Quadratic function and its graph. Quadratic function and its graph

The function y=x^2 is called a quadratic function. The graph of a quadratic function is a parabola. General view The parabola is shown in the figure below.

Quadratic function

Fig 1. General view of the parabola

As can be seen from the graph, it is symmetrical about the Oy axis. The Oy axis is called the axis of symmetry of the parabola. This means that if you draw a straight line on the graph parallel to the Ox axis above this axis. Then it will intersect the parabola at two points. The distance from these points to the Oy axis will be the same.

The axis of symmetry divides the graph of a parabola into two parts. These parts are called branches of the parabola. And the point of a parabola that lies on the axis of symmetry is called the vertex of the parabola. That is, the axis of symmetry passes through the vertex of the parabola. The coordinates of this point are (0;0).

Basic properties of a quadratic function

1. At x =0, y=0, and y>0 at x0

2. The quadratic function reaches its minimum value at its vertex. Ymin at x=0; It should also be noted that maximum value the function does not exist.

3. The function decreases on the interval (-∞;0] and increases on the interval

Let us illustrate the fact that the same function value can be achieved with several argument values.

A function of the form where is called quadratic function.

Graph of a quadratic function – parabola.

Let's consider the cases:

I CASE, CLASSICAL PARABOLA

That is , ,

To construct, fill out the table by substituting the x values into the formula:

Mark the points (0;0); (1;1); (-1;1), etc. on coordinate plane(the smaller the step we take the x values (in this case step 1), and the more x values we take, the smoother the curve will be), we get a parabola:

It is easy to see that if we take the case , , , that is, then we get a parabola that is symmetrical about the axis (oh). It’s easy to verify this by filling out a similar table:

II CASE, “a” IS DIFFERENT FROM UNIT

What will happen if we take , , ? How will the behavior of the parabola change? With title="Rendered by QuickLaTeX.com" height="20" width="55" style="vertical-align: -5px;"> парабола изменит форму, она “похудеет” по сравнению с параболой (не верите – заполните соответствующую таблицу – и убедитесь сами):!}

In the first picture (see above) it is clearly visible that the points from the table for the parabola (1;1), (-1;1) were transformed into points (1;4), (1;-4), that is, with the same values, the ordinate of each point is multiplied by 4. This will happen to all key points of the original table. We reason similarly in the cases of pictures 2 and 3.

And when the parabola “becomes wider” than the parabola:

Let's summarize:

1)The sign of the coefficient determines the direction of the branches. With title="Rendered by QuickLaTeX.com" height="14" width="47" style="vertical-align: 0px;"> ветви направлены вверх, при - вниз. !}

2) Absolute value coefficient (modulus) is responsible for the “expansion” and “compression” of the parabola. The larger , the narrower the parabola; the smaller |a|, the wider the parabola.

III CASE, “C” APPEARS

Now let's introduce into the game (that is, consider the case when), we will consider parabolas of the form . It is not difficult to guess (you can always refer to the table) that the parabola will shift up or down along the axis depending on the sign:

IV CASE, “b” APPEARS

When will the parabola “break away” from the axis and finally “walk” along the entire coordinate plane? When will it stop being equal?

Here to construct a parabola we need formula for calculating the vertex: , .

So at this point (as at point (0;0) new system coordinates) we will build a parabola, which we can already do. If we are dealing with the case, then from the vertex we put one unit segment to the right, one up, - the resulting point is ours (similarly, a step to the left, a step up is our point); if we are dealing with, for example, then from the vertex we put one unit segment to the right, two - upward, etc.

For example, the vertex of a parabola:

Now the main thing to understand is that at this vertex we will build a parabola according to the parabola pattern, because in our case.

When constructing a parabola after finding the coordinates of the vertex veryIt is convenient to consider the following points:

1) parabola will definitely pass through the point . Indeed, substituting x=0 into the formula, we obtain that . That is, the ordinate of the point of intersection of the parabola with the axis (oy) is . In our example (above), the parabola intersects the ordinate at point , since .

2) axis of symmetry parabolas is a straight line, so all points of the parabola will be symmetrical about it. In our example, we immediately take the point (0; -2) and build it symmetrical relative to the axis of symmetry of the parabola, we get the point (4; -2) through which the parabola will pass.

