Graph the function x squared. Graph of function y x squared

Textbook:

• Makarychev Yu. N., Mindyuk N. R. Mathematics. 7th grade

Goals:

I. Student survey

1. What is a function called?

(A function is the dependence of one variable on another, in which each value of the independent variable corresponds to a single value of the dependent variable)

2. What is the domain of a function?

(All values ​​that the independent variable (argument) takes form the domain of the function.)

3. What is the range of a function?

(All values ​​that the dependent variable takes are called function values)

4. What functions did we get to know?

a) with a linear function of the form y = kx + b,

direct proportionality of the form y = kx

b) with functions of the form y = x 2, y = x 3

5. What is the graph of a linear function? ( straight). How many points are needed to construct this graph?

Without performing construction, determine the relative position of the graphs of functions given by the following formulas:

A ) y = 3x + 2; y = 1.2x + 5;

b) y = 1.5x + 4; y = -0.2x + 4; y = x + 4;

With) y = 2x + 5; y = 2x - 7; y = 2x

Picture 1

The figure shows graphs of linear functions ( Each student is given a sheet of paper with the graphs at their desk.). Write a formula for each graph

What function graphs are we still familiar with? ( y = x 2; y = x 3 )

1. What is the graph of a function y = x 2 (parabola).
2. How many points do we need to construct to depict a parabola? ( 7, one of which is the vertex of a parabola).

Let's construct the parabola given by the formula y = x 2

Figure 2

What properties does the graph of a function have? y = x 3 ?

1. If x = 0 , That y = 0 - vertex of the parabola (0;0)
2. Domain: X - any number, D (y) = (- ?; ?) D (y) = R
3. Range of values at ? 0
4. E (y) =
5. The function increases over the interval

   The function increases over the interval 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

   The graph of a function is visual representation behavior of some function on the coordinate plane. Graphs help you understand various aspects of a function that cannot be determined from the function itself. You can build graphs of many functions, and each of them will be given a specific formula. The graph of any function is built using a specific algorithm (if you have forgotten the exact process of graphing a specific function).

   Steps

   Graphing a Linear Function

   Determine whether the function is linear. The linear function is given by a formula of the form F (x) = k x + b (\displaystyle F(x)=kx+b) or y = k x + b (\displaystyle y=kx+b)(for example, ), and its graph is a straight line. Thus, the formula includes one variable and one constant (constant) without any exponents, root signs, or the like. If a function of a similar type is given, it is quite simple to plot a graph of such a function. Here are other examples of linear functions:

   Use a constant to mark a point on the Y axis. The constant (b) is the “y” coordinate of the point where the graph intersects the Y axis. That is, it is a point whose “x” coordinate is equal to 0. Thus, if x = 0 is substituted into the formula, then y = b (constant). In our example y = 2 x + 5 (\displaystyle y=2x+5) the constant is equal to 5, that is, the point of intersection with the Y axis has coordinates (0.5). Place this point on coordinate plane.

   Find the slope of the line. It is equal to the multiplier of the variable. In our example y = 2 x + 5 (\displaystyle y=2x+5) with the variable “x” there is a factor of 2; thus, the slope coefficient is equal to 2. The slope coefficient determines the angle of inclination of the straight line to the X axis, that is, the greater the slope coefficient, the faster the function increases or decreases.

   Write the slope as a fraction. The angular coefficient is equal to the tangent of the angle of inclination, that is, the ratio of the vertical distance (between two points on a straight line) to the horizontal distance (between the same points). In our example, the slope is 2, so we can state that the vertical distance is 2 and the horizontal distance is 1. Write this as a fraction: 2 1 (\displaystyle (\frac (2)(1))).

   • If the slope is negative, the function is decreasing.

   1. From the point where the straight line intersects the Y axis, plot a second point using vertical and horizontal distances. A linear function can be graphed using two points. In our example, the intersection point with the Y axis has coordinates (0.5); From this point, move 2 spaces up and then 1 space to the right. Mark a point; it will have coordinates (1,7). Now you can draw a straight line.

      Using a ruler, draw a straight line through two points. To avoid mistakes, find the third point, but in most cases the graph can be plotted using two points. Thus, you have plotted a linear function.

      Plotting points on the coordinate plane

      1. Define a function. The function is denoted as f(x). All possible values ​​of the variable "y" are called the domain of the function, and all possible values ​​of the variable "x" are called the domain of the function. For example, consider the function y = x+2, namely f(x) = x+2.

         Draw two intersecting perpendicular lines. The horizontal line is the X axis. The vertical line is the Y axis.

         Label the coordinate axes. Divide each axis into equal segments and number them. The intersection point of the axes is 0. For the X axis: positive numbers are plotted to the right (from 0), and negative numbers to the left. For the Y axis: positive numbers are plotted on top (from 0), and negative numbers on the bottom.

         Find the values ​​of "y" from the values ​​of "x". In our example, f(x) = x+2. Substitute specific x values ​​into this formula to calculate the corresponding y values. If given a complex function, simplify it by isolating the “y” on one side of the equation.

         • -1: -1 + 2 = 1
         • 0: 0 +2 = 2
         • 1: 1 + 2 = 3

      2. Plot the points on the coordinate plane. For each pair of coordinates, do the following: find the corresponding value on the X axis and draw a vertical line (dotted); find the corresponding value on the Y axis and draw a horizontal line (dashed line). Mark the intersection point of the two dotted lines; thus, you have plotted a point on the graph.

         Erase the dotted lines. Do this after plotting all the points on the graph on the coordinate plane. Note: the graph of the function f(x) = x is a straight line passing through the coordinate center [point with coordinates (0,0)]; the graph f(x) = x + 2 is a line parallel to the line f(x) = x, but shifted upward by two units and therefore passing through the point with coordinates (0,2) (because the constant is 2).

      Graphing a Complex Function

      Find the zeros of the function. The zeros of a function are the values ​​of the x variable where y = 0, that is, these are the points where the graph intersects the X-axis. Keep in mind that not all functions have zeros, but they are the first step in the process of graphing any function. To find the zeros of a function, equate it to zero. For example:

      Find and mark the horizontal asymptotes. An asymptote is a line that the graph of a function approaches but never intersects (that is, in this region the function is not defined, for example, when dividing by 0). Mark the asymptote with a dotted line. If the variable "x" is in the denominator of a fraction (for example, y = 1 4 − x 2 (\displaystyle y=(\frac (1)(4-x^(2))))), set the denominator to zero and find “x”. In the obtained values ​​of the variable “x” the function is not defined (in our example, draw dotted lines through x = 2 and x = -2), because you cannot divide by 0. But asymptotes exist not only in cases where the function contains a fractional expression. Therefore, it is recommended to use common sense:

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

   hence the obvious fundamental difference - in the 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 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 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 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.