Y x 2 is a quadratic function. GIA

Name the coordinates of points symmetrical to these points
relative to the y-axis:
y
(- 2; 6)
(2; 6)
(- 1; 4)
(1; 4)
(0; 0)
(0; 0)
(- 3; - 5)
(3; - 5)
X

The graph shows that the OY axis divides the parabola into symmetrical
left and right parts (parabola branches), at the point with coordinates (0; 0)
(vertex of the parabola) the value of the function x 2 is the smallest.
The function is not of greatest importance. The vertex of a parabola is
the point of intersection of the graph with the axis of symmetry OY.
In the section of the graph for x ∈ (– ∞; 0 ] the function decreases,
and for x ∈ [ 0; + ∞) increases.

The graph of the function y = x 2 + 3 is the same parabola, but its
the vertex is at the point with coordinates (0; 3) .

Find the value of the function
y = 5x + 4 if:
x=-1
y = - 1 y = 19
x=-2
y=-6
y=29
x=3
x=5

Specify
function domain:
y = 16 – 5x
10
y
X
x – any
number
x≠0
1
y
x 7
4x 1
y
5
x≠7

Graph the functions:
1).U=2X+3
2).U=-2X-1;
3).

10.

Mathematical
study
Topic: Function y = x2

11.

Build
schedule
functions
y = x2

12.

Algorithm for constructing a parabola..
1.Fill out the table of X and Y values.
2. Mark points in the coordinate plane,
whose coordinates are indicated in the table.
3.Connect these points with a smooth line.

13.

Incredible
but it's a fact!
Parabola Pass

14.

Did you know?
The trajectory of a stone thrown under
angle to the horizon, will fly along
parabola.

15. Properties of the function y = x2

*
Function properties
y=
2
x

16.

*Domain
functions D(f):
x – any number.
*Value range
functions E(f):
all values ​​of y ≥ 0.

17.

*If
x = 0, then y = 0.
Graph of a function
goes through
origin.

18.

II
I
*If
x ≠ 0,
then y > 0.
All graph points
functions other than point
(0; 0), located
above the x axis.

19.

*Opposite
x values
matches one
and the same value for y.
Graph of a function
symmetrical
relative to the axis
ordinate

20.

Geometric
properties of a parabola
*Has symmetry
*The axis cuts the parabola into
two parts: branches
parabolas
*Point (0; 0) – vertex
parabolas
*The parabola touches the axis
abscissa
Axis
symmetry

21.

Find y if:
“Knowledge is a tool,
not the goal"
L. N. Tolstoy
x = 1.4
- 1,4
y = 1.96
x = 2.6
-2,6
y = 6.76
x = 3.1
- 3,1
y = 9.61
Find x if:
y=6
y=4
x ≈ 2.5 x ≈ -2.5
x=2 x=-2

22.

build in one
coordinate system
graphs of two functions
1. Case:
y=x2
Y=x+1
2. case:
Y=x2
y= -1

23.

Find
multiple values
x, for which
function values:
less than 4
more than 4

24.

Does the graph of the function y = x2 belong to the point:
P(-18; 324)
R(-99; -9081)
belongs
do not belong
S(17; 279)
do not belong
Without performing calculations, determine which of the
points do not belong to the graph of the function y = x2:
(-1; 1)
*
(-2; 4)
(0; 8)
(3; -9)
(1,8; 3,24)
At what values ​​of a does the point P(a; 64) belong to the graph of the function y = x2.
a = 8; a = - 8
(16; 0)

25.

Algorithm for solving the equation
graphically
1. Build in one system
coordinates of the graphics of the functions standing
on the left and right sides of the equation.
2. Find the abscissa of the intersection points
graphs. These will be the roots
equations
3. If there are no points of intersection, then
the equation has no roots

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

hence the obvious fundamental difference- in a linear function X stands in the first degree, and in that new feature, which we begin 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 take negative values, since

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 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 the 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

Rice. 3. Graph of the function y = x 2 , x ∈ [-3; 2]

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

Lesson on the topic: "Graph and properties of the function $y=x^2$. Examples of plotting graphs"

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Function is the dependence of one variable on another.

Graph of a function– graphical representation of the function.

Function properties

• Function Domain– all values ​​that the independent variable can take.
• Function Range– all values ​​that the dependent variable can take.
• Function Zeros – Value independent variable such that the dependent variable equals 0.
• Minimum function value– minimum value of the dependent variable.
• Maximum function value– maximum value of the dependent variable.