3) Equating to , we find out the points of intersection of the parabola with the axis (oh). To do this, we solve the equation. Depending on the discriminant, we will get one (, ), two ( title="Rendered by QuickLaTeX.com" height="14" width="54" style="vertical-align: 0px;">, ) или нИсколько () точек пересечения с осью (ох) !} . In the previous example, our root of the discriminant is not an integer; when constructing, it doesn’t make much sense for us to find the roots, but we clearly see that we will have two points of intersection with the axis (oh) (since title="Rendered by QuickLaTeX.com" height="14" width="54" style="vertical-align: 0px;">), хотя, в общем, это видно и без дискриминанта.!}

So let's work it out

Algorithm for constructing a parabola if it is given in the form

1) determine the direction of the branches (a>0 – up, a<0 – вниз)

2) we find the coordinates of the vertex of the parabola using the formula , .

3) we find the point of intersection of the parabola with the axis (oy) using the free term, construct a point symmetrical to this point with respect to the symmetry axis of the parabola (it should be noted that it happens that it is unprofitable to mark this point, for example, because the value is large... we skip this point...)

4) At the found point - the vertex of the parabola (as at the point (0;0) of the new coordinate system) we construct a parabola. If title="Rendered by QuickLaTeX.com" height="20" width="55" style="vertical-align: -5px;">, то парабола становится у’же по сравнению с , если , то парабола расширяется по сравнению с !}

5) We find the points of intersection of the parabola with the axis (oy) (if they have not yet “surfaced”) by solving the equation

Example 1

Example 2

Note 1. If the parabola is initially given to us in the form , where are some numbers (for example, ), then it will be even easier to construct it, because we have already been given the coordinates of the vertex . Why?

Let's take a quadratic trinomial and isolate the complete square in it: Look, we got that , . You and I previously called the vertex of a parabola, that is, now,.

For example, . We mark the vertex of the parabola on the plane, we understand that the branches are directed downward, the parabola is expanded (relative to ). That is, we carry out points 1; 3; 4; 5 from the algorithm for constructing a parabola (see above).

Note 2. If the parabola is given in a form similar to this (that is, presented as a product of two linear factors), then we immediately see the points of intersection of the parabola with the axis (ox). In this case – (0;0) and (4;0). For the rest, we act according to the algorithm, opening the brackets.

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“Quadratic function” - 1 Definition of a quadratic function 2 Properties of a function 3 Graphs of a function 4 Quadratic inequalities 5 Conclusion. Properties: Inequalities: Prepared by 8A class student Andrey Gerlitz. Plan: Graph: -Intervals of monotonicity for a > 0 for a< 0. Quadratic function. Quadratic functions have been used for many years.

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Previously, we studied other functions, for example linear, let us recall its standard form:

hence the obvious fundamental difference - in linear function X stands in the first degree, and in the new function we are beginning to study, X stands to the second power.

Recall that the graph of a linear function is a straight line, and the graph of a function, as we will see, is a curve called a parabola.

Let's start by finding out where the formula came from. The explanation is this: if we are given a square with side A, then we can calculate its area like this:

If we change the length of the side of a square, then its area will change.

So, this is one of the reasons why the function is studied

Recall that the variable X- this is an independent variable, or argument; in a physical interpretation, it can be, for example, time. Distance is, on the contrary, a dependent variable; it depends on time. The dependent variable or function is a variable at.

This is the law of correspondence, according to which each value X a single value is assigned at.

Any correspondence law must satisfy the requirement of uniqueness from argument to function. In a physical interpretation, this looks quite clear using the example of the dependence of distance on time: at each moment of time we are at a certain distance from the starting point, and it is impossible to be both 10 and 20 kilometers from the beginning of the journey at the same time at time t.

At the same time, each function value can be achieved with several argument values.

So, we need to build a graph of the function, for this we need to make a table. Then study the function and its properties using the graph. But even before constructing a graph based on the type of function, we can say something about its properties: it is obvious that at cannot accept negative values, because

So, let's make a table:

Rice. 1

From the graph it is easy to note the following properties:

Axis at- this is the axis of symmetry of the graph;

The vertex of the parabola is point (0; 0);

We see that the function only accepts non-negative values;

In the interval where the function decreases, and on the interval where the function increases;

The function acquires its smallest value at the vertex, ;

There is no greatest value of a function;

Example 1

Condition:

Solution:

Because X by condition changes on a specific interval, we can say about the function that it increases and changes on the interval . The function has a minimum value and a maximum value on this interval

Rice. 2. Graph of the function y = x 2 , x ∈

Example 2

Condition: Find the greatest and smallest value Features:

Solution:

X changes over the interval, which means at decreases on the interval while and increases on the interval while .

So, the limits of change X, and the limits of change at, and, therefore, on a given interval there is both a minimum value of the function and a maximum