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An algebra lesson in grade 9 on the topic "Graphing a function whose analytical expression contains an absolute value sign" was built on the basis of computer technology, using research learning activities.

Lesson Objectives: Educational: Visually demonstrate to students the possibilities of using a computer when plotting function graphs with modules; for self-control, saving time when plotting functions of the form y=f|(x)| , y = | f(x)| , y=|f |(x)| |.

Developing: Development of intellectual skills and mental operations - analysis and synthesis, comparison, generalization. Formation of ICT competence of students.

Educational: Raising cognitive interest in the subject by introducing the latest learning technologies. Education of independence in solving educational problems.

Equipment: Equipment: computer class, interactive whiteboard, presentation on the topic "Graphing a function whose analytical expression contains an absolute value sign", handouts: cards for working with a graphical model of functions, sheets for recording the results of research functions, personal computers. Self-control sheet.

Software: Microsoft PowerPoint presentation "Graphing a function whose analytical expression contains an absolute value sign"

During the classes

1. Organizational moment

2. Repetition, generalization and systematization. This stage of the lesson is accompanied by a computer presentation.

Function Graph y=f|(x)|

y=f |(x)| is an even function, because | x | = | -x |, then f |-x| = f | x |

The graph of this function is symmetrical about the coordinate axis.

Therefore, it suffices to plot the function y=f(x) for x>0, and then complete its left side, symmetrically to the right side with respect to the coordinate axis.

For example, let the graph of the function y=f(x) is the curve shown in Fig. 1, then the graph of the function y=f|(x)| there will be a curve shown in Fig.2.

1. Study of the graph of the function y= |x|

Thus, the desired graph is a broken line, composed of two half-lines. (Fig.3)

From a comparison of two graphs: y=x and y= |x|, students will conclude that the second one is obtained from the first by mirroring the part of the first graph that lies under the x-axis with respect to OX. This position follows from the definition of the absolute value.

From a comparison of two graphs: y \u003d x and y \u003d -x, they will conclude: the function y \u003d f ( | x |) is obtained from the graph y \u003d f (x) at x 0 symmetrical display about the y-axis.

Can this plotting method be applied to any function containing an absolute value?

Slide 3 and 4.

1. Plot the function y=0.5 x 2 - 2|x| - 2.5

1) Because |x| = x at x 0, y \u003d 0.5 x 2 - 2x - 2.5. If x<0, то поскольку х 2 = |х| 2 , |х|=-х и требуемый график совпадает с параболой y \u003d 0.5 x 2 + 2x - 2.5.

2) If we consider the graph y \u003d 0.5 x 2 -2x - 2.5 at x

Is it possible to use this plotting method for a quadratic function, for inverse proportionality plots containing an absolute value?

1) Because |x| = x at x 0, the desired graph is the same as a parabola y \u003d 0.25 x 2 - x - 3. If x<0, то поскольку х 2 = |х| 2 , |х|=-х и требуемый график совпадает с параболой y \u003d 0.25 x 2 + x - 3.

2) If we consider the graph y \u003d 0.25 x 2 - x - 3 at x0 and display it relative to the y-axis, we get the same graph.

(0; - 3) coordinates of the point of intersection of the graph of the function with the y-axis.

y \u003d 0, x 2 -x -3 \u003d 0

x 2 -4x -12 = 0

We have x 1 = - 2; x 2 = 6.

(-2; 0) and (6; 0) - coordinates of the point of intersection of the graph of the function with the OX axis.

If x<0, ордината точки требуемого графика такая же, как и у точки параболы, но с положительной абсциссой, равной |х|. Такие точки симметричны относительно оси ОУ(например, вершины (2; -4) и -(2; -4).

This means that the part of the required graph corresponding to the values ​​of x<0, симметрична относительно оси ОУ его же части, соответствующей значениям х>0.

b) Therefore, I complete for x<0 часть графика, симметричную построенной относительно оси ОУ.

On notebooks, students prove that the graph of the function y \u003d f | (x) | coincides with the graph of the function y = f (x) on the set of non-negative values ​​of the argument and is symmetrical to it with respect to the y-axis on the set of negative values ​​of the argument.

