The relationship between the parabola and the vertex coordinates, what are the vertex coordinates of

Vertex coordinates. It is used to represent a quadratic function.

The reference index for the position of the parabolic vertice, vertex formula: y=a(x-h) +k (a≠0, k is constant) vertex coordinates: [-b 2a,(4ac-b) 4a].

When h>0, the graph of y=a(x-h) can be represented by the parabola y=ax2; Move h units parallel to the right to obtain;

When h<0, move | parallel to the lefth|units get;

When h>0 and k>0, move the parabola y=ax parallel to the right by h units, and then move k units upwards to obtain the image of y=a(x-h)+k. Waiting.

Points and lines of a parabola.

The focus is not on the alignment.

Above. A parabola is the trajectory of a point in that plane that is equidistant from the alignment and focus.

Another description of a parabola is as a conic section, formed by the intersection of a conical surface and a plane parallel to the tapered busbar. The third description is algebra.

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."

The point on the parabola that intersects the axis of symmetry is called the "vertex" and is the point where the parabola bends most sharply. The distance between the vertex and the focal point, measured along the axis of symmetry, is the focal length. "Straight lines" are parallel lines of a parabola.

And through the focus.

The parabola can be opened up, down, left, right or in any other direction. Any parabola can be repositioned and repositioned to fit any other parabola - that is, all old parabolas are geometrically similar.

Vertex coordinates. is used to indicateQuadratic functionsThe reference index for the position of the parabolic vertice, vertex formula: y=a(x-h) +k (a≠0, k is constant) vertex coordinates: [-b 2a, (4ac-b) 4a].

When h>0, the image of y=a(x-h) can be changed by the parabola y=ax2; Move h units parallel to the right to obtain;

When h<0, move | parallel to the lefth|A single balance calendar bit obtained.

When h>0 and k>0, move the parabola y=ax parallel to the right by h units, and then move k units upwards to obtain the image of y=a(x-h)+k.

The primary term coefficient b and the quadratic term coefficient.

a. Jointly determine the axis of symmetry.

location. When a>0 and b have the same sign (i.e., ab>0), the axis of symmetry is left on the y-axis; Because the axis of symmetry is on the left, the axis of symmetry is less than 0, that is, - b 2a<0, so b 2a should be greater than 0, so a and b should have the same sign.

When a>0 and b (i.e., ab<0), the axis of symmetry is to the right of the y-axis. Because the axis of symmetry is on the right, the axis of symmetry should be greater than 0, that is, - b 2a>0, so b 2a should be less than 0, so a and b should have different signs.

It can be simply remembered as the left is the same as the right, that is, when the axis of symmetry is on the left of the y-axis, a and b have the same sign (i.e., a>0, b>0 or a<0, b "rotten state 0"; When the axis of symmetry is on the right side of the y-axis, a is different from b (i.e., a0 or a>0, b<0) (ab<0).

In fact, b has its own geometric meaning: a quadratic function image.

The tangent of this quadratic function image at the intersection with the y-axis.

(a one-time function.

The value of the slope k. It can be obtained by finding a derivative of the quadratic function.

Parabolic vertex coordinates formula:

The vertex coordinates formula for y=ax +bx+c(a≠0) is (-b 2a, (4ac-b) 4a).

The vertex coordinates of y=ax +bx are (-b 2a, -b 4a).

Parabolic standard equations.

Right opening parabola: y 2 = 2px.

Left opening parabola: y 2 = -2px.

Upper opening parabola: x 2 = 2py y = ax 2 (a is greater than or equal to 0).

Lower opening parabola: x 2 = -2py y = ax 2 (a less than or equal to 0).

p is the focal distance (p>0)].

Peculiarity. In the parabola y 2 = 2px, the focus is (p 2,0) and the equation for the alignment is x = -p 2, eccentricity e = 1, range: x 0.

In the parabola y 2 = 2px, the focus is (p 2,0) and the equation for the alignment is x = -p 2, eccentricity e = 1, range: x 0.

