If the square-2x-1 = 0 of the quadratic equation NX has no real root, then the image of the linear function y = (n + 1) x-n does not pass through the image line

If the square-2x-1 = 0 of the quadratic equation NX has no real root, then the image of the linear function y = (n + 1) x-n does not pass through the image line

Because it is a quadratic equation of one variable, so n ≠ 0, otherwise it is a linear equation. The equation has no real root, that is, the parabola corresponding to the equation has no intersection with the horizontal axis, so &; = (- 2) ^ 2 - 4N (- 1) < 0

If the quadratic equation nx-2x-1 = 0 with respect to X has no real root, then the image of the linear function y = (n + 1) x-n will not be ignored
If the quadratic equation nx-2x-1 = 0 with respect to X has no real root, then the image of the linear function y = (n + 1) x-n does not pass through the quadrant

When n = 0, the equation has roots n ≠ 0, Δ = 4 + 4N

The intersection coordinates of the image of function y = x & # 178; - 2x-3 and X axis are__________ ,_________ ,
The opening of the image________ The vertex coordinates are________ ,
The axis of symmetry is a straight line_________ ,
When x=_______ The function has a minimum value_________ .

The intersection coordinates of the image of function y = x & # 178; - 2x-3 and X axis are_____ （-1,0）_____ ,_____ （3,0）____ ,
The opening of the image____ Up____ The vertex coordinates are___ (1,-4)_____ ,
The axis of symmetry is a straight line____ x=1_____ ,
When x=____ 1___ The function has a minimum value____ -4_____ .

It is known that the minimum value of the quadratic function is - 3, and the abscissa of the intersection of the image and the x-axis is - 2 and 3 respectively

Analysis: the abscissa of the intersection of the image and the X axis are - 2 and 3 respectively. From the symmetry, we know that the axis of symmetry x = (- 2 + 3) / 2 = 0.5, which is also the abscissa of the vertex, and the minimum value of the quadratic function is - 3, so the vertex coordinate is (0.5, - 3). Let the quadratic function be y = a (x + 2) (x-3), and substitute the vertex coordinate (0.5, - 3) into

For quadratic function y = - 4x2 + 8x-3 (1) opening direction, symmetry axis equation, vertex coordinates; (2) finding the maximum or minimum value of function; (3) analyzing the monotonicity of function

(1) ∵ the coefficient of quadratic term of quadratic function is less than zero, the opening of parabola is downward; the axis of symmetry is x = 1; the vertex coordinates are (1,1); (2) according to the opening downward of parabola, the parabola has the highest point, the function has the maximum value at the axis of symmetry, and the maximum value of function is 1; there is no minimum value; (3) according to the opening downward of parabola, and the axis of symmetry of parabola, the parabola has the highest point The function is increasing on (- ∞, 1) and decreasing on (1, + ∞)

If the image of quadratic function y = 2x ^ 2 + 4x + 7 is symmetric with respect to the line y = 2, the corresponding function expression of the image is?

The axis of symmetry is y = 2
Then y is replaced by 2 × 2-y, that is 4-y
So 4-y = 2x & # 178; + 4x + 7
y=-2x²-4x-3

When the image of quadratic function y = - 2 (x-1) + 2 is symmetric with respect to the line y = 2, the corresponding function expression of the image is obtained

Only find the symmetric point of vertex
(1,2) symmetric points (3,2) on the line y = 2
The function expression corresponding to the image y = - 2 (x-3) + 2

Quadratic function y = ax square + BX + C

The first coefficient B and the second coefficient a jointly determine the position of the axis of symmetry
(1) When a > 0 and B have the same sign (i.e. AB > 0), the axis of symmetry is on the left side of the Y axis; because the axis of symmetry is on the left side, the axis of symmetry is less than 0, that is - B / 2a0; when B has a different sign (i.e. Ab0), B / 2a is less than 0, so a and B have different signs
To sum up, it can be simply remembered as the same left but different right, that is, when a and B have the same sign (i.e. AB > 0), the axis of symmetry is on the left of Y axis; when a and B have different signs (i.e. AB > 0), the axis of symmetry is on the left of Y axis

How to decompose quadratic function y = ax square + BX + C

It seems that many people have asked this question. There is an internal relationship between factoring and finding the root of equation
because
General formula: y = ax ^ 2 + BX + C (a, B, C are constants, a ≠ 0)
Let x1.x2 be the root of the equation AX ^ 2 + BX + C = 0
Y = ax ^ 2 + BX + C can be decomposed into y = a (x-x1) (x-x2)
Namely
y=a{x-[-b-√(b^2-4ac)]/2/a}*{x-[-b+√(b^2-4ac)]/2/a}

Quadratic function y = ax square + BX = C
When the x-axis intersects A. B, the y-axis intersects C, and the triangle ABC is a right triangle, write out a quadratic function that meets the requirements

If the two points intersecting with X axis are hypotenuse,
Then the length of the hypotenuse is the sum of two absolute values, and the right angle side is the distance from point C on the Y axis to two points a and B respectively
y=a*x^2 +bx +c
Point x = 0 (0, c) on the y-axis
Points y = 0 (m, 0) and (n, 0) on the x-axis
Then according to the square of the hypotenuse = the square of one right angle side + the square of the other right angle side
Calculate m, N and C