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Quadratic function of ninth grade mathematics
Problem solving 1: put x = 1 and y = 5; respectively; X=- 1, y= 1 and y=ax? +bx can get the equations about a and b.

a+b=5

a-b= 1

Solve the equation to get a = 3 and b = 2.

The analytical formula of this parabola is y=3x? +2 times

The symmetry axis is x=-b/2a=-2/6=- 1/3.

Solution 2: Put X = 0 and Y = 0; respectively; x=2,y = 8; X=-2, y=0 and y=ax? +bx+c can get the equations about a, b and c;

c=0

4a+2b+c=8

4a-2b+c=0

Solve the equation to get a = 1, b = 2, c = 0.

The analytical formula of this parabola is y=x? +2 times

Solution 3:

(1): replace y=ax with X =-2 and Y = 4? Get:

4a=4

a= 1

(2): The analytical formula of parabola is y=x?

When x=-3 and y=(-3)? =9

(3): When x=- 1, y=(- 1)? = 1, so the parabola does not pass through the point (-1, 2).

Solution 4: Substitute X =-2 and Y = 0 into y=x? -2x+m available:

4+4+m=0

m=-8

The analytical formula of parabola is y=x? -2x-8, let y=0 to get the equation:

x? -2x-8=0

(x+2)(x-4)=0

X+2=0 or x-4=0.

X=-2 or x=4.

The coordinate of another intersection of parabola and X axis is (4,0).

Solve five problems:

(1): x = 4 and y = 0;, respectively; X= 1, y=3 and y=-x? +bx+c can get the equations about b and c;

- 16+4b+c=0

- 1+b+c=3

Solve the equations to get b = 4 and c = 0.

The analytical formula of parabola is y=-x? +4x

Make the analytical expression of parabola into vertex;

y=-x? +4x

=-(x? -4x+4)+4

=-(x-2)? +4

The symmetry axis is x=2, and the vertex coordinates are (2,4).