Graph y x2. Functions and graphs. Cubic function properties

Putties 21.11.2021

Putties

Let us choose a rectangular coordinate system on the plane and plot the values of the argument on the abscissa axis NS, 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 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 NS, at which satisfy the relation y = f (x).

In fig. 45 and 46 are graphs of functions y = 2x + 1 and y = x 2 - 2x.

Strictly speaking, one should distinguish between the graph of the function (the exact mathematical definition of which was given above) and the 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 part of the plane). In what follows, however, we will usually say "graph" rather than "sketch graph".

Using the graph, you can find the value of a function at a point. Namely, if the point x = a belongs to the domain of the function y = f (x), then to find the number f (a)(i.e., the values of the function at the point x = a) you should do this. It is necessary through a point with an abscissa x = a draw a straight line parallel to the ordinate; 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.

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

To plot the function f (x) you need to find all points of the plane, coordinates NS,at which satisfy the equation y = f (x)... In most cases, this cannot be done, since there are infinitely many such points. Therefore, the graph of the function is depicted approximately - with more or less accuracy. The simplest is the multi-point graphing method. It consists in the fact that the argument NS give a finite number of values - say, x 1, x 2, x 3, ..., x k and make up a table containing the selected values of the function.

The table looks like this:

Having compiled such a table, we can outline several points of 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 designated points and its behavior outside the segment between the extreme of the points taken remains unknown.

Example 1... To plot the function y = f (x) someone made 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 by a dotted line). Can this conclusion be considered reliable? If there are no 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 just described by the above table. However, the graph of this function is not at all a straight line (it is shown in Fig. 49). Another example is the function y = x + l + sinπx; its values are also described in the table above.

These examples show that the pure multi-point charting method is unreliable. Therefore, to build a graph of a given function, as a rule, proceed as follows. First, we study the properties of this function, with which you can build a sketch of the graph. Then, calculating the values of the function at several points (the choice of which depends on the set 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.

Some (the most simple and often used) properties of functions used to find a sketch of a graph, we will consider later, and now we will analyze some of the most commonly used methods of plotting.

The graph of the function y = | f (x) |.

Often you have to plot a function y = | f (x)|, where f (x) - given function. Let us recall how this is done. By the definition of the absolute value of a number, you can write

This means that the graph of the function y = | f (x) | can be obtained from graph, function y = f (x) as follows: all points of the graph of the function y = f (x) for which the ordinates are non-negative should be left unchanged; further, instead of the points of the graph of the function y = f (x) with negative coordinates, you should build the corresponding points of 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 NS, should be symmetrically reflected about the axis NS).

Example 2. Plot function y = | x |.

We take the graph of the function y = x(Fig. 50, a) and part of this graph at NS< 0 (lying under the axis NS) symmetrically reflect about the axis NS... As a result, we get the graph of the function y = | x |(Fig. 50, b).

Example 3... Plot 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 upwards, the vertex of the parabola has coordinates (1; -1), its graph intersects the abscissa axis at points 0 and 2. On the interval (0; 2), the function takes negative values, therefore it is this part of the graph reflect symmetrically about 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 plotting the function y = f (x) + g (x). if function graphs are given y = f (x) and y = g (x).

Note that the domain 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, that is, this domain is the intersection of domains, 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) point (x n, y 1 + y 2), where y 2 = g (x n), i.e., by the shift of 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 NS 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 the addition of the graphs of the functions y = f (x) and y = g (x)

Example 4... In the figure, by adding graphs, a graph of the function is plotted
y = x + sinx.

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

Lesson: how to construct a parabola or a quadratic function?

THEORETICAL PART

A parabola is a graph of a function described by the formula ax 2 + bx + c = 0.
To build a parabola, you need to follow a simple algorithm of actions:

1) Parabola formula y = ax 2 + bx + c,
if a> 0 then the branches of the parabola are directed up,
otherwise the branches of the parabola are directed way down.
Free member c this point intersects the parabola with the OY axis;

2), it is found by the formula x = (- b) / 2a, we substitute the found x into the parabola equation and find y;

3)Function zeros or otherwise the points of intersection of the parabola with the OX axis, they are also called the roots of the equation. To find the roots, we equate the equation to 0 ax 2 + bx + c = 0;

Types of equations:

a) The complete quadratic equation is ax 2 + bx + c = 0 and is decided by the discriminant;
b) Incomplete quadratic equation of the form ax 2 + bx = 0. To solve it, you need to put x outside the brackets, then equate each factor to 0:
ax 2 + bx = 0,
x (ax + b) = 0,
x = 0 and ax + b = 0;
c) Incomplete quadratic equation of the form ax 2 + c = 0. To solve it, you need to move the unknown in one direction, and the known in the other. x = ± √ (c / a);

4) Find some additional points to build the function.