Proof: If x 0, then f |(x)|= f(x), i.e. on the set of non-negative values ​​of the graph argument of the function y = f(x) and y = f |(x)| match. Since y = f |(x)| is an even function, then its graph is symmetrical with respect to the OS.

Thus, the graph of the function y = f |(x)| can be obtained from the graph of the function y \u003d f (x) as follows:

1. build a graph of the function y \u003d f (x) for x> 0;

2. For x<0, симметрично отразить построенную часть относительно оси ОУ.

Conclusion: To plot the function y = f |(x)|

1. build a graph of the function y \u003d f (x) for x> 0;

2. For x<0, симметрично reflect the built part

about the y-axis.

slide 5

4. Research work on plotting the function y = | f(x)|

Plot the function y = |x 2 - 2x|

Let's get rid of the modulus sign by definition

If x 2 - 2x0, i.e. if x
0 and x2, then | x 2 - 2x | \u003d x 2 - 2x

If x 2 - 2x<0, т.е. если 0<х< 2, то |х 2 - 2х|=- х 2 + 2х

We see that on the set x
0 and x2 function graphs

y \u003d x 2 - 2x and y \u003d | x 2 - 2x | coincide, and on the set (0; 2)

graphs of the function y \u003d -x 2 + 2x and y \u003d |x 2 - 2x | match. Let's build them.

Graph of the function y = | f(x)| consists of a part of the graph of the function y \u003d f (x) for y? 0 and a symmetrically reflected part y \u003d f (x) for y<0 относительно оси ОХ.

Plot a function y = |x 2 - X - 6|

1) If x 2 - x -6 0, i.e. if x
-2 and x3, then | x 2 - x -6 | = x 2 - x -6.

If x 2 - x -6<0, т.е. если -2<х< 3, то |х 2 - х -6|= -х 2 + х +6.

Let's build them.

2) Let's build y \u003d x 2 - x -6. Bottom of the chart

is displayed symmetrically with respect to OX.

Comparing 1) and 2), we see that the graphs are the same.

Work on notebooks.

Let us prove that the graph of the function y = | f(x)| coincides with the graph of the function y \u003d f (x) for f (x) > 0 and the symmetrically reflected part y \u003d f (x) for y<0 относительно оси ОХ.

Indeed, by the definition of the absolute value, this function can be considered as a set of two lines:

y = f(x) if f(x) 0; y = - f(x) if f(x)<0

For any function y = f(x), if f(x) >0, then

| f(x)| = f(x), so in this part the graph of the function

y = | f(x)| coincides with the graph of the function itself

If f(x)<0, то | f (х)| = - f(х),т.е. точка (х; - f(х)) symmetrical to the point (x; f (x)) about the OX axis. Therefore, to obtain the required graph, we reflect the "negative" part of the graph y \u003d f (x) symmetrically about the OX axis.

Conclusion: valid for plotting a function graph y = |f(x) | enough:

1. Construct a graph of the function y \u003d f (x);

F(x)<0, симметрично отражаем относительно оси абсцисс. (Рис.5)

Conclusion: To plot a function graph y=|f(x) |

1. Graph the function y=f(X) ;

2. In areas where the graph is located in the lower half-plane, i.e., where f(X)<0, строим кривые, симметричные построенным графикам относительно оси абсцисс.

Slides 8-13.

5. Research work on plotting functions y=|f|(x)| |

Applying the definition of the absolute value and the previously considered examples, we will plot the graphs of the function:

y = |2|x| - 3|

y = |x 2 - 5|x||

y = | | x 2 | - 2| and drew conclusions.

In order to plot the function y = | f |(x)| necessary:

1. Graph the function y = f(x) for x>0.

2. Build the second part of the graph, i.e. reflect the constructed graph symmetrically with respect to the OS, because this function is even.

3. The sections of the resulting graph located in the lower half-plane should be converted to the upper half-plane symmetrically to the OX axis.

Plot the function y = | 2|x | - 3| (1st way to define the module)

1. We build y = 2|x | - 3, for 2 |x| - 3 > 0 , | x |>1.5 i.e. X< -1,5 и х>1,5

a) y = 2x - 3, for x>0

b) for x<0, симметрично отражаем построенную часть относительно оси ОУ.

