Plot the function y ax2 bx c. Presentation “Function y = ax2, its graph and properties. The best way to learn

Square three-term is called a polynomial of the 2nd degree, that is, an expression of the form ax 2 + bx + c , where a ≠ 0, b, c - (usually given) real numbers, called its coefficients, x - variable.

Note: coefficient a can be any real number other than zero. Indeed, if a= 0, then ax 2 + bx + c = 0 x 2 + bx + c = 0 + bx + c = bx + c. In this case, there is no square left in the expression, so it cannot be counted square three-term. However, such expressions are binomial as, for example, 3 x 2 − 2x or x 2 + 5 can be considered as square trinomials, if we supplement them with missing monomials with zero coefficients: 3x 2 − 2x = 3x 2 − 2x + 0 and x 2 + 5 = x 2 + 0x + 5.

If the task is to determine the values ​​of the variable NS at which the square trinomial takes zero values, i.e. ax 2 + bx + c = 0, then we have quadratic equation.

If there are valid roots x 1 and x 2 of some quadratic equation, then the corresponding the trinomial can be decomposed into linear factors: ax 2 + bx + c = a(x − x 1)(x − x 2)

Comment: If the square trinomial is considered on the set of complex numbers C, which, perhaps, you have not studied yet, then it can always be decomposed into linear factors.

When there is another task, determine all the values ​​that the result of calculating the square trinomial can take for different values ​​of the variable NS, i.e. define y from expression y = ax 2 + bx + c, then we are dealing with quadratic function.

Wherein quadratic roots are zeros of the quadratic function .

A square trinomial can also be represented as

This representation is useful for plotting and studying the properties of the quadratic function of a real variable.

Quadratic function is the function given by the formula y = f(x), where f(x) is a square trinomial. Those. by a formula of the form

y = ax 2 + bx + c,

Where a ≠ 0, b, c- any real numbers. Or a transformed formula of the form

.

The graph of a quadratic function is a parabola, the vertex of which is at the point .

Note: It is not written here that the graph of the quadratic function was called a parabola. It says here that the graph of a function is a parabola. This is because mathematicians discovered and called such a curve a parabola earlier (from the Greek παραβολή - comparison, comparison, similarity), to the stage of a detailed study of the properties and graph of a quadratic function.

Parabola - the line of intersection of a straight circular cone by a plane that does not pass through the apex of the cone and is parallel to one of the generatrices of this cone.

Parabola has another interesting property, which is also used as its definition.

Parabola is a set of points on the plane, the distance from which to a certain point on the plane, called the focus of the parabola, is equal to the distance to a certain straight line, called the directrix of the parabola.

Draw a sketch of the graph a quadratic function can by characteristic points .
For example, for the function y = x 2 take points

Connecting them by hand, we build the right half of the parabola. The left one is obtained by symmetric reflection about the ordinate axis.

For building sketch of the general form of a quadratic function graph as characteristic points, it is convenient to take the coordinates of its vertex, the zeros of the function (roots of the equation), if any, the point of intersection with the ordinate axis (for x = 0, y = c) and a point symmetric to it with respect to the parabola axis (- b / a; c).

But in any case, only a sketch of the graph of a quadratic function can be plotted by points, i.e. approximate graph. To build a parabola exactly, you need to use its properties: focus and directories.
Equip yourself with paper, a ruler, a square, two buttons and a strong thread. Stick one button approximately in the center of the sheet of paper - at the point that will be the focal point of the parabola. Attach the second button to the vertex of the smaller corner of the square. On the bases of the buttons, fasten the thread so that its length between the buttons is equal to the large leg of the square. Draw a straight line that does not go through the focus of the future parabola - the headmistress of the parabola. Attach the ruler to the directrix and the square to the ruler as shown in the figure. Move the square along the ruler while pressing the pencil against the paper and against the square. Make sure the thread is taut.