PRACTICAL PART

And so now, using an example, we will analyze everything according to the actions:
Example # 1:
y = x 2 + 4x + 3
c = 3 means the parabola intersects OY at the point x = 0 y = 3. The branches of the parabola look upward since a = 1 1> 0.
a = 1 b = 4 c = 3 x = (- b) / 2a = (- 4) / (2 * 1) = - 2 y = (-2) 2 +4 * (- 2) + 3 = 4- 8 + 3 = -1 the vertex is at the point (-2; -1)
Find the roots of the equation x 2 + 4x + 3 = 0
Find the roots by the discriminant
a = 1 b = 4 c = 3
D = b 2 -4ac = 16-12 = 4
x = (- b ± √ (D)) / 2a
x 1 = (- 4 + 2) / 2 = -1
x 2 = (- 4-2) / 2 = -3

Take some arbitrary points that are near the vertex x = -2

x -4 -3 -1 0
y 3 0 0 3

Substitute x into the equation y = x 2 + 4x + 3 values
y = (- 4) 2 +4 * (- 4) + 3 = 16-16 + 3 = 3
y = (- 3) 2 +4 * (- 3) + 3 = 9-12 + 3 = 0
y = (- 1) 2 +4 * (- 1) + 3 = 1-4 + 3 = 0
y = (0) 2 + 4 * (0) + 3 = 0-0 + 3 = 3
It can be seen from the values of the function that the parabola is symmetric with respect to the straight line x = -2

Example # 2:
y = -x 2 + 4x
c = 0 means the parabola intersects OY at the point x = 0 y = 0. The branches of the parabola look down as a = -1 -1 Find the roots of the equation -x 2 + 4x = 0
Incomplete quadratic equation of the form ax 2 + bx = 0. To solve it, you need to take x out of the brackets, then equate each factor to 0.
x (-x + 4) = 0, x = 0 and x = 4.

Take some arbitrary points that are near the vertex x = 2
x 0 1 3 4
y 0 3 3 0
Substitute the x into the equation y = -x 2 + 4x values
y = 0 2 + 4 * 0 = 0
y = - (1) 2 + 4 * 1 = -1 + 4 = 3
y = - (3) 2 + 4 * 3 = -9 + 13 = 3
y = - (4) 2 + 4 * 4 = -16 + 16 = 0
It can be seen from the values of the function that the parabola is symmetric with respect to the straight line x = 2

Example No. 3
y = x 2 -4
c = 4 means the parabola intersects OY at the point x = 0 y = 4. The branches of the parabola look upward since a = 1 1> 0.
a = 1 b = 0 c = -4 x = (- b) / 2a = 0 / (2 * (1)) = 0 y = (0) 2 -4 = -4 the vertex is at the point (0; -4 )
Find the roots of the equation x 2 -4 = 0
Incomplete quadratic equation of the form ax 2 + c = 0. To solve it, you need to move the unknown in one direction, and the known in the other. x = ± √ (c / a)
x 2 = 4
x 1 = 2
x 2 = -2

Take some arbitrary points that are near the vertex x = 0
x -2 -1 1 2
y 0 -3 -3 0
Substitute the x into the equation y = x 2 -4 values
y = (- 2) 2 -4 = 4-4 = 0
y = (- 1) 2 -4 = 1-4 = -3
y = 1 2 -4 = 1-4 = -3
y = 2 2 -4 = 4-4 = 0
It can be seen from the values of the function that the parabola is symmetric with respect to the straight line x = 0

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How to graph the function y = x squared + 2x-5? and got the best answer

Answer from Alexey Popov (Ocean) [guru]
The quadratic function and its graph is a parabola. Let's find the coordinates of the vertex of this parabola X = -2 / 2 = -1 Y = 1-2-5 = -6 (it is necessary to substitute "-1" instead of X in the formula y = x squared + 2x-5 and calculate). Mark the vertex of the parabola A (-1; -6) in the coordinate system. And from this point (from point A) we mark the points found by the formula y = x squared, that is, points (1; 1) (-1; 1) (2; 4) (-2; 4) (3; 9) ( -3; 9) Attention! We postpone all these points from the top of the parabola, from point A (and not from point O, the origin)

Answer from Yergey Cherevan[master]
Take x = 0 - this will be the beginning of the graph, and then take 4 points x = 1, x = -1, x = 2 and x = -2 and build a graph, called a parabola

Answer from Elena Fedyukina[guru]
quadratic function, graph is a parabola, wind upward. Vertices along the x-axis = -1, along the y-axis = -5.