2. We build y \u003d - 2 |x| + 3, for 2|x | - 3 < 0. т.е. -1,5<х<1,5

a) y = - 2x + 3, for x>0

b) for x<0, симметрично отражаем построенную часть относительно оси ОУ.

Y = | 2|x | - 3|

1) We build y \u003d 2x-3, for x> 0.

2) We build a straight line, symmetrical to the one built with respect to the OS axis.

3) The sections of the graph located in the lower half-plane are displayed symmetrically about the OX axis.

Comparing both graphs, we see that they are the same.

y = | X 2 - 5|x| |

1. We build y \u003d x 2 - 5 | x |, for x 2 - 5 | x | > 0 i.e. x >5 and x<-5

a) y \u003d x 2 - 5 x, for x> 0

b) for x<0, симметрично отражаем построенную часть относительно оси ОУ.

2. We build y \u003d - x 2 + 5 | x | , for x 2 - 5 |x|< 0. т.е. -5х5

a) y \u003d - x 2 + 5 x, for x> 0

b) for x<0, симметрично отражаем построенную часть относительно оси ОУ.

Y = | x 2 - 5|x| |

a) We build a graph of the function y \u003d x 2 - 5 x for x> 0.

B) We build a part of the graph, symmetrical to the one built with respect to the y-axis

c) I transform the part of the graph located in the lower half-plane to the upper half-plane symmetrically to the OX axis.

Comparing both graphs, we see that they are the same. (Fig.10)

3. Summing up the lesson.

14.15 slides.

y=f|(x)|

1. Graph the function y=f(x) for x>0;

2.Build for x<0 часть графика, симметричную построенной относительно оси ОУ.

Algorithm for plotting a function graph y=|f(x) |

1. Graph the function y=f(X) ;

2. In areas where the graph is located in the lower half-plane, i.e., where f(X)<0, строить кривые, симметричные построенным графикам относительно оси абсцисс.

Algorithm for plotting a function graph y=|f|(x)| |

1. Graph the function y=f(x) for x>0.

2. Construct a graph curve symmetrical to the one constructed with respect to the OS axis, since this function is even.

3. Plots of the graph located in the lower half-plane should be converted to the upper half-plane symmetrically to the OX axis.

Today we will carefully study the functions whose graph is a straight line.

Write the topic of the lesson in your notebook

"Linear function and direct proportionality".

Complete all tasks carefully and
try to remember new definitions for you.

Remember the definition:
A linear function is a function that can be defined by a formula of the form
y = kx + b, where x is an independent variable, k and b are some numbers.

For example: if k = 0.5 and b = -2, then y = 0.5x - 2.

Exercise:
Plot a linear function graph y \u003d 0.5x - 2.

Make a table of values ​​of pairs (x, y).
Mark them on the coordinate plane.
Connect the dots with a line.

Check out the solution:
Let's plot a linear function graph y \u003d 0.5x - 2.

X	-4	0	2	4
at	-4	-2	-1	0

To build a graph y \u003d -x + 3, we calculate the coordinates of two points

X	-2	4
at	5	-1

We mark two points on the coordinate plane and connect them with a straight line.

Can you determine:
does the point A(36; 5) belong to the graph of the linear function ?

Yes

Not

Now compare these two graphs and see that the linear function has y \u003d kx + b,
even before its construction, you can "predict" the location of a straight line on the coordinate plane!

How?
You just need to look carefully at the numbers k and b...

And they tell us a lot!

Try to guess...

Function y \u003d 0.5x - 2	Function y = -x + 3

So, we observe and draw conclusions:
1) The first one crosses the y-axis at the point (0; -2), and the second one at (0; 3)
!!! the first has b = -2, and the second has b = 3
Conclusion: by the number b in the formula y \u003d kx + b, we will determine at what point the line will intersect the y-axis.

2) The first is inclined to the positive direction of the OX axis at an acute angle, and the second at an obtuse angle.
!!! for the first function k > 0, and for the second function k
Conclusion: if in the formula y \u003d kx + b we see that the number k\u003e 0, then the graph is inclined to the positive direction of the x-axis at an acute angle;
if the number k The number k (coefficient at x) is called for this - the slope.
Remember it all! We will need this kind of knowledge again and again.