Properties of the function $y=x^2$

Let's describe the properties of this function:

1. x is an independent variable, y is a dependent variable.

2. Domain of definition: it is obvious that for any value of the argument (x) there is a value of the function (y). Accordingly, the domain of definition of this function is the entire number line.

3. Range of values: y cannot be less than 0, since the square of any number is a positive number.

4. If x=0, then y=0.

5. Please note that for opposite values ​​of the argument the function takes the same value. For a pair of numbers x = 1 and x = -1, the value of the function will be 1, i.e. y = 1. For a pair of numbers x = 2 and x = – 2; y = 4, etc.
$y = x^2 =(-x)^2$.

Graph of the function $y=x^2$

Let's look carefully at the formula y = x 2 and try to describe in words the approximate appearance of the future graph.

1. Since y ≥ 0, the entire graph cannot be located below the OX axis.

2. The graph is symmetrical about the OY axis. All we need to do is plot the graph for positive values ​​of x and then mirror it for negative values ​​of x.

Let's find several values ​​of y:

Let's plot these points (see Fig. 1).

If we try to connect them with a dotted line, as shown in Fig. 1, then some function values ​​will not fall on these lines, for example, points A (x = 0.5; y = 0.25) and B (x = 2.5; y = 6.25). Even if we construct a lot of points and connect them with small straight segments, there will always be y values ​​that do not fall on these segments. Therefore, the points must be connected with a smooth curved line (see Fig. 2).

Now it remains to mirror the graph for negative values ​​of x (see Fig. 3). Such a curve is called a parabola. Point O (0;0) is called the vertex of the parabola. Symmetrical curves are called branches of a parabola.

Examples

I. The designer needs to paint a part of the wall of a house in the shape of a square with sides of 2.7 meters. Special paint for walls is sold in packaging at the rate of one can per 1 m2. Without doing any calculations, find out how many cans of paint you need to buy so that after painting there are no extra unopened cans left.

Solution:
1. Let's build a parabola.
2. Find point A on the parabola, whose coordinate is x=2.7 (see Fig. 4).
3. We see that at this point the value of the function is greater than 7, but less than 8. This means that the designer will need at least 8 cans of paint.

II. Construct a graph of the function y = (x + 1) 2.

Let's find several values ​​of y.

Let's construct these points and a straight line x= -1 parallel to the OY axis. It is obvious that the constructed points are symmetrical about this line. As a result, we will get the same parabola, only shifted to the left along the OX axis (see Fig. 5).

How to build a parabola? There are several ways to graph a quadratic function. Each of them has its pros and cons. Let's consider two ways.

Let's start by plotting a quadratic function of the form y=x²+bx+c and y= -x²+bx+c.

Example.

Graph the function y=x²+2x-3.

Solution:

y=x²+2x-3 is a quadratic function. The graph is a parabola with branches up. Parabola vertex coordinates

From the vertex (-1;-4) we build a graph of the parabola y=x² (as from the origin of coordinates. Instead of (0;0) - vertex (-1;-4). From (-1;-4) we go to the right by 1 unit and up by 1 unit, then left by 1 and up by 1; then: 2 - right, 4 - up, 2 - left, 3 - up; 9 - up, 3 - left, 9 - up If. these 7 points are not enough, then 4 to the right, 16 to the top, etc.).

The graph of the quadratic function y= -x²+bx+c is a parabola, the branches of which are directed downward. To construct a graph, we look for the coordinates of the vertex and from it we construct a parabola y= -x².

Example.

Graph the function y= -x²+2x+8.

Solution:

y= -x²+2x+8 is a quadratic function. The graph is a parabola with branches down. Parabola vertex coordinates

From the top we build a parabola y= -x² (1 - to the right, 1- down; 1 - left, 1 - down; 2 - right, 4 - down; 2 - left, 4 - down, etc.):

This method allows you to build a parabola quickly and does not cause difficulties if you know how to graph the functions y=x² and y= -x². Disadvantage: if the coordinates of the vertex are fractional numbers, it is not very convenient to build a graph. If you need to know exact values points of intersection of the graph with the Ox axis, you will have to additionally solve the equation x²+bx+c=0 (or -x²+bx+c=0), even if these points can be directly determined from the drawing.

Another way to construct a parabola is by points, that is, you can find several points on the graph and draw a parabola through them (taking into account that the line x=xₒ is its axis of symmetry). Usually for this they take the vertex of the parabola, the points of intersection of the graph with the coordinate axes and 1-2 additional points.