Measure the distance between the focus and the directrix (I remind you that the distance between a point and a straight line is determined by the perpendicular). This is the focal parameter of the parabola p... In the coordinate system shown in the right figure, the equation of our parabola is: y = x 2/ 2p... On the scale of my drawing, I got a graph of the function y = 0,15x 2.

Comment: to build a given parabola at a given scale, you need to do the same thing, but in a different order. You need to start with the coordinate axes. Then draw the headmistress and determine the position of the focus of the parabola. And only then construct a tool from a square and a ruler. For example, to build a parabola on checkered paper, the equation of which is at = x 2, you need to place the focus at a distance of 0.5 cells from the directrix.

Function properties at = x 2

1. The domain of the function is the whole number line: D(f) = R = (−∞; ∞).
2. The range of values ​​of the function is a positive half-line: E(f) =, and the function increases over the interval. The values ​​of this function cover the entire positive part of the real axis, it is equal to zero at the point, and does not have the greatest value.

   Slide 15 describes the properties of the function y = ax 2 if negative. It is noted that its graph also passes through the origin, but all of its points, except, lie in the lower half-plane. The symmetry of the graph about the axis is noted, and equal values ​​of the function correspond to opposite values ​​of the argument. The function increases on the interval, decreases on. The values ​​of this function lie in the interval, it is equal to zero at the point, and does not have the least value.

   
   

   Summarizing the considered characteristics, slide 16 shows that the branches of the parabola are directed downward at, and upward - at. The parabola is symmetrical about the axis, and the apex of the parabola is located at the point of its intersection with the axis. The parabola y = ax 2 has a vertex - the origin.

   Also, an important conclusion about parabola transformations is displayed on slide 17. It shows the options for transforming the graph of a quadratic function. It is noted that the graph of the function y = ax 2 is transformed by symmetric display of the graph about the axis. It is also possible to compress or stretch the graph about the axis.

   On the last slide, general conclusions are made about the transformations of the function graph. Conclusions are presented that the graph of the function is obtained by symmetric transformation about the axis. A function graph is obtained by compressing or stretching the original graph from the axis. In this case, stretching from the axis by times is observed in the case when. By shrinking to the axis 1 / a times, the graph is formed in the case.

   
   

   The presentation "Function y = ax 2, its graph and properties" can be used by the teacher as a visual aid in an algebra lesson. Also, this manual covers the topic well, giving an in-depth understanding of the subject, therefore it can be offered for independent study by students. Also, this material will help the teacher to explain in the course of distance learning.

   Abstract of a lesson in algebra for the 8th grade of a secondary school

   Lesson topic: Function

   
   

   The purpose of the lesson:

   · Educational: define the concept of a quadratic function of the form (compare the graphs of functions and), show the formula for finding the coordinates of the vertex of a parabola (teach how to apply this formula in practice); to form the ability to determine the properties of a quadratic function on a graph (finding the axis of symmetry, coordinates of the vertex of a parabola, coordinates of points of intersection of the graph with the coordinate axes).

   · Developing: development of mathematical speech, the ability to correctly, consistently and rationally express your thoughts; developing the skill of correctly writing a mathematical text using symbols and notations; development of analytical thinking; development of students' cognitive activity through the ability to analyze, systematize and generalize material.

   · Educational: education of independence, the ability to listen to others, the formation of accuracy and attention in written mathematical speech.

   Lesson type: learning new material.

   Teaching methods:

   generalized reproductive, inductive heuristic.

   Requirements for knowledge and skills of students

   know what a quadratic function of the form, the formula for finding the coordinates of the vertex of a parabola; to be able to find the coordinates of the vertex of the parabola, the coordinates of the points of intersection of the graph of the function with the coordinate axes, to determine the properties of the quadratic function from the graph of the function.

   Equipment:

   
   

   Lesson plan

   I. Organizational moment (1-2 min)

   II. Knowledge update (10 min)

   III. Presentation of new material (15 min)

   IV. Securing new material (12 min)

   V. Summing up (3 min)

   Vi. Homework (2 min)

   
   

   During the classes

   I. Organizational moment

   Greetings, checking for absentees, collecting notebooks.

