foot is m. is there a point P, so that the triangle with vertices a, P, and M is similar to △ OAC? If it exists, the coordinates of the qualified point P are requested; if it does not exist, please explain the reason
- 2. It's very urgent. The parabola y = ax-3ax + 3 intersects the x-axis at a (- 1,0) B (4,0) intersects the y-axis at C The square of parabola y = ax - 3ax + 3 intersects X axis at a (- 1,0) B (4,0) intersects Y axis at C. 1. Find the analytical formula of parabola 2. Translate the parabola along the Y-axis and intersect the y-axis at point D. when BD = 2CD, find the coordinates of point D 3. If the parabola is folded along the straight line x = m, make the folded parabola intersecting straight line BC at two points P and Q, and P and Q are symmetrical about point C, draw the value graph of m by yourself. Because I don't have a picture
- 3. It is known that the image with inverse scale function y = 1 / X and primary function y = 2x-1 has an intersection point a (1, a). Whether there is a point P on the x-axis makes the triangle POA isosceles Triangle? If it exists, find out the coordinates of point P. if not, explain the reason
- 4. As shown in the figure, in the plane rectangular coordinate system, a (- 3,0), point C is on the positive half axis of Y axis, BC ‖ X axis, and BC = 5, AB intersects Y axis at point D, OD = 32 (1) (2) the parabola passing through three points a, C and B intersects with the x-axis at point E and connects with be. If the moving point m starts from point a and moves in the positive direction of x-axis, and the moving point n starts from point E and moves at a constant speed on the straight line EB, the moving speed is one unit length per second. When the moving time t is, the △ mon is a right triangle
- 5. Parabola y = 4x, straight line y = X-1 and parabola intersect at two points a and B, calculate triangle OAB area, O is the origin
- 6. If the distance between a point m on the parabola y ^ 2 = 3x and the coordinate origin o is 2, then the distance between the point m and the focus of the parabola is 2_____ .
- 7. It is known that the square of parabola y = ax + BX + C (a is less than 0) passes through point a (- 2,0), O (0,0) We know the square of parabola y = ax + BX + C (a)
- 8. If the parabola y = ax square + BX + C passes through (2,0) (0,2) (- 1,0) 3 points, its analytical formula is
- 9. Given the parabola y = ax square + BX + C, | a | = 1 / 2, the coordinates of the highest point are (* 1,5 / 2), find the value of a.b.c
- 10. The parabola y = ax square + BX + C, when x = 2, x = 0, the value of Y is equal The parabola y = ax square + BX + C, when x = 2, x = 0, the value of Y is equal, the vertex m of the parabola is on the straight line y = 3x-7, and the abscissa of the intersection point with the straight line is 4 Find the analytic formula of this parabola
- 11. Given the parabola y = 3ax ^ 2 + 2bx + C, if a = b = 1, and when - 1 is less than x is less than 1, there is only one common point between the parabola and the x-axis, find the value range of C If a + B + C = 0 and X1 = 0, the corresponding Y1 > 0, X2 = 1, the corresponding Y2 > 0, try to judge when 0
- 12. It is known that a, B and C are all positive integers, and the parabola y = AX2 + BX + C has two different intersections A and B with the x-axis. If the distance from a and B to the origin is less than 1, Finding the minimum of a + B + C
- 13. If y = ax ^ 2 + BX + C satisfies 4a-2b + C, then the parabola must pass through the point It satisfies 4a-2b + C = 2
- 14. If the parabola C1: y = ax ^ 2 + BX + C and C2: y = x ^ 2-5x + 2 are symmetric with respect to point m (3.2),
- 15. Given the plane rectangular coordinate system xoy, the parabola y = - x2 + BX + C passes through points a (4,0), B (1,3). (1) find the expression of the parabola, and write out the symmetry axis and vertex coordinates of the parabola; (2) note that the symmetry axis of the parabola is a straight line L, let the point P (m, n) on the parabola be in the fourth quadrant, the symmetry point of point P about line L is e, and the symmetry point of point e about y axis is f, if The area of oapf is 20, and the values of M and N are obtained
- 16. The parabola y = AX2 + BX + 3 intersects with the X axis at points a (1,0) and B (- 3, O), and intersects with the Y axis at point C (1). Find the analytical formula of the parabola (2) let the symmetry axis of the parabola and X
- 17. Given that the vertex coordinates of the parabola y = AX2 + BX + C are (1,4), and when x = 2, y = 1, find the values of a, B, C
- 18. Given that the parabola y = - 3x & # 178; + BX + C passes through points (3,1) and (0,4), then the value of B + C is?
- 19. For the quadratic function y = AX2 + BX + C, if y is an integer when x takes any integer, then we call the image of the function an integral point parabola (for example: y = x2 + 2x + 2). (1) please write an analytic expression of the integral point parabola whose absolute value of the quadratic coefficient is less than 1______ (need not prove) (2) please explore: is there an integral parabola with the absolute value of the quadratic coefficient less than 12? If it exists, please write down the analytical formula of one parabola; if not, please explain the reason
- 20. If the image of a linear function y = - 2x + 1 passes through the vertex of the parabola y = x2 + MX + 1 (m ≠ 0), then M=______ ．