EntranceFunction y = square root of x, its properties and graph. Mathematics lesson “Function y = √x, its properties and graph Lesson summary: function y root of x
Basic goals:
1) form an idea of the feasibility of a generalized study of the dependencies of real quantities using the example of quantities related by the relation y=
2) to develop the ability to construct a graph y= and its properties;
3) repeat and consolidate the techniques of oral and written calculations, squaring, extracting square roots.
Equipment, demonstration material: handouts.
1. Algorithm:
2. Sample for completing the task in groups:
3. Sample for self-test of independent work:
4. Card for the reflection stage:
1) I understood how to graph the function y=.
2) I can list its properties using a graph.
3) I did not make mistakes in independent work.
4) I made mistakes in my independent work (list these mistakes and indicate their reason).
During the classes
1. Self-determination for educational activities
Purpose of the stage:
1) include students in educational activities;
2) determine the content of the lesson: we continue to work with real numbers.
Organization of the educational process at stage 1:
– What did we study in the last lesson? (We studied the set of real numbers, operations with them, built an algorithm to describe the properties of a function, repeated functions studied in 7th grade).
– Today we will continue to work with a set of real numbers, a function.
2. Updating knowledge and recording difficulties in activities
Purpose of the stage:
1) update educational content that is necessary and sufficient for the perception of new material: function, independent variable, dependent variable, graphs
y = kx + m, y = kx, y =c, y =x 2, y = - x 2,
2) update mental operations necessary and sufficient for the perception of new material: comparison, analysis, generalization;
3) record all repeated concepts and algorithms in the form of diagrams and symbols;
4) record an individual difficulty in activity, demonstrating at a personally significant level the insufficiency of existing knowledge.
Organization of the educational process at stage 2:
1. Let's remember how you can set dependencies between quantities? (Using text, formula, table, graph)
2. What is a function called? (A relationship between two quantities, where each value of one variable corresponds to a single value of another variable y = f(x)).
What is the name of x? (Independent variable - argument)
What is the name of y? (Dependent variable).
3. In 7th grade did we study functions? (y = kx + m, y = kx, y =c, y =x 2, y = - x 2,).
Individual task:
What is the graph of the functions y = kx + m, y =x 2, y =?
3. Identifying the causes of difficulties and setting goals for activities
Purpose of the stage:
1) organize communicative interaction, during which the distinctive property of the task that caused difficulty in learning activities is identified and recorded;
2) agree on the purpose and topic of the lesson.
Organization of the educational process at stage 3:
-What's special about this task? (The dependence is given by the formula y = which we have not yet encountered.)
– What is the purpose of the lesson? (Get acquainted with the function y =, its properties and graph. Use the function in the table to determine the type of dependence, build a formula and graph.)
– Can you formulate the topic of the lesson? (Function y=, its properties and graph).
– Write the topic in your notebook.
4. Construction of a project for getting out of a difficulty
Purpose of the stage:
1) organize communicative interaction to build a new method of action that eliminates the cause of the identified difficulty;
2) fix a new method of action in a symbolic, verbal form and with the help of a standard.
Organization of the educational process at stage 4:
Work at this stage can be organized in groups, asking the groups to construct a graph y =, then analyze the results. Groups can also be asked to describe the properties of a given function using an algorithm.
5. Primary consolidation in external speech
The purpose of the stage: to record the studied educational content in external speech.
Organization of the educational process at stage 5:
Construct a graph of y= - and describe its properties.
Properties y= - .
1.Domain of definition of a function.
2. Range of values of the function.
3. y = 0, y> 0, y<0.
y =0 if x = 0.
y<0, если х(0;+)
4.Increasing, decreasing functions.
The function decreases as x.
Let's build a graph of y=.
Let's select its part on the segment. Note that we have = 1 for x = 1, and y max. =3 at x = 9.
Answer: at our name. = 1, y max. =3
6. Independent work with self-test according to the standard
The purpose of the stage: to test your ability to apply new educational content in standard conditions based on comparing your solution with a standard for self-test.
Organization of the educational process at stage 6:
Students complete the task independently, conduct a self-test against the standard, analyze, and correct errors.
Let's build a graph of y=.
Using a graph, find the smallest and largest values of the function on the segment.
7. Inclusion in the knowledge system and repetition
The purpose of the stage: to train the skills of using new content together with previously studied: 2) repeat the educational content that will be required in the next lessons.
Organization of the educational process at stage 7:
Solve the equation graphically: = x – 6.
One student is at the blackboard, the rest are in notebooks.
8. Reflection of activity
Purpose of the stage:
1) record new content learned in the lesson;
2) evaluate your own activities in the lesson;
3) thank classmates who helped get the result of the lesson;
4) record unresolved difficulties as directions for future educational activities;
5) discuss and write down your homework.
Organization of the educational process at stage 8:
- Guys, what was our goal today? (Study the function y=, its properties and graph).
– What knowledge helped us achieve our goal? (Ability to look for patterns, ability to read graphs.)
– Analyze your activities in class. (Cards with reflection)
Homework
paragraph 13 (before example 2) № 13.3, 13.4
Solve the equation graphically:
Construct a graph of the function and describe its properties.
Consider the function y=√x. The graph of this function is shown in the figure below.
Graph of the function y=√x
As you can see, the graph resembles a rotated parabola, or rather one of its branches. We get a branch of the parabola x=y^2. It can be seen from the figure that the graph touches the Oy axis only once, at the point with coordinates (0;0).
Now it is worth noting the main properties of this function.
Properties of the function y=√x
1. The domain of definition of a function is a ray)