Graph y ax2. How to build a parabola? What is a parabola? How are quadratic equations solved? Problems to solve independently

Algebra lesson notes for 8th grade secondary secondary school

Lesson topic: Function

The purpose of the lesson:

· Educational: define the concept of a quadratic function of the form (compare graphs of functions and), show the formula for finding the coordinates of the vertex of a parabola (teach how to use this formula on practice); to develop the ability to determine the properties of a quadratic function from a graph (finding the axis of symmetry, the coordinates of the vertex of a parabola, the coordinates of the points of intersection of the graph with the coordinate axes).

· Developmental: development of mathematical speech, the ability to correctly, consistently and rationally express one’s thoughts; developing the skill of correctly writing mathematical text using symbols and notations; development analytical thinking; development cognitive activity students through the ability to analyze, systematize and generalize material.

· Educational: fostering independence, the ability to listen to others, developing accuracy and attention in written mathematical speech.

Lesson type: learning new material.

Teaching methods:

generalized reproductive, inductive heuristic.

Requirements for students' knowledge and skills

know what a quadratic function of the form is, the formula for finding the coordinates of the vertex of a parabola; be able to find the coordinates of the vertex of a parabola, the coordinates of the points of intersection of the graph of a function with the coordinate axes, and use the graph of a function to determine the properties of a quadratic function.

Equipment:

Lesson Plan

I. Organizing time(1-2 min)

II. Updating knowledge (10 min)

III. Presentation of new material (15 min)

IV. Consolidating new material (12 min)

V. Summing up (3 min)

VI. Homework assignment (2 min)

During the classes

I. Organizational moment

Greeting, checking absentees, collecting notebooks.

II. Updating knowledge

Teacher: In today's lesson we will study new topic: "Function". But first, let's repeat the previously studied material.

Frontal survey:

1) What is called a quadratic function? (Function where given real numbers, , a real variable, is called a quadratic function.)

2) What is the graph of a quadratic function? (The graph of a quadratic function is a parabola.)

3) What are the zeros of a quadratic function? (The zeros of a quadratic function are the values ​​at which it becomes zero.)

4) List the properties of the function. (The values ​​of the function are positive at and equal to zero at; the graph of the function is symmetrical with respect to the ordinate axes; at - the function increases, at - decreases.)

5) List the properties of the function. (If , then the function takes positive values ​​at , if , then the function takes negative values ​​at , the value of the function is only 0; the parabola is symmetrical about the ordinate axis; if , then the function increases at and decreases at , if , then the function increases at , decreases – at .)

III. Presentation of new material

Teacher: Let's start learning new material. Open your notebooks, write down the date and topic of the lesson. Pay attention to the board.

Writing on the board: Number.

Function.

Teacher: On the board you see two graphs of functions. The first graph, and the second. Let's try to compare them.

You know the properties of the function. Based on them, and comparing our graphs, we can highlight the properties of the function.

So, what do you think will determine the direction of the branches of the parabola?

Students: The direction of the branches of both parabolas will depend on the coefficient.

Teacher: Absolutely right. You can also notice that both parabolas have an axis of symmetry. In the first graph of the function, what is the axis of symmetry?

Students: For a parabola, the axis of symmetry is the ordinate axis.

Teacher: Right. What is the axis of symmetry of a parabola?

Students: The axis of symmetry of a parabola is the line that passes through the vertex of the parabola, parallel to the ordinate axis.

Teacher: Right. So, the axis of symmetry of the graph of a function will be called a straight line passing through the vertex of the parabola, parallel to the ordinate axis.

And the vertex of a parabola is a point with coordinates . They are determined by the formula:

Write the formula in your notebook and circle it in a frame.

Writing on the board and in notebooks

Coordinates of the vertex of the parabola.

Teacher: Now, to make it more clear, let's look at an example.

Example 1: Find the coordinates of the vertex of the parabola.

Solution: According to the formula

Teacher: As we have already noted, the axis of symmetry passes through the vertex of the parabola. Look at the blackboard. Draw this picture in your notebook.

Write on the board and in notebooks:

Teacher: In the drawing: - equation of the axis of symmetry of a parabola with the vertex at the point where the abscissa is the vertex of the parabola.

Let's look at an example.

Example 2: Using the graph of the function, determine the equation for the axis of symmetry of the parabola.

The equation for the axis of symmetry has the form: , which means the equation for the axis of symmetry of this parabola is .

Answer: - equation of the axis of symmetry.

IV. Consolidation of new material

Teacher: The tasks that need to be solved in class are written on the board.

Writing on the board: № 609(3), 612(1), 613(3)

Teacher: But first, let's solve an example not from the textbook. We will decide at the board.

Example 1: Find the coordinates of the vertex of a parabola

Solution: According to the formula

Answer: coordinates of the vertex of the parabola.

Example 2: Find the coordinates of the intersection points of the parabola with coordinate axes.

Solution: 1) With axis:

Those.

According to Vieta's theorem:

The points of intersection with the x-axis are (1;0) and (2;0).

2) With axle:

The point of intersection with the ordinate axis (0;2).

Answer: (1;0), (2;0), (0;2) – coordinates of the points of intersection with the coordinate axes.

Methodological development of an algebra lesson in 9th grade.

A bad teacher presents the truth, a good teacher teaches how to obtain it.

A.Disterweg

Teacher: Netikova Margarita Anatolyevna, mathematics teacher, GBOU school No. 471, Vyborg district of St. Petersburg.

Lesson topic: “Graph of a functiony= ax 2 »

Lesson type: lesson in learning new knowledge.

Target: teach students to graph a function y= ax 2 .

