Quadratic function. Graph of a quadratic function

Quadratic function. Graph of a quadratic function
Quadratic function. Graph of a quadratic function
21.09.2019

A quadratic function is a function of the form:
y=a*(x^2)+b*x+c,
where a is the coefficient at the highest degree of the unknown x,
b - coefficient at unknown x,
and c is a free member.
The graph of a quadratic function is a curve called a parabola. General form parabola is shown in the figure below.

Fig.1 General view of the parabola.

There are several various ways plotting a quadratic function. We will consider the main and most general of them.

Algorithm for plotting a graph of a quadratic function y=a*(x^2)+b*x+c

1. Build a coordinate system, mark a single segment and sign coordinate axes.

2. Determine the direction of the branches of the parabola (up or down).
To do this, you need to look at the sign of the coefficient a. If plus - then the branches are directed upwards, if minus - then the branches are directed downwards.

3. Determine the x-coordinate of the top of the parabola.
To do this, you need to use the formula Tops = -b / 2 * a.

4. Determine the coordinate at the top of the parabola.
To do this, substitute in the equation of the Top = a * (x ^ 2) + b * x + c instead of x, the value of the Top found in the previous step.

5. Put the obtained point on the graph and draw an axis of symmetry through it, parallel to the coordinate axis Oy.

6. Find the points of intersection of the graph with the x-axis.
This requires solving quadratic equation a*(x^2)+b*x+c = 0 by one of known ways. If the equation has no real roots, then the graph of the function does not intersect the x-axis.

7. Find the coordinates of the point of intersection of the graph with the Oy axis.
To do this, we substitute the value x = 0 into the equation and calculate the value of y. We mark this and the point symmetrical to it on the graph.

8. Find the coordinates of an arbitrary point A (x, y)
To do this, we choose an arbitrary value of the x coordinate, and substitute it into our equation. We get the value of y at this point. Put a point on the graph. And also mark a point on the graph that is symmetrical to the point A (x, y).

9. Connect the obtained points on the chart with a smooth line and continue the chart for extreme points, to the end of the coordinate axis. Sign the graph either on the callout, or, if space permits, along the graph itself.

An example of plotting a graph

As an example, let's plot a quadratic function given by the equation y=x^2+4*x-1
1. Draw coordinate axes, sign them and mark a single segment.
2. The values ​​of the coefficients a=1, b=4, c= -1. Since a \u003d 1, which is greater than zero, the branches of the parabola are directed upwards.
3. Determine the X coordinate of the top of the parabola Tops = -b/2*a = -4/2*1 = -2.
4. Determine the coordinate At the top of the parabola
Tops = a*(x^2)+b*x+c = 1*((-2)^2) + 4*(-2) - 1 = -5.
5. Mark the vertex and draw an axis of symmetry.
6. We find the points of intersection of the graph of a quadratic function with the Ox axis. We solve the quadratic equation x^2+4*x-1=0.
x1=-2-√3 x2 = -2+√3. We mark the obtained values ​​on the graph.
7. Find the points of intersection of the graph with the Oy axis.
x=0; y=-1
8. Choose an arbitrary point B. Let it have a coordinate x=1.
Then y=(1)^2 + 4*(1)-1= 4.
9. We connect the received points and sign the chart.

This lesson in algebra is conducted as a recapitulation-generalizing in preparation for the GIA in grade 9. This is a lesson in the complex application of knowledge. The lesson should form the basic concepts of a quadratic function, its properties, graph. Students should know the definition of a quadratic function, be able to plot a quadratic function, transform it, and apply this knowledge when solving quadratic inequalities

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MOU "Secondary School No. 3 of Ershov, Saratov Region"

Grade 9

Subject: " quadratic function, its graph and properties"

The motto of the lesson: "Difficult to make easy, easy habitual, habitual pleasant"

Teacher: E.I. Kormilina

2010 - 2011 academic year.

Quadratic function, its properties and graph.

Lesson type: Lesson of complex application of knowledge.

Lesson Objectives:

  1. To reveal the degree of formation of the concept of a quadratic function among students, its properties for solving inequalities, the features of its graph.
  2. Create conditions for the formation of the ability to analyze, compare, classify graphs of quadratic functions.
  3. Continue developing the culture of plotting a quadratic function.
  4. Cultivate a sense of camaraderie, delicacy and discipline.

Lesson logic:

  1. Knowledge update
  2. Repetition
  3. Showing a sample application of a set of knowledge
  4. Independent application of knowledge
  5. Control, self-control
  6. Correction

Lesson structure:

  1. Organizational
  2. Update
  3. Application of knowledge, skills and abilities

4. Control, self-control

5. Correction

6. Information about homework

7. Summing up

8. Reflection


Slides captions:

Quadratic function, its graph and properties Our motto is: “Make the difficult easy, the easy familiar, the familiar pleasant!”

y x 0 Graph of the function y = a x , 2 for a=1 for a= -1 1 2 3 4 5 6 Х -3 -2 -1 0 1 2 3 y - 9 - 4 - 1 0 - 1 - 4 - 9 - 6 -5-4-3-2-1 1 4 9 -9 -4

Transforming a Graph of a Quadratic Function

Plotting functions y=x 2 and y=x 2 + m.

0 m X Y m 1 1 y \u003d x 2 + m, m>0

0 X Y m 1 1 m y \u003d x 2 + m, m

Plotting functions y \u003d x 2 and y \u003d (x + l) 2.

