Math masters come and find the analytic formula of the parabola

The coordinates should be drawn by themselves, and at the moment when the shot put ball is shot, the perpendicular line of the lead ball to the ground is the y-axis, the vertical foot is the coordinate origin, and the projection of the flight trajectory of the shot put on the ground is the x-axis as the coordinate axis.

Let the analytic formula of this parabola be y=ax 2+bx+c

According to the meaning of the question, we can know that there are three points (0,5 3),(b 2a,3), (10,0) in the parabola y=ax 2+bx+c, and after substitution, we get 3 equations:

c=3 b^2/4a-b^2/2a+5/3=3

100a+10b+5/3=0

The solved parabola: y=-1 12x 2+2 3x+5 3

Supplement: 1. According to the nature of the parabola, this parabola opening indicates a<0 downward. The axis of symmetry is on the positive axis of x, which means that x=-2a b>0, and the coordinates of the highest point are (-b 2a,3), and b is greater than 0, and the final result b has 2 solutions, and the value less than 0 should be rounded.

2. My answer is the same as that of the classmate upstairs, but the expression is different, mine is more detailed, I don't know if the picture of this question can be drawn? This is very important, and everything is easy to do when the picture is drawn! Any parabola y=ax 2+bx+c, its fixed-point coordinates are [-b 2a,(4ac-b 2) 4a].

Let the straight line of the point at the time of the shot be the y-axis and the ground be the x-axis, then let y=ax square + bx square + 1 and two-thirds, and then there are two lights: 1When y=0 is x=10, we get 0=10a+10b+1 and two-thirds is 4ac-b squared 4a=3 (c is 1 and two-thirds), then just solve it, do the math yourself.

Parabolic axis of symmetry formula: x=-b 2a. A line perpendicular to the alignment and passing through the focal point (i.e., a line that splits the parabola through the middle) is called the "axis of symmetry".

y=ax²+bx+c。

a(x²+b/ax)+c。

a（x²+b/ax+b²/4a²）+c-b²/4a。

a（x+b/2a）²-4ac+b²）/4a）

vertices (-b 2a, (4ac-b) 4a).

axis of symmetry x=-b 2a.

Analytical solution for parabola:

1. Know that the parabola passes through three points (x1, y1) (x2, y2) (x3, y3) and let the parabola equation be y=ax +bx+c, substitute the coordinates of each point to get a ternary system of equations, and solve the values of a, b, and c to obtain the analytical formula.

2. Know the two intersection points of the parabola and the x-axis (x1,0), (x2,0), and know that the parabola passes through a certain point (m,n), let the equation of the parabola be y=a(x-x1)(x-x2), and then substitute the point (m,n) to obtain the quadratic term coefficient a.

3. Knowing the axis of symmetry x=k, let the parabolic equation be y=a(x-k) +b, and then combine other conditions to determine the value of a,c.

4. Knowing that the minimum value of the quadratic function is p, the parabolic equation is y=a(x-k) +p,a,k should be determined according to other conditions.

Methods for finding parabolic analysis:

1. The parabola is known to pass three points.

Let the parabolic equation.

It is a standard quadratic type.

equation, substituting the coordinates of each point into the equation, to get a three-element system of equations, solve the value, that is, get the analytic formula.

2. The two intersection points of the parabola and the x-axis are known, and the parabola passes through a certain certain point.

Let the equation for the parabola be a two-point formula.

equation, substituting the determined point into the equation, solving the coefficient value, that is, obtaining the analytic formula.

3. The axis of symmetry is known.

Let the parabolic closed-line family calendar equation be an oblique truncated equation, and the value is determined by combining other conditions to obtain the analytical formula.

Problem 1: How to find the analytic formula of the parabola Knowing the three points, let y=ax 2+bx+c(a≠0), and substitute the three points to solve a, b, and c

Knowing the vertex (h, k) and another point of resistance, let y=a(x-h) 2+k(a≠0), and substitute the other point to solve a, and put parentheses.

Knowing the intersection point with the x-axis (m,0)(n,0), let y=a(x-m)(x-n)(a≠0), substitute another point, solve a, and put parentheses.

Problem 2: How to find the vertices of the parabola analytically Solve the vertices of the parabola as (h, k).

Let the parabolic equation be y=a(x-h) 2+k

Problem 3: Finding the Parabola Analytic Formula for the Coordinates of Known Points How to Find the Parabola Knowing that the parabola passes through three points (x1, y1) (x2, y2) (x3, y3) and setting the parabola Fang Chang to hold the equation as y=ax2+bx+c

Substituting the coordinates of each point to obtain a system of ternary equations, the values of a, b, and c are solved to obtain the analytic formula.

The parabolic bright plexus equation is: y 2=2px, the focal coordinates are (p 2,0), and the alignment equation is x=-p 2, so the distance from the parabolic focus to the alignment is p key draft 2-(-p 2)=p