## The image of the known primary function passes through the vertex of the parabola y = (x + 1) square + 2 and the coordinate origin to find the relationship of the primary function

Fixed point (- 1,2)

So the primary function is y = - 2x;

### The graph of the first-order function is known to find the relationship between the vertex of the parabola y = (x + 1) square + 2 and the coordinate origin (1) (2) Judge whether the point (- 2,5) is on the image of this parabola

Not in a parabolic upward Hello! Solution (1) the coordinates of two points c and D can be obtained as shown in the figure ∵ a (√ 3,0) ∵ OA = √ 3, and the radius of ∵ A is 2 √ 3, connecting ad ∵ AC = ad = AB = 2 √ 3 ∵ co = 3 √ 3, B0 = √ 3 at RT Δ In AOD, do = √ (AD ^ 2-ao ^ 2) = 3 ‡ D (0, - 3), C (3 √ 3,0), B (- √ 3,0) put

### Given the square of the parabola y = - X - 2x + the square of a - 1 / 2, determine the quadrant of the vertex of the parabola. Suppose the parabola passes through the origin, find the vertex coordinates

y=-x ²- 2x+a ²- 1/2=-(x+1) ²+ a ²+ 1/2

So the vertex coordinates are (- 1, a) ²+ 1/2)

Because a ²+ 1/2＞0

So the vertex is in the second quadrant

If the parabola passes through the origin

So a ²- 1/2=0

So a ²= 1/2

So the vertex coordinates are (- 1,1)

### It is known that the square of quadratic function y = x - (M-3) x-m is a parabola 1. When finding the value of M, the distance between the two intersections of the parabola and the x-axis is 3 2. When m is what value, the two roots of square - (M-3) x-m = 0 of equation x are negative

1. When finding the value of M, the distance between the two intersections of the parabola and the x-axis is 3

（m-3）^2+4m=3^2=9

m^2-2m=0

m=0 or m=2

2. When m is what value, the two roots of square - (M-3) x-m = 0 of equation x are negative

x1*x2=-m＞0 m＜0

x1+x2=m-3＜0 m＜3

When m < 0, the two roots of the square - (M-3) x-m = 0 of equation x are negative

### As shown in the figure, it is known that the image of the quadratic function y = X2 - (M-3) x-m is a parabola (1) When trying to find the value of M, the distance between the two intersections of the parabola and the x-axis is 3? (2) When m is what value, the two roots of equation X2 - (M-3) x-m = 0 are negative? (3) Let the vertex of the parabola be m and the intersection P and Q with the x-axis, and find the area of the shortest △ MPQ when PQ

(1) According to the meaning of the question

(m-3)2-4•(-m)

1=3，

The solution is M1 = 0, M2 = 2,

That is, when m is 0 or 2, the distance between the two intersections of the parabola and the x-axis is 3;

（2）∵△=（m-3）2-4•（-m）=m2-2m+9=（m-1）2+8＞0，

The equation X2 - (M-3) x-m = 0 has two real roots,

Let the two roots of equation X2 - (M-3) x-m = 0 be x1, X2,

Then X1 + x2 = M-3 ＜ 0, x1 • x2 = - M ＞ 0,

∴m＜0；

（3）∵PQ=

(m-3)2-4•(-m)=

(m-1)2+8，

When m = 1, PQ is the shortest, and the shortest value is

8=2

2. At this time, the parabola analytical formula is y = x2 + 2x-1 = (x + 1) 2-2,

The coordinates of point m are (- 1, - 2),

△ area of MPQ = 1

two × two × two

2=2

2．

### As shown in the figure, it is known that the image of the quadratic function y = X2 - (M-3) x-m is a parabola (1) When trying to find the value of M, the distance between the two intersections of the parabola and the x-axis is 3? (2) When m is what value, the two roots of equation X2 - (M-3) x-m = 0 are negative? (3) Let the vertex of the parabola be m and the intersection P and Q with the x-axis, and find the area of the shortest △ MPQ when PQ

(1) According to the meaning of the question

(m-3)2-4•(-m)

1=3，

The solution is M1 = 0, M2 = 2,

That is, when m is 0 or 2, the distance between the two intersections of the parabola and the x-axis is 3;

（2）∵△=（m-3）2-4•（-m）=m2-2m+9=（m-1）2+8＞0，

The equation X2 - (M-3) x-m = 0 has two real roots,

Let the two roots of equation X2 - (M-3) x-m = 0 be x1, X2,

Then X1 + x2 = M-3 ＜ 0, x1 • x2 = - M ＞ 0,

∴m＜0；

（3）∵PQ=

(m-3)2-4•(-m)=

(m-1)2+8，

When m = 1, PQ is the shortest, and the shortest value is

8=2

2. At this time, the parabola analytical formula is y = x2 + 2x-1 = (x + 1) 2-2,

The coordinates of point m are (- 1, - 2),

△ area of MPQ = 1

two × two × two

2=2

2．