Topicma2a7127cde3848dd_1528449000663_0Topic

Quadratic monomial and its properties. Translating the plot of the quadratic monomial along axis of the coordinate system

Levelma2a7127cde3848dd_1528449084556_0Level

Third

Core curriculumma2a7127cde3848dd_1528449076687_0Core curriculum

V. Function. The Student:

4) reads from the graph of the function: the domain, the range, roots, monotonic intervals, intervals in which the function takes values not greater (not smaller) or smaller (not greater) than a given number, greatest and smallest values of the function (if they exist) in the closed interval and arguments for which the function takes greatest and smallest values;

7) draws the plot of the quadratic function given by a formula;

9) identifies the formula of a quadratic function based on information about this function or about its plot;

12) based on the plot of the functionplot of the functionplot of the function y = f(x) draws plot of functions y = f(x - a), 
y = f(x)+b, y =  - f(x), y = f( - x).

Timingma2a7127cde3848dd_1528449068082_0Timing

45 minutes

General objectivema2a7127cde3848dd_1528449523725_0General objective

Using mathematical objects, interpreting mathematical concepts.

Specific objectivesma2a7127cde3848dd_1528449552113_0Specific objectives

1. Drawing the plot of a monomial and identifying its properties.

2. Drawing plots of functions $y=a(x-p{)}^2$and $y=a{x}^2+q$ based on the plot of the function $y=a{x}^2$.

3. Communicating in English, developing basic mathematical, computer and scientific competences, developing learning skills.

Learning outcomesma2a7127cde3848dd_1528450430307_0Learning outcomes

The Student:

- draws the plot of a monomial and identifying its properties,

- draws plots of functions $y=a(x-p{)}^2$ and $y=a{x}^2+q$ based on the plot of the function $y=a{x}^2$.

Methodsma2a7127cde3848dd_1528449534267_0Methods

1. Situational analysis.

2. Expert stations.

Forms of workma2a7127cde3848dd_1528449514617_0Forms of work

1. Individual work.

2. Group work.

Lesson stages

Introductionma2a7127cde3848dd_1528450127855_0Introduction

Six students create three expert groups and before the class prepare information about one of the following subject:

I. Quadratic monomial – the general formula, the plot.

II. Properties of the quadratic monomial.

III. Transformations of plots of functions.

Procedurema2a7127cde3848dd_1528446435040_0Procedure

Students – experts present prepare information one by one. After the presentation, they answer questions from the other students and clarify doubts.

Information that should be included in presentations:

I expert group

Quadratic monomial – the general formula, the plot.

General formula of the exponential function: $f\left(x\right)=a{x}^2$ where $x\in R$.

The plot of the functionplot of the functionplot of the function $f\left(x\right)=x^2$ where x ∈ R.

[Illustration 1]

- the plot of the function $f\left(x\right)=x^2$ where x ∈ R, is called a parabola,
- the point O = (0, 0) is called the vertex of the parabolavertex of the parabolavertex of the parabola,
- the vertex divides the parabola into two parts called arms of the parabola,
- the line x = 0 is the parabola’s axis of symmetryparabola’s axis of symmetryparabola’s axis of symmetry,
- the greater the absolute value of the coefficient acoefficient acoefficient a, the closer the arms are to the axis Y,
- if the coefficient a > 0, then the parabola opens up,
-if the coefficient a < 0, then the parabola opens down.

II expert group

Properties of the quadratic monomial.

- the domain is the set of real numbers,
- the range is the interval $\langle 0,\infty )$,
- the axis of symmetry of the plot is the line x = 0,
- it has one root x = 0,
- it is decreasing in the interval $(-\infty ,0)$ and decreasing in the interval $(0,\infty )$,
- it reaches the smallest value equal to 0 for the argument 0, it does not reach the greatest value,
- it is not injective.ma2a7127cde3848dd_1527752256679_0- the domain is the set of real numbers,
- the range is the interval $\langle 0,\infty )$,
- the axis of symmetry of the plot is the line x = 0,
- it has one root x = 0,
- it is decreasing in the interval $(-\infty ,0)$ and decreasing in the interval $(0,\infty )$,
- it reaches the smallest value equal to 0 for the argument 0, it does not reach the greatest value,
- it is not injective.