Answer from Anna Egorova[guru]
y = x squared + 2x-5 is a graph-parabola, the branches of which are directed upwards (a = 1 is greater than zero), you find the vertex of the parabola: m = -b divided by 2a - this is the coordinate along the x-axis - it will be -1; the y coordinate: you substitute it into your function: it will be -6, which means the vertex of the parabola (-1; -6) then draw a table with the values of x and y, for example, when x = -3, y = -2; x = -2, y = -5; x = -1, y = -6; x = 0, y = -5; x = 1, y = -2; x = 2, y = 3 then mark these points on the coordinate plane and connect)))

Answer from Bibi[guru]
y = x in sq. + 2x-5, highlighting the square of the binomial, we get y = (x +1) in sq. -6 hence it follows that the top is (-1; -6). The graph of the function is a parabola. The branches of the parabola are directed vertically upward, since there is no minus in front of the bracket (a).

Answer from 2 answers[guru]

Hey! Here is a selection of topics with answers to your question: How to graph the function y = x squared + 2x-5?

A function graph is a visual representation of the behavior of a function on a coordinate plane. Graphs help you understand various aspects of a function that cannot be identified from the function itself. You can plot graphs of many functions, and each of them will be given by a certain formula. The graph of any function is built according to a certain algorithm (if you forgot the exact process of plotting a graph of a specific function).

Steps

Plotting a Linear Function

Determine if 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, and the like. Given a function of a similar type, it is quite easy to plot such a function. Here are other examples of linear functions:

Use a constant to mark a point on the Y axis. Constant (b) is the “y” coordinate of the point of intersection of the graph with the y-axis. That is, it is the point whose “x” coordinate is 0. Thus, if you substitute x = 0 in the formula, then y = b (constant). In our example y = 2 x + 5 (\ displaystyle y = 2x + 5) the constant is 5, that is, the y-intercept has coordinates (0.5). Draw this point on the 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) at the variable "x" there is a factor of 2; thus, the slope is 2. The slope determines the angle of inclination of the straight line to the X-axis, that is, the larger the slope, the faster the function increases or decreases.

Write down the slope as a fraction. The slope is equal to the tangent of the slope, 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.

From the intersection of the line with the Y-axis, draw a second point using the vertical and horizontal distances. A linear function graph can be plotted from two points. In our example, the y-intercept has coordinates (0.5); from this point, move 2 divisions up, and then 1 division to the right. Mark the point; it will have coordinates (1,7). Now you can draw a straight line.

Use a ruler to draw a straight line through two points. Find the third point to avoid mistakes, but in most cases you can plot a graph from two points. Thus, you have plotted a linear function.

Placing 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 range of values of the function, and all possible values of the variable "x" are called the range 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 point of intersection 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 above (from 0), and negative numbers below.
  
  Find the y-values by the x-values. In our example, f (x) = x + 2. Plug in the specific x-values into this formula to calculate the corresponding y-values. If you have 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. Draw 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 line); find the corresponding value on the Y-axis and draw a horizontal line (dotted line). Draw the intersection point of the two dashed lines; thus you have plotted a point on the graph.
  
  Erase the dotted lines. Do this after plotting all the points of the graph on the coordinate plane. Note: the graph of the function f (x) = x is a straight line passing through the center of coordinates [point with coordinates (0,0)]; the graph f (x) = x + 2 is a straight line parallel to the straight line f (x) = x, but shifted two units up and therefore passing through the point with coordinates (0,2) (because the constant is 2).
Plotting a Complex Function

The construction of graphs of functions containing modules usually causes considerable difficulties for schoolchildren. However, things are not so bad. It is enough to remember several algorithms for solving such problems, and you can easily build a graph of even the most seemingly complex function. Let's see what these algorithms are.

1. Plotting the function y = | f (x) |

Note that the set of values of the functions y = | f (x) | : y ≥ 0. Thus, the graphs of such functions are always located completely in the upper half-plane.

Plotting the function y = | f (x) | consists of the following simple four steps.

1) Construct accurately and carefully the graph of the function y = f (x).

2) Leave unchanged all points of the graph that are above the 0x axis or on it.

3) The part of the graph that lies below the 0x axis, display symmetrically about the 0x axis.

Example 1. Display the graph of the function y = | x 2 - 4x + 3 |

1) We build a graph of the function y = x 2 - 4x + 3. Obviously, the graph of this function is a parabola. Find the coordinates of all points of intersection of the parabola with the coordinate axes and the coordinates of the vertex of the parabola.

x 2 - 4x + 3 = 0.

x 1 = 3, x 2 = 1.