Draw a graph of the function y=x²+5x+4.

Solution:

y=x²+5x+4 is a quadratic function. The graph is a parabola with branches up. Parabola vertex coordinates

that is, the vertex of the parabola is the point (-2.5; -2.25).

Are looking for . At the point of intersection with the Ox axis y=0: x²+5x+4=0. Roots quadratic equation x1=-1, x2=-4, that is, we got two points on the graph (-1; 0) and (-4; 0).

At the point of intersection of the graph with the Oy axis x=0: y=0²+5∙0+4=4. We got the point (0; 4).

To clarify the graph, you can find an additional point. Let's take x=1, then y=1²+5∙1+4=10, that is, another point on the graph is (1; 10). We mark these points on the coordinate plane. Taking into account the symmetry of the parabola relative to the straight line passing through its vertex, we mark two more points: (-5; 6) and (-6; 10) and draw a parabola through them:

Graph the function y= -x²-3x.

Solution:

y= -x²-3x is a quadratic function. The graph is a parabola with branches down. Parabola vertex coordinates

The vertex (-1.5; 2.25) is the first point of the parabola.

At the points of intersection of the graph with the abscissa axis y=0, that is, we solve the equation -x²-3x=0. Its roots are x=0 and x=-3, that is (0;0) and (-3;0) - two more points on the graph. The point (o; 0) is also the point of intersection of the parabola with the ordinate axis.

At x=1 y=-1²-3∙1=-4, that is (1; -4) is an additional point for plotting.

Constructing a parabola from points is a more labor-intensive method compared to the first one. If the parabola does not intersect the Ox axis, more additional points will be required.

Before continuing to construct graphs of quadratic functions of the form y=ax²+bx+c, let us consider the construction of graphs of functions using geometric transformations. It is also most convenient to construct graphs of functions of the form y=x²+c using one of these transformations—parallel translation.

Category: |

Let us choose a rectangular coordinate system on the plane and plot the values ​​of the argument on the abscissa axis X, and on the ordinate - the values ​​of the function y = f(x).

Function graph y = f(x) is the set of all points whose abscissas belong to the domain of definition of the function, and the ordinates are equal to the corresponding values ​​of the function.

In other words, the graph of the function y = f (x) is the set of all points of the plane, coordinates X, at which satisfy the relation y = f(x).

In Fig. 45 and 46 show graphs of functions y = 2x + 1 And y = x 2 - 2x.

Strictly speaking, one should distinguish between a graph of a function (the exact mathematical definition of which was given above) and a drawn curve, which always gives only a more or less accurate sketch of the graph (and even then, as a rule, not the entire graph, but only its part located in the final parts of the plane). In what follows, however, we will generally say “graph” rather than “graph sketch.”

Using a graph, you can find the value of a function at a point. Namely, if the point x = a belongs to the domain of definition of the function y = f(x), then to find the number f(a)(i.e. the function values ​​at the point x = a) you should do this. It is necessary through the abscissa point x = a draw a straight line parallel to the ordinate axis; this line will intersect the graph of the function y = f(x) at one point; the ordinate of this point will, by virtue of the definition of the graph, be equal to f(a)(Fig. 47).

For example, for the function f(x) = x 2 - 2x using the graph (Fig. 46) we find f(-1) = 3, f(0) = 0, f(1) = -l, f(2) = 0, etc.

A function graph clearly illustrates the behavior and properties of a function. For example, from consideration of Fig. 46 it is clear that the function y = x 2 - 2x accepts positive values at X< 0 and at x > 2, negative - at 0< x < 2; наименьшее значение функция y = x 2 - 2x accepts at x = 1.

To graph a function f(x) you need to find all the points of the plane, coordinates X,at which satisfy the equation y = f(x). In most cases, this is impossible to do, since there are an infinite number of such points. Therefore, the graph of the function is depicted approximately - with greater or lesser accuracy. The simplest is the method of plotting a graph using several points. It consists in the fact that the argument X give a finite number of values ​​- say, x 1, x 2, x 3,..., x k and create a table that includes the selected function values.

The table looks like this:

Having compiled such a table, we can outline several points on the graph of the function y = f(x). Then, connecting these points with a smooth line, we get an approximate view of the graph of the function y = f(x).

It should be noted, however, that the multi-point plotting method is very unreliable. In fact, the behavior of the graph between the intended points and its behavior outside the segment between the extreme points taken remains unknown.

