How to describe the properties of the graph of a parabola function. Quadratic function and its graph

In the lessons of mathematics at school, you have already become acquainted with the simplest properties and the graph of a function y=x2. Let's expand our knowledge quadratic function.

Exercise 1.

Plot a function y=x2. Scale: 1 = 2 cm. Mark a point on the Oy axis F(0; 1/4). Using a compass or strip of paper, measure the distance from the point F to some point M parabolas. Then pin the strip at point M and rotate it around this point so that it becomes vertical. The end of the strip will fall slightly below the x-axis (Fig. 1). Mark on the strip how far it goes beyond the x-axis. Take now another point on the parabola and repeat the measurement again. How much has the edge of the strip now dropped beyond the x-axis?

Result: no matter what point on the parabola y \u003d x 2 you take, the distance from this point to the point F (0; 1/4) will be greater than the distance from the same point to the x-axis always by the same number - by 1/4.

It can be said differently: the distance from any point of the parabola to the point (0; 1/4) is equal to the distance from the same point of the parabola to the line y = -1/4. This wonderful point F(0; 1/4) is called focus parabolas y \u003d x 2, and the straight line y \u003d -1/4 - headmistress this parabola. Each parabola has a directrix and a focus.

Interesting properties of a parabola:

1. Any point of the parabola is equidistant from some point, called the focus of the parabola, and some line, called its directrix.

2. If you rotate a parabola around the axis of symmetry (for example, a parabola y \u003d x 2 around the Oy axis), you get a very interesting surface, which is called a paraboloid of revolution.

The surface of a liquid in a rotating vessel has the shape of a paraboloid of revolution. You can see this surface if you stir hard with a spoon in an incomplete glass of tea, and then remove the spoon.

3. If you throw a stone in the void at a certain angle to the horizon, then it will fly along a parabola (Fig. 2).

4. If you intersect the surface of the cone with a plane parallel to any one of its generators, then in the section you get a parabola (Fig. 3).

5. In amusement parks, they sometimes arrange a funny attraction called the Paraboloid of Wonders. To each of those standing inside the rotating paraboloid, it seems that he is standing on the floor, and the rest of the people, by some miracle, keep on the walls.

6. In mirror telescopes, parabolic mirrors are also used: the light of a distant star, traveling in a parallel beam, falling on the telescope mirror, is collected in focus.

7. For spotlights, the mirror is usually made in the form of a paraboloid. If you place a light source at the focus of a paraboloid, then the rays, reflected from the parabolic mirror, form a parallel beam.

Plotting a Quadratic Function

In the lessons of mathematics, you studied how to get graphs of functions of the form from the graph of the function y \u003d x 2:

1) y=ax2– expansion of the graph y = x 2 along the Oy axis in |a| times (for |a|< 0 – это сжатие в 1/|a| раз, rice. 4).

2) y=x2+n– graph shift by n units along the Oy axis, and if n > 0, then the shift is up, and if n< 0, то вниз, (или же можно переносить ось абсцисс).

3) y = (x + m)2– graph shift by m units along the Ox axis: if m< 0, то вправо, а если m >0, then to the left, (Fig. 5).

4) y=-x2- symmetrical display about the Ox axis of the graph y = x 2 .

Let's dwell on plotting a function graph in more detail. y = a(x - m) 2 + n.

A quadratic function of the form y = ax 2 + bx + c can always be reduced to the form

y \u003d a (x - m) 2 + n, where m \u003d -b / (2a), n \u003d - (b 2 - 4ac) / (4a).

Let's prove it.

Really,

y = ax 2 + bx + c = a(x 2 + (b/a) x + c/a) =

A(x 2 + 2x (b/a) + b 2 /(4a 2) - b 2 /(4a 2) + c/a) =

A((x + b/2a) 2 - (b 2 - 4ac)/(4a 2)) = a(x + b/2a) 2 - (b 2 - 4ac)/(4a).

Let us introduce new notation.

Let be m = -b/(2a), a n \u003d - (b 2 - 4ac) / (4a),

then we get y = a(x - m) 2 + n or y - n = a(x - m) 2 .

Let's make some more substitutions: let y - n = Y, x - m = X (*).

Then we get the function Y = aX 2 , whose graph is a parabola.

The vertex of the parabola is at the origin. x=0; Y = 0.

Substituting the coordinates of the vertex in (*), we obtain the coordinates of the vertex of the graph y = a(x - m) 2 + n: x = m, y = n.

Thus, in order to plot a quadratic function represented as

y = a(x - m) 2 + n

by transformation, you can proceed as follows:

a) build a graph of the function y = x 2 ;

b) by parallel translation along the Ox axis by m units and along the Oy axis by n units - transfer the top of the parabola from the origin to the point with coordinates (m; n) (Fig. 6).

Write transformations:

y = x 2 → y = (x - m) 2 → y = a(x - m) 2 → y = a(x - m) 2 + n.

Example.

Using transformations, construct a graph of the function y = 2(x - 3) 2 in the Cartesian coordinate system – 2.

Decision.

Chain of transformations:

y=x2 (1) → y = (x - 3) 2 (2) → y = 2(x – 3) 2 (3) → y = 2(x - 3) 2 - 2 (4) .

The construction of the graph is shown in rice. 7.

You can practice quadratic function plotting by yourself. For example, build a graph of the function y = 2(x + 3) 2 + 2 in one coordinate system using transformations. If you have any questions or want to get advice from a teacher, then you have the opportunity to conduct free 25-minute lesson with an online tutor after . For further work with the teacher, you can choose the one that suits you

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