If in the formula y = kx + b, we take b = 0, then we get the formula y = kx.

Remember the definition:
A function that can be specified by the formula y \u003d kx, where k is a number not equal to 0, x is a variable, is called direct proportionality.

Complete the task in your notebook:
Come up with several formulas of direct proportionality with different coefficients k and build their graphs in the same coordinate plane.

Since the direct proportionality has b \u003d 0, then the graph will cross the y-axis at the point (0; 0).

On one coordinate plane, we can draw several graphs!

A linear function has a graph that is a straight line.
Lines can be parallel or intersect at the same point...
Interestingly, before plotting graphs, only by looking (attentively!) at their formulas, we can conclude:

The graphs of these functions will intersect,
the graphs of these functions are arranged in parallel.

Hello David.

The graph of a function is its geometric image. It shows where on the coordinate plane there is a point whose coordinates (X and Y) are connected by a certain mathematical expression (function).

Before you start plotting functions, you first need to draw the coordinate axes OX and OY. It is best to use scale - coordinate paper for this. Next, you need to determine the type of function, because the graphs of different functions are very different. For example, the linear function, which will be discussed below, has a graph in the form of a straight line. After that, you need to define the scope of the functions, i.e. restrictions for the values ​​of X and Y. For example, if X is in the denominator of a fraction, then its value cannot be equal to 0. Next, you need to find the zeros of the function, that is, the intersections of the graph of the function with the coordinate axes.

Let's start plotting the function specified in paragraph a) of your question.

Function y= - 6x + 4, which you want to plot in the first problem of your question, is a linear function, because linear functions are represented by the expression y = kx + m. The domain of definition of a linear function is considered to be the entire line OX. The m parameter in the linear function determines the point at which the graph of the linear function intersects the OY axis.

In order to build a graph of a linear function, it is enough to determine at least two of its points, because the graph of a function is a straight line. If you find more points, you can build a more accurate graph. In general, when plotting a linear function graph, it is necessary to determine the points at which the graph will cross the X, Y coordinate axes.

So, in your case, the intersection points of the function graph with the coordinate axes will be like this:

With X=0, Y= -6*0+4=4 Thus, we have obtained the value of the parameter m in a linear function.

Y \u003d 0, that is, 0 \u003d -6 * X + 4, that is, 6x \u003d 4, therefore X \u003d 4 / 6 \u003d 0.667

With X= -1, Y=-6*-1+4=10

With X=1, Y= -6*1+4=-2

With X=2, Y= -6*2+4=-8

Having received all the above points, you just have to mark them on the coordinate plane, connect them with a straight line, as shown in the example in the figure attached to this article.

Now let's build a graph of the function indicated in paragraph b) of your question.

It is immediately obvious that function y \u003d 0.5x, from the second problem, is also a linear function. Unlike the first example, this expression does not contain the value m, which means that the graph of the function y \u003d 0.5x passes through the origin of the coordinate axes, that is, at their zero point.

At X=0, Y= 0.5*0=0

At X=1, Y=0.5*1=0.5

At X=2, Y= 0.5*2=1

At X=3, Y=0.5*3=1.5

With X \u003d -1, Y \u003d 0.5 * -1 \u003d -0.5

With X \u003d -2, Y \u003d 0.5 * -2 \u003d -1

With X \u003d -3, Y \u003d 0.5 * 3 \u003d -1.5

Now, having all the above values ​​​​of X and Y, you can easily put these points on the coordinate plane, connect them with a straight line using a ruler, and you will get a graph of a linear function y \u003d 0.5x

Below I have provided a link, by clicking on which you can find lessons in mathematics, algebra, geometry and Russian. I would encourage you to read a few topics that deal with plotting functions. This tutorial shows very clearly how you can plot linear functions, and in the topics below you can see examples of plotting other functions. Everything is written in sufficient detail, so it will be clear not only to those who have long graduated from school and have an idea of ​​​​how to plot a function graph, but also to those who are just starting to comprehend the basics of science. I believe that having seen clearly on specific examples how function graphs are built, then you can easily solve any problem of plotting functions without any problems.