Image shape of quadratic function What does the shape of the image of quadratic function have to do with? y=ax^2+bx+c

Image shape of quadratic function What does the shape of the image of quadratic function have to do with? y=ax^2+bx+c

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Quadratic function: y = ax ^ 2 + BX + C (a, B, C are constants, and a is not equal to 0)
a> 0 opening up
A0, ax ^ 2 + BX + C = 0 has two unequal real roots
The analytic formula is y = a (x + B / 2A + D) ^ 2 + (4ac-b ^ 2) / 4A
The function moves up D (d > 0) units, the analytic formula is y = a (x + B / 2a) ^ 2 + (4ac-b ^ 2) / 4A + D, and down is subtraction
When a > 0, the opening is upward, the parabola is above the y-axis (the vertex is on the x-axis), and extends upward infinitely; when a < 0, the opening is downward, the parabola is below the x-axis (the vertex is on the x-axis), and extends downward infinitely
4. When drawing a parabola y = AX2, you should first list, then trace the points, and finally connect the lines. When selecting the independent variable x value in the list, always take 0 as the center, and select the integer value that is convenient for calculation and tracing the points. When tracing the lines of the points, you must connect them with smooth curves, and pay attention to the change trend
Several forms of analytic expression of quadratic function
(1) General formula: y = AX2 + BX + C (a, B, C are constants, a ≠ 0)
(2) Vertex formula: y = a (X-H) 2 + K (a, h, K are constants, a ≠ 0)
(3) Two formulas: y = a (x-x1) (x-x2), where X1 and X2 are the abscissa of the intersection of the parabola and X axis, that is, the two roots of the quadratic equation AX2 + BX + C = 0, a ≠ 0
Note: (1) any quadratic function can be transformed into vertex formula y = a (X-H) 2 + K by formula. When the vertex coordinates of the parabola are (h, K), H = 0, the vertex of the parabola y = AX2 + k is on the y-axis; when k = 0, the vertex of the parabola a (X-H) 2 is on the x-axis; when h = 0 and K = 0, the vertex of the parabola y = AX2 is on the origin
(2) When the parabola y = AX2 + BX + C has an intersection with the X axis, that is to say, the quadratic equation AX2 + BX + C = 0 has real roots X1 and x1
When x2 exists, according to the decomposition formula AX2 + BX + C = a (x-x1) (x-x2), the quadratic function y = AX2 + BX + C can be transformed into two formulas y = a (x-x1) (x-x2)
The method of finding the vertex, symmetry axis and maximum value of parabola
① Collocation method: the analytic formula is changed into the form of y = a (X-H) 2 + K, vertex coordinates (h, K), symmetry axis is a straight line x = h, if a > 0, y has a minimum value, when x = h, y has a minimum value = k, if a < 0, y has a maximum value, when x = h, y has a maximum value = K
② Formula method: directly use the vertex coordinate formula (-), to find its vertex; the symmetry axis is a straight line x = -, if a > 0, y has the minimum value, when x = - y has the minimum value, if a < 0, y has the maximum value, when x = - y has the maximum value
6. How to draw the image of quadratic function y = AX2 + BX + C
Because the image of quadratic function is a parabola and an axisymmetric figure, the simplified point tracing method and five point method are commonly used in drawing
(1) First, find out the vertex coordinates and draw the axis of symmetry;
(2) Find out the four points on the parabola about the axis of symmetry (such as the intersection with the coordinate axis, etc.);
(3) Connect the above five points with a smooth curve from left to right