III expert group 

Transformation of the plot of the functionplot of the functionplot of the function.

- by translating the plot of the function $y=a{x}^2$ by p units along the X axis in accordance with the direction of the axis, we obtain the plot of the function  $y=a(x-p{)}^2$,
- by translating the plot of the function $y=a{x}^2$ by q units along the Y axis in accordance with the direction of the axis, we obtain the plot of the function  $y=a{x}^2+q$.

Students work individually, using computers. Their task is to analyse how the plot of monomials changes in discussed translations and how the plot of the functionplot of the functionplot of the function looks in various cases.

[Geogebra applet]

The teacher divides students into group. Each task group solves exercises prepared by the teacher. Experts help students, clarify doubts. The teacher supervises students’ work.

Task 1

Knowing that $f\left(x\right)=x^2$, $g\left(x\right)=-x^2$ and $h\left(x\right)=2x^2$, calculate:

a) $f\left(\frac{1}{3}\right)$

b) $f(-3)$

c) $g\left(2\right)$

d) $g(-2)$

e) $h\left(\frac{1}{2}\right)$

f) $h(-1)$

Task 2

Draw plots of functions:

a) $y=2x^2-2$

b) $y=\frac{1}{2}(x-2{)}^2$

Task 3

Give formulas of functions presented in the plot and then give their properties.

[Illustration 2]

An extra task:

Prove that if the function f is defined by the formula $f\left(x\right)=x^2$, where $x\in R$ then for each natural number n, the difference $f(n+3)-f(n+1)$ is a number that can be divided by four.

Lesson summaryma2a7127cde3848dd_1528450119332_0Lesson summary

Students do the revision exercises. Then together they sum‑up the classes, by formulating the conclusions to memorise.

- the plot of the function $f\left(x\right)=x^2$ where x ∈ R, is called a parabola,
- the point O = (0, 0) is called the vertex of the parabola,
- the vertex divides the parabola into two parts called arms of the parabola,
- the line x = 0 is the parabola’s axis of symmetry,
- the greater the absolute value of the coefficient a, the closer the arms are to the axis Y,
- if the coefficient a > 0, then the parabola opens up,
-if the coefficient a < 0, then the parabola opens down,
- by translating the plot of the function $y=a{x}^2$ by p units along the X axis in accordance with the direction of the axis, we obtain the plot of the function $y=a(x-p{)}^2$,
- by translating the plot of the function $y=a{x}^2$ by q units along the Y axis in accordance with the direction of the axis, we obtain the plot of the function $y=a{x}^2+q$.ma2a7127cde3848dd_1527752263647_0- the plot of the function $f\left(x\right)=x^2$ where x ∈ R, is called a parabola,
- the point O = (0, 0) is called the vertex of the parabola,
- the vertex divides the parabola into two parts called arms of the parabola,
- the line x = 0 is the parabola’s axis of symmetry,
- the greater the absolute value of the coefficient a, the closer the arms are to the axis Y,
- if the coefficient a > 0, then the parabola opens up,
-if the coefficient a < 0, then the parabola opens down,
- by translating the plot of the function $y=a{x}^2$ by p units along the X axis in accordance with the direction of the axis, we obtain the plot of the function $y=a(x-p{)}^2$,
- by translating the plot of the function $y=a{x}^2$ by q units along the Y axis in accordance with the direction of the axis, we obtain the plot of the function $y=a{x}^2+q$.

Selected words and expressions used in the lesson plan

coefficient acoefficient acoefficient a

parabola’s armsparabola’s armsparabola’s arms

parabola’s axis of symmetryparabola’s axis of symmetryparabola’s axis of symmetry

plot of the functionplot of the functionplot of the function

translation of the plot along the X axistranslation of the plot along the X axistranslation of the plot along the X axis

translation of the plot of the functiontranslation of the plot of the functiontranslation of the plot of the function

vertex of the parabolavertex of the parabolavertex of the parabola