Therefore, the parabola intersects the 0x axis at points (3, 0) and (1, 0).

y = 0 2 - 4 0 + 3 = 3.

Therefore, the parabola intersects the 0y axis at the point (0, 3).

Parabola vertex coordinates:

x in = - (- 4/2) = 2, y in = 2 2 - 4 2 + 3 = -1.

Therefore, point (2, -1) is the vertex of this parabola.

Draw a parabola using the received data (fig. 1)

2) The part of the graph that lies below the 0x axis is displayed symmetrically about the 0x axis.

3) We get the graph of the original function ( rice. 2, depicted by a dotted line).

2. Plotting the function y = f (| x |)

Note that functions of the form y = f (| x |) are even:

y (-x) = f (| -x |) = f (| x |) = y (x). This means that the graphs of such functions are symmetric about the 0y axis.

Plotting the function y = f (| x |) consists of the following simple chain of actions.

1) Construct a graph of the function y = f (x).

2) Leave that part of the graph for which x ≥ 0, that is, the part of the graph located in the right half-plane.

3) Display the part of the graph indicated in paragraph (2) symmetrically to the 0y axis.

4) Select the union of the curves obtained in paragraphs (2) and (3) as the final graph.

Example 2. Display the graph of the function y = x 2 - 4 · | x | + 3

Since x 2 = | x | 2, then the original function can be rewritten as follows: y = | x | 2 - 4 · | x | + 3. Now we can apply the algorithm proposed above.

1) We construct accurately and carefully the graph of the function y = x 2 - 4 x + 3 (see also rice. 1).

2) We leave that part of the graph for which x ≥ 0, that is, the part of the graph located in the right half-plane.

3) Display the right side of the graph symmetrically to the 0y axis.

(fig. 3).

Example 3. Display the graph of the function y = log 2 | x |

We apply the scheme given above.

1) Plot the function y = log 2 x (fig. 4).

3. Plotting the function y = | f (| x |) |

Note that functions of the form y = | f (| x |) | are also even. Indeed, y (-x) = y = | f (| -x |) | = y = | f (| x |) | = y (x), and therefore, their graphs are symmetric about the 0y axis. The set of values of such functions: y ≥ 0. Hence, the graphs of such functions are located completely in the upper half-plane.

To plot the function y = | f (| x |) |, you need:

1) Construct accurately the graph of the function y = f (| x |).

2) Leave the part of the graph that is above or on the 0x axis unchanged.

3) The part of the graph, located below the 0x axis, display symmetrically about the 0x axis.

4) Select the union of the curves obtained in paragraphs (2) and (3) as the final graph.

Example 4. Display the graph of the function y = | -x 2 + 2 | x | - 1 |.

1) Note that x 2 = | x | 2. Hence, instead of the original function y = -x 2 + 2 | x | - 1

you can use the function y = - | x | 2 + 2 | x | - 1, since their graphs are the same.

We build a graph y = - | x | 2 + 2 | x | - 1. For this we use algorithm 2.

a) Plot the function y = -x 2 + 2x - 1 (fig. 6).

b) Leave the part of the graph that is located in the right half-plane.

c) Display the resulting part of the graph symmetrically to the 0y axis.

d) The resulting graph is shown in the figure with a dotted line. (fig. 7).

2) There are no points above the 0x axis, we leave the points on the 0x axis unchanged.

3) The part of the graph located below the 0x axis is displayed symmetrically about 0x.

4) The resulting graph is shown in the figure with a dotted line (fig. 8).

Example 5. Construct a graph of the function y = | (2 | x | - 4) / (| x | + 3) |

1) First, you need to plot the function y = (2 | x | - 4) / (| x | + 3). To do this, we return to Algorithm 2.

a) Carefully plot the function y = (2x - 4) / (x + 3) (fig. 9).

Note that this function is linear-fractional and its graph is a hyperbola. To plot the curve, you first need to find the asymptotes of the graph. Horizontal - y = 2/1 (the ratio of the coefficients at x in the numerator and denominator of the fraction), vertical - x = -3.

2) Leave the part of the graph above or on the 0x axis unchanged.

3) The part of the graph, located below the 0x axis, will be displayed symmetrically about 0x.

4) The final graph is shown in the figure (fig. 11).

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Graph y x2. Functions and graphs. Cubic function properties

THEORETICAL PART

PRACTICAL PART

Steps

Plotting a Linear Function

Placing points on the coordinate plane

Plotting a Complex Function

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