   II. Knowledge update

   Teacher: In today's lesson we will explore a new topic: "Function". But first, let's repeat the previously studied material.

   Frontal poll:

   1) What is called a quadratic function? (A function where given real numbers,, real variable, is called a quadratic function.)

   2) What is the graph of a quadratic function? (The graph of a quadratic function is a parabola.)

   3) What are the zeros of a quadratic function? (The zeros of a quadratic function are the values ​​at which it vanishes.)

   4) List the properties of the function. (The values ​​of the function are positive at and equal to zero at; the graph of the function is symmetric with respect to the axes of the ordinates; at the function increases, at - decreases.)

   5) List the properties of the function. (If, then the function takes positive values ​​at, if, then the function takes negative values ​​at, the value of the function is only 0; the parabola is symmetric about the ordinate; if, then the function increases at and decreases at, if, then the function increases at, decreases - at .)

   
   

   III. Presentation of new material

   Teacher: Let's start learning new material. Open your notebooks, write down the number and topic of the lesson. Pay attention to the board.

   Writing on the blackboard: Number.

   Function.

   

   Teacher: On the board, you see two function graphs. The first is the graph and the second. Let's try to compare them.

   You know the properties of the function. Based on them, and comparing our graphs, we can highlight the properties of the function.

   So what do you think the direction of the branches of the parabola will depend on?

   Students: The direction of the branches of both parabolas will depend on the coefficient.

   Teacher: Quite right. You can also notice that both parabolas have an axis of symmetry. The first graph of the function, what is the axis of symmetry?

   Students: For a parabola of the form, the axis of symmetry is the ordinate axis.

   Teacher: Right. And what is the axis of symmetry of the parabola

   
   

   Students: The axis of symmetry of a parabola is a line that passes through the apex of the parabola, parallel to the ordinate axis.

   Teacher: Right. So, the axis of symmetry of the graph of the function will be called the line passing through the vertex of the parabola, parallel to the ordinate axis.

   And the vertex of the parabola is the point with coordinates. They are determined by the formula:

   Write the formula down in a notebook and frame it.

   Writing on the board and in notebooks

   The coordinates of the vertex of the parabola.

   Teacher: Now, to make it clearer, let's look at an example.

   Example 1: Find the coordinates of the vertex of the parabola .

   Solution: By formula

   
   

   Teacher: As we have already noted, the axis of symmetry passes through the apex of the parabola. Look at the blackboard. Draw this drawing in your notebook.

   Writing on the board and in notebooks:

   

   Teacher: In the drawing: - the equation of the axis of symmetry of the parabola with the vertex at the point where the abscissa of the vertex of the parabola.

   Let's look at an example.

   Example 2: From the graph of the function, determine the equation of the axis of symmetry of the parabola.

   

   
   

   The equation of the axis of symmetry has the form:, therefore, the equation of the axis of symmetry of the given parabola.

   Answer: - the equation of the axis of symmetry.

   IV. Securing new material

   Teacher: Written on the board are tasks that need to be solved in class.

   Writing on the blackboard: № 609(3), 612(1), 613(3)

   Teacher: But first, let's solve a non-textbook example. We will decide at the blackboard.

   Example 1: Find the coordinates of the vertex of a parabola

   

   Solution: By formula

   Answer: the coordinates of the vertex of the parabola.

   Example 2: Find the coordinates of the intersection points of a parabola with coordinate axes.

   Solution: 1) With axis:

   
   

   Those.

   By Vieta's theorem:

   The points of intersection with the abscissa axis (1; 0) and (2; 0).

   2) With axle:

   The point of intersection with the y-axis (0; 2).

   Answer: (1; 0), (2; 0), (0; 2) - coordinates of points of intersection with coordinate axes.