The construction of graphs of functions containing modules usually causes considerable difficulties for schoolchildren. However, everything is not so bad. It is enough to remember several algorithms for solving such problems, and you can easily plot 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 function values 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 carefully and carefully the graph of the function y = f(x).
2) Leave unchanged all points of the graph that are above or on the 0x axis.
3) The part of the graph that lies below the 0x axis, display symmetrically about the 0x axis.
Example 1. Draw a graph of the function y = |x 2 - 4x + 3|
1) We build a graph of the function y \u003d x 2 - 4x + 3. It is obvious that 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 \u003d 0 2 - 4 0 + 3 \u003d 3.
Therefore, the parabola intersects the 0y axis at the point (0, 3).
Parabola vertex coordinates:
x in \u003d - (-4/2) \u003d 2, y in \u003d 2 2 - 4 2 + 3 \u003d -1.
Therefore, the point (2, -1) is the vertex of this parabola.
Draw a parabola using the received data (Fig. 1)
2) The part of the graph lying below the 0x axis is displayed symmetrically with respect to the 0x axis.
3) We get the graph of the original function ( rice. 2, is shown 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 symmetrical about the 0y axis.
Plotting the function y = f(|x|) consists of the following simple chain of actions.
1) Plot 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 specified in paragraph (2) symmetrically to the 0y axis.
4) As the final graph, select the union of the curves obtained in paragraphs (2) and (3).
Example 2. Draw a 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. And now we can apply the algorithm proposed above.
1) We build carefully and carefully the graph of the function y \u003d x 2 - 4 x + 3 (see also rice. one).
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 right side graphics symmetrical to the 0y axis.
(Fig. 3).
Example 3. Draw a graph of the function y = log 2 |x|
We apply the scheme given above.
1) We 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 symmetrical 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 to:
1) Construct a neat graph of the function y = f(|x|).
2) Leave unchanged the part of the graph that is above or on the 0x axis.
3) The part of the graph located below the 0x axis should be displayed symmetrically with respect to the 0x axis.
4) As the final graph, select the union of the curves obtained in paragraphs (2) and (3).
Example 4. Draw a 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| - one
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) We plot the function y \u003d -x 2 + 2x - 1 (Fig. 6).
b) We leave that part of the graph, which 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 with respect to 0x.
4) The resulting graph is shown in the figure by a dotted line (Fig. 8).
Example 5. Plot 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).
notice, that given function is linear-fractional and its graph is a hyperbola. To build a curve, you first need to find the asymptotes of the graph. Horizontal - y \u003d 2/1 (the ratio of the coefficients at x in the numerator and denominator of a fraction), vertical - x \u003d -3.
2) The part of the chart that is above or on the 0x axis will be left unchanged.
3) The part of the chart located below the 0x axis will be displayed symmetrically with respect to 0x.
4) The final graph is shown in the figure (Fig. 11).
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Lesson on the topic: "Graph and properties of the function $y=x^3$. Examples of plotting"
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Properties of the function $y=x^3$
Let's describe the properties of this function:
1. x is the independent variable, y is the dependent variable.
2. Domain of definition: it is obvious that for any value of the argument (x) it is possible to calculate the value of the function (y). Accordingly, the domain of definition of this function is the entire number line.
3. Range of values: y can be anything. Accordingly, the range is also the entire number line.
4. If x= 0, then y= 0.
Graph of the function $y=x^3$
1. Let's make a table of values:
2. For positive values of x, the graph of the function $y=x^3$ is very similar to a parabola, the branches of which are more "pressed" to the OY axis.
3. Since the function $y=x^3$ has opposite values for negative values of x, the graph of the function is symmetrical with respect to the origin.
Now let's mark the points on coordinate plane and build a graph (see Fig. 1).
This curve is called a cubic parabola.
Examples
I. Completely finished on a small ship fresh water. It is necessary to bring enough water from the city. Water is ordered in advance and paid for a full cube, even if you fill it a little less. How many cubes should be ordered so as not to overpay for an extra cube and completely fill the tank? It is known that the tank has the same length, width and height, which are equal to 1.5 m. Let's solve this problem without performing calculations.
Solution:
1. Let's plot the function $y=x^3$.
2. Find point A, coordinate x, which is equal to 1.5. We see that the function coordinate is between the values 3 and 4 (see Fig. 2). So you need to order 4 cubes.
Into the golden age information technologies Few people will buy a graph paper and spend hours drawing a function or an arbitrary set of data, and why do such a chore when you can graph the function online. In addition, it is almost impossible and difficult to calculate millions of expression values for correct display, and despite all efforts, you will get a broken line, not a curve. Because the computer in this case - indispensable assistant.
What is a function graph
A function is a rule according to which each element of one set is associated with some element of another set, for example, the expression y = 2x + 1 establishes a connection between the sets of all x values and all y values, therefore, this is a function. Accordingly, the graph of the function will be called the set of points whose coordinates satisfy the given expression.
In the figure we see a graph of the function y=x. This is a straight line and each of its points has its own coordinates on the axis X and on the axis Y. Based on the definition, if we substitute the coordinate X some point into this equation, then we obtain the coordinate of this point on the axis Y.
Services for plotting function graphs online
Consider several popular and best services that allow you to quickly draw a graph of a function.
Opens the list of the most common service that allows you to plot a function graph using an online equation. Umath contains only the necessary tools, such as zooming, moving along the coordinate plane, and viewing the coordinate of the point where the mouse is pointing.
Instruction:
- Enter your equation in the box after the "=" sign.
- Click the button "Build Graph".
As you can see, everything is extremely simple and accessible, the syntax for writing complex mathematical functions: with a modulus, trigonometric, exponential - is given right below the graph. Also, if necessary, you can set the equation by the parametric method or build graphs in the polar coordinate system.
Yotx has all the functions of the previous service, but at the same time it contains such interesting innovations as the creation of a function display interval, the ability to build a graph using tabular data, and also display a table with entire solutions.
Instruction:
- Select the desired schedule method.
- Enter an equation.
- Set the interval.
- Click the button "Build".
For those who are too lazy to figure out how to write down certain functions, this position presents a service with the ability to select the one you need from the list with one click of the mouse.
Instruction:
- Find the function you need from the list.
- Click on it with the left mouse button
- If necessary, enter the coefficients in the field "Function:".
- Click the button "Build".
In terms of visualization, it is possible to change the color of the graph, as well as hide it or delete it altogether.
Desmos is by far the most sophisticated service for building equations online. By moving the cursor with the left mouse button held down on the graph, you can see in detail all the solutions of the equation with an accuracy of 0.001. The built-in keyboard allows you to quickly write degrees and fractions. The most important plus is the ability to write the equation in any state, without leading to the form: y = f(x).
Instruction:
- In the left column, right-click on a free line.
- In the lower left corner, click on the keyboard icon.
- On the panel that appears, type the desired equation (to write the names of the functions, go to the "A B C" section).
- The graph is built in real time.
The visualization is just perfect, adaptive, it is clear that the designers worked on the application. Of the pluses, one can note a huge abundance of opportunities, for the development of which you can see examples in the menu in the upper left corner.
There are a lot of sites for plotting functions, but everyone is free to choose for themselves based on the required functionality and personal preferences. The list of the best has been compiled to meet the requirements of any mathematician, young and old. Good luck to you in comprehending the "queen of sciences"!