Tasks:

Educational: develop the ability to construct a parabola y= ax 2 and establish a pattern between the graph of the function y= ax 2

and coefficient A.

Educational: development of cognitive skills, analytical and comparative thinking, mathematical literacy, ability to generalize and draw conclusions.

Educators: nurturing interest in the subject, accuracy, responsibility, demandingness towards oneself and others.

Planned results:

Subject: be able to use a formula to determine the direction of the branches of a parabola and construct it using a table.

Personal: be able to defend your point of view and work in pairs and in a team.

Metasubject: be able to plan and evaluate the process and result of their activities, process information.

Pedagogical technologies: elements of problem-based and advanced learning.

Equipment: interactive whiteboard, computer, handouts.

1.Formula of roots of quadratic equation and expansion quadratic trinomial by multipliers.

2. Reduction of algebraic fractions.

3.Properties and graph of the function y= ax 2 , dependence of the direction of the branches of the parabola, its “stretching” and “compression” along the ordinate axis on the coefficient a.

Lesson structure.

1.Organizational part.

2.Updating knowledge:

Examination homework

Oral work based on finished drawings

3.Independent work

4.Explanation of new material

Preparing to study new material (creating a problem situation)

Primary assimilation of new knowledge

5. Fastening

Application of knowledge and skills in a new situation.

6. Summing up the lesson.

7.Homework.

8. Lesson reflection.

Technological map of an algebra lesson in 9th grade on the topic: “Graph of a functiony= ax 2 »

Lesson steps	Stage tasks	Teacher activities	Student activities	UUD
1.Organizational part 1 minute	Creating a working mood at the beginning of the lesson	Greets students checks their preparation for the lesson, notes those absent, writes the date on the board.	Getting ready to work in class, greeting the teacher	Regulatory: organization of educational activities.
2.Updating knowledge 4 minutes	Check homework, repeat and summarize the material learned in previous lessons and create conditions for successful independent work.	Collects notebooks from six students (selectively two from each row) to check homework for assessment (Annex 1), then works with the class on the interactive whiteboard (Appendix 2).	Six students hand in their homework notebooks for inspection, then answer front-end survey questions. (Appendix 2).	Cognitive: bringing knowledge into the system. Communicative: the ability to listen to the opinions of others. Regulatory: evaluating the results of your activities. Personal: assessing the level of mastery of the material.
3.Independent work 10 minutes	Test your ability to factor a quadratic trinomial and reduce algebraic fractions and describe some properties of functions based on its graph.	Hands out cards to students with individual differentiated tasks (Appendix 3). and solution sheets.	Execute independent work, independently choosing the difficulty level of exercises based on points.	Cognitive: Personal: assessing the level of mastery of the material and one’s capabilities.
4.Explanation of new material Preparing to study new material Primary assimilation of new knowledge	Creating a favorable environment for getting out of a problematic situation, perception and comprehension of new material, independent coming to the right conclusion	So, you know how to graph a function y= x 2 (graphs are pre-built on three boards). Name the main properties of this function: 3. Vertex coordinates 5. Periods of monotony What's in in this case coefficient is equal at x 2 ? Using the example of the quadratic trinomial, you saw that this is not at all necessary. What sign could he be? Give examples. You will have to find out for yourself what parabolas with other coefficients will look like. The best way to study something is to discover for yourself. D.Poya We divide into three teams (in rows), choose captains who come to the board. The task for the teams is written on three boards, the competition begins! Construct function graphs in one coordinate system 1 team: a)y=x 2 b)y= 2x 2 c)y= x 2 Team 2: a)y= - x 2 b)y=-2x 2 c)y= - x 2 Team 3: a)y=x 2 b)y=4x 2 c)y=-x 2 Mission accomplished! (Appendix 4). Find functions that have the same properties. Captains consult with their teams. What does this depend on? But how do these parabolas differ and why? What determines the “thickness” of a parabola? What determines the direction of the branches of a parabola? We will conventionally call graph a) “initial”. Imagine a rubber band: if you stretch it, it becomes thinner. This means that graph b) was obtained by stretching the original graph along the ordinate. How was graph c) obtained? So, when x 2 there can be any coefficient that affects the configuration of the parabola. This is the topic of our lesson: "Graph of a functiony= ax 2 »	1. R 4. Branches up 5. Decreases by (- Increases by $. Solution. Let's build a graph of this function, select the required interval and find the lowest and highest points of our graph. Let's find the coordinates of the vertex of the parabola: $x_(c)=-\frac(b)(2a)=-\frac(6)(-2)=3$. $y_(в)=-1*(3)^2+6*3+4=-9+18+4=13$. At the point with coordinates $(3;13)$ we construct a parabola $y=-x^2$. Let's select the required interval. The lowest point has a coordinate of -3, the highest point has a coordinate of 13. $y_(name)=-3$; $y_(maximum)=13$. Problems to solve independently 1. Without graphing the function $y=-3x^2+12x-4$, answer next questions: a) Identify the straight line that serves as the axis of the parabola. b) Find the coordinates of the vertex. c) Which way does the parabola point (up or down)? 2. Construct a graph of the function: $y=2x^2-6x+2$. 3. Graph the function: $y=-x^2+8x-4$. 4. Find the greatest and smallest value functions: $y=x^2+4x-3$ on the interval $[-5;2]$. Site navigation Health Psychology Relationship Job Entertainment Latest articles Allelic genes, their properties How is a peptide bond formed in humans? How lithospheric plates move Rocks and minerals Pre-professional training at school Contents of the pre-professional training system How volume and surface are related in biology What does surface area mean?