Example 1. To graph a function y = f(x) someone compiled a table of argument and function values:

The corresponding five points are shown in Fig. 48.

Based on the location of these points, he concluded that the graph of the function is a straight line (shown in Fig. 48 with a dotted line). Can this conclusion be considered reliable? Unless there are additional considerations to support this conclusion, it can hardly be considered reliable. reliable.

To substantiate our statement, consider the function

.

Calculations show that the values ​​of this function at points -2, -1, 0, 1, 2 are exactly described by the table above. However, the graph of this function is not a straight line at all (it is shown in Fig. 49). Another example would be the function y = x + l + sinπx; its meanings are also described in the table above.

These examples show that in its “pure” form the method of plotting a graph using several points is unreliable. Therefore, to plot a graph of a given function, one usually proceeds as follows. First, the properties of this function are studied, with the help of which you can build a sketch of the graph. Then, by calculating the values ​​of the function at several points (the choice of which depends on the established properties of the function), the corresponding points of the graph are found. And finally, a curve is drawn through the constructed points using the properties of this function.

We will look at some (the simplest and most frequently used) properties of functions used to find a graph sketch later, but now we will look at some commonly used methods for constructing graphs.

Graph of the function y = |f(x)|.

It is often necessary to plot a function y = |f(x)|, where f(x) - given function. Let us remind you how this is done. A-priory absolute value numbers can be written

This means that the graph of the function y =|f(x)| can be obtained from the graph, function y = f(x) as follows: all points on the graph of the function y = f(x), whose ordinates are non-negative, should be left unchanged; further, instead of the points of the graph of the function y = f(x) having negative coordinates, you should construct the corresponding points on the graph of the function y = -f(x)(i.e. part of the graph of the function
y = f(x), which lies below the axis X, should be reflected symmetrically about the axis X).

Example 2. Graph the function y = |x|.

Let's take the graph of the function y = x(Fig. 50, a) and part of this graph at X< 0 (lying under the axis X) symmetrically reflected relative to the axis X. As a result, we get a graph of the function y = |x|(Fig. 50, b).

Example 3. Graph the function y = |x 2 - 2x|.

First, let's plot the function y = x 2 - 2x. The graph of this function is a parabola, the branches of which are directed upward, the vertex of the parabola has coordinates (1; -1), its graph intersects the x-axis at points 0 and 2. In the interval (0; 2) the function takes negative values, therefore this part of the graph symmetrically reflected relative to the abscissa axis. Figure 51 shows the graph of the function y = |x 2 -2x|, based on the graph of the function y = x 2 - 2x

Graph of the function y = f(x) + g(x)

Consider the problem of constructing a graph of a function y = f(x) + g(x). if function graphs are given y = f(x) And y = g(x).

Note that the domain of definition of the function y = |f(x) + g(x)| is the set of all those values ​​of x for which both functions y = f(x) and y = g(x) are defined, i.e. this domain of definition is the intersection of the domains of definition, functions f(x) and g(x).

Let the points (x 0 , y 1) And (x 0, y 2) respectively belong to the graphs of functions y = f(x) And y = g(x), i.e. y 1 = f(x 0), y 2 = g(x 0). Then the point (x0;. y1 + y2) belongs to the graph of the function y = f(x) + g(x)(for f(x 0) + g(x 0) = y 1 +y2),. and any point on the graph of the function y = f(x) + g(x) can be obtained this way. Therefore, the graph of the function y = f(x) + g(x) can be obtained from function graphs y = f(x). And y = g(x) replacing each point ( x n, y 1) function graphics y = f(x) dot (x n, y 1 + y 2), Where y 2 = g(x n), i.e. by shifting each point ( x n, y 1) function graph y = f(x) along the axis at by the amount y 1 = g(x n). In this case, only such points are considered X n for which both functions are defined y = f(x) And y = g(x).

This method of plotting a function y = f(x) + g(x) is called addition of graphs of functions y = f(x) And y = g(x)

Example 4. In the figure, a graph of the function was constructed using the method of adding graphs
y = x + sinx.

When plotting a function y = x + sinx we thought that f(x) = x, A g(x) = sinx. To plot the function graph, we select points with abscissas -1.5π, -, -0.5, 0, 0.5,, 1.5, 2. Values f(x) = x, g(x) = sinx, y = x + sinx Let's calculate at the selected points and place the results in the table.