0 l l X Y 1 1 y \u003d (x + l) 2, l\u003e 0

0 l l X Y 1 1 y \u003d (x + l) 2, l

Plot the function graphs in one coordinate plane:

Find the coordinates of the vertex of the parabola: Y=2(x-4)² +5 Y=-6(x-1)² Y=-x²+12 Y= x²+4 Y= (x+7)² - 9 Y=6 x² (4;5) (1;0) (0;12) (0;4) (-7;-9) (0;0)

Graph of a quadratic function, its properties

A quadratic function is a function that can be specified by a formula of the form y=ax² + bx+c, where x is an independent variable, a, b and c are some numbers (moreover, a ≠ 0). For example: y \u003d 5x ² + 6x + 3, y \u003d -7x ² + 8x-2, y \u003d 0.8x ² +5, y \u003d ¾ x ² -8x, y \u003d -12x ² quadratic functions

The graph of a quadratic function is a parabola whose branches are directed upwards (if a > 0) or downwards (if a 0). y \u003d -7 x ² -x + 3 - the graph is a parabola, the branches of which are directed downwards (because a \u003d -7, and

Determine the coordinate of the parabola vertex using the formulas: Mark this point on the coordinate plane. Draw the axis of symmetry of the parabola through the vertex of the parabola Find the zeros of the function and mark them on the number line Find the coordinates of two additional points and those symmetrical to them Draw a curve of the parabola. Solution algorithm

Build a graph of the function y \u003d 2x ² + 4x-6, describe its properties

X Y 1 1 -2 2 3 -1 1. D(y) = R 2. y=0 if x= 1; -3 3. y > 0 if x 4. y ↓ if x y if x 5. y naim = -8 if x= -1 y naib does not exist. 6. E (y): Check yourself: y

Solving a Quadratic Inequality Using a Graph of a Quadratic Function

Definition: Inequality, left side which is a polynomial of the second degree, and the right one is zero, is called an inequality of the second degree. All quadratic inequalities can be reduced to one of the following types: 1) ax 2 + bx + c >0; 2) ax 2 + bx + c

Which of the inequalities would you call inequalities of the second degree: 1) 6x 2 -13x>0; 2) x 2 -3 x -14>0; 3) (5+ x)(x -4)>7; 4) ; 5) 6) 8 x 2 >0; 7) (x -5) 2 -25>0;

Which numbers are solutions to the inequality? 1 -3 0 -1 5 -4 -2 0.5 ? ? ? ? ? ? ? ?

What is the number of roots of the equation a x 2 + b x + c \u003d 0 and the sign of the coefficient a if the graph of the corresponding quadratic function is located as follows: f a b c d e

Name the intervals of sign constancy of a function if its graph is located in the indicated way: Ι variant. Ι I option. c b a a c b

Name the intervals of constant sign of the function if its graph is located in the indicated way: Ι variant f(x)>0 for x Є R f(x) 0 for x Є (-∞ ;1) U (2.5;+∞); f(x)

Name the intervals of constancy of the function if its graph is located in the indicated way: Ι option f(x)>0 for x Є (-∞ ;-3) U (-3;+∞) f(x) 0 for x Є (-∞ ; 0.5) U (0.5;+∞) f(x)

Name the intervals of constancy of the function if its graph is located in the indicated way Ι option f (x)> 0 for x Є (-∞ ;-4) U (3; + ∞); f(x) 0 __________ ; f(x)

Algorithm for solving second degree inequalities with one variable 5x 2 +9x-2 0 (a x 2 + b x+ c 0 (y

Algorithm for solving second degree inequalities with one variable 5x 2 +9x-2 0 (a x 2 + b x+ c 0 (y 0 (y

In table 1, find the correct solution to inequality 1, in table 2 - the solution to inequality 2: 1. 2. Table 1 a c c d a b c d Table 2

In table 1, find the correct solution to inequality 1, in table 2, the solution to inequality 2: 1. 2. Table 1 a c c d a b c d Table 2

In table 1, find the correct solution to inequality 1, in table 2, the solution to inequality 2: 1. 2. Table 1 a c c d a b c d Table 2

Lesson summary When solving these tasks, we managed to systematize knowledge about the use of a quadratic function. Mathematics is meaningful, exciting and available field activities that give the student rich food for thought. The properties of the quadratic function underlie the solution square inequalities. Many physical relationships are expressed by a quadratic function; for example, a stone thrown upwards with speed v 0 is at the moment t at a distance s (t)=- q \2 t 2+ v 0 t from the earth's surface (here q is the acceleration of gravity); the amount of heat Q released during the passage of current in a conductor with resistance R is expressed in terms of current strength I by the formula Q \u003d RI 2. Knowledge of the properties of a quadratic function allows you to calculate the flight range of a body thrown vertically upwards or at a certain angle. This is used in the defense industry.

Unfinished Sentence Task: Complete one of the three sentences that best fits your condition. “It is difficult for me to complete tasks and solve problems, because ...” “It is easy for me to complete tasks and solve problems, because ...” “It is pleasant and interesting for me to complete tasks and solve problems, because ...”

Homework Textbook No. 142; №190


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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 reflecting 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 the paraboloid, then the rays, reflected from 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 registration . For further work With a teacher, you can choose the tariff plan that suits you.

Do you have any questions? Don't know how to graph a quadratic function?
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