In the parabola x 2=2py, the focal point is (0, p 2), the equation for the alignment is y = -p 2, the eccentricity e = 1, and the range: y 0.

In the parabola x 2=2py, the focal point is (0, p 2), the equation for the alignment is y = -p 2, the eccentricity e = 1, and the range: y 0.

To require the vertex coordinates of a parabola, the following formula can be used: for the general form of the parabolic equation y = ax 2 + bx + c, where a, b, c are constants, and the x coordinates of the vertices can be obtained by the formula x = b 2a).

There are also several ways to solve for the vertex coordinates of a parabola

Method 1: Use a perfectly squared formula.

For the parabolic equation of the general form y = ax 2 + bx + c, where a, b, c are constants, the x-coordinates of the vertices can be found quietly by the formula x = b 2a). Then, the obtained x-coordinate is substituted into the parabolic equation to calculate the corresponding y-coordinate.

For example, for the parabolic equation y = 2x 2 + 4x + 1, first calculate the x coordinates: x = b 2a) =4 2*2) =1 and then substitute x = 1 into the parabolic equation to calculate the y coordinates: y = 2*(-1) 2 + 4*(-1) +1 = 2 + 4) +1 = 1 So, the vertex coordinates of the parabola are (-1, -1).

Method 2: Complete the square.

For the parabolic equation of the general form y = ax 2 + bx + c, it can be written as the standard form y = a(x - h) 2 + k, where (h, k) are the vertex coordinates. First, the parabolic equation is squared, i.e., the coefficients of the x 2 and x terms are moved to one side of the equation respectively, giving y - c = a (x 2 + bx a). Then, divide the coefficient of the x 2 term by a and square half of the coefficient of the x term to get y - c = a (x 2 + bx a + b 2a) 2).

Then square the contents in the right parentheses to get y - c = a(x + b 2a) 2 + b 2 - 4ac) 4a. Finally, move the constant term on the right to the side of the equation and get y = a(x + b 2a) 2 + b 2 - 4ac) 4a + c. From this standard form, you can read the vertex coordinates directly as (-b 2a, (b 2 - 4ac) 4a + c).

For example, for the parabolic equation y = 2x 2 + 4x + 1, the coordinates of the vertices can be obtained as (-4 (2*2), 4 2 - 4*2*1) (4*2) +1) =1, -1 according to the standard formula. So, the vertex coordinates of the parabola are (-1, -1).

These are common ways to solve parabolic vertex coordinates, and depending on the situation, you can choose the most suitable method for the calculation.

The basic knowledge points of parabola are as follows:

1. The parabola is an axisymmetric figure.

The axis of symmetry is the straight line x=—b 2a, and the only intersection point between the axis of symmetry and the parabola is the vertex p of the parabola, in particular, when b = 0, the axis of symmetry of the parabola is the y-axis (i.e., the straight line x=0).

2. The old reed parabola has a vertex p

The coordinates are: p(—b 2a,(4ac—b 2) 4a) when —b 2a=0, p is on the y axis; When b 2—4ac=0, p is on the x-axis.

3. The quadratic term coefficient a determines the direction and size of the opening of the parabola.

When a0, the parabola opens upwards; When a0, the parabola opens downward, a|The larger it is, the smaller the opening of the parabola.

4. The primary term coefficient b and the quadratic term coefficient a jointly determine the position of the symmetry axis.

When A and B have the same sign (i.e., ab0), the axis of symmetry is to the left of the y-axis; When A and B have different signs (i.e., ab0), the axis of symmetry is to the right of the y-axis.

5. The constant term c determines the intersection of the parabola and the y-axis.

The parabola intersects with the y-axis at (0,c).

6. The number of intersection points between the parabola and the x-axis.

b 2—4ac0, the parabola and the x-axis have two intersections.

b 2—4ac=0, the parabola has 1 intersection point with the x-axis.

b 2-4ac0, there is no intersection between the parabola and the x-axis. The value of x is an imaginary number (the opposite of the value of x=-bb 2-4ac, multiplied by the imaginary number i, and the whole equation is divided by 2a).