Topicm81161e1b19e9adb2_1528449000663_0Topic

Roots of the quadratic function. Factored form of the quadratic function

Levelm81161e1b19e9adb2_1528449084556_0Level

Third

Core curriculumm81161e1b19e9adb2_1528449076687_0Core curriculum

V. Function.

Basic level. 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 functionquadratic functionquadratic function given by a formula;

8. Interprets coefficients occurring in the quadratic function is standard, vertex and factored form (if it exists).

Timingm81161e1b19e9adb2_1528449068082_0Timing

45 minutes

General objectivem81161e1b19e9adb2_1528449523725_0General objective

Using mathematical objects, interpreting mathematical concepts.

Specific objectivesm81161e1b19e9adb2_1528449552113_0Specific objectives

1. Drawing the plot of the quadratic function based on its formula and reading roots from the plot of the function.

2. Identifying the number of roots based on the discriminant of the quadratic polynomialdiscriminant of the quadratic polynomialdiscriminant of the quadratic polynomial.

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

Learning outcomem81161e1b19e9adb2_1528450430307_0Learning outcome

The Student:

- draws the plot of the quadratic functionquadratic functionquadratic function based on its formula and reading roots from the plot of
the function,

- identifies the number of roots based on the discriminant of the quadratic polynomial.

Methodsm81161e1b19e9adb2_1528449534267_0Methods

1. Situational analysis.

2. JIGSAW.

Forms of workm81161e1b19e9adb2_1528449514617_0Forms of work

1. Individual work.

2. Group work.

Lesson stages

Introductionm81161e1b19e9adb2_1528450127855_0Introduction

Students revise terms: the root of the functionroot of the functionroot of the function, the quadratic function, the discriminant of the quadratic polynomialdiscriminant of the quadratic polynomialdiscriminant of the quadratic polynomial.

- The root of the function is such an argument for which the function takes the value 0. Graphically, it is the point of intersection of the plot with the axis X,
- The quadratic function is a function defined by the formula $f\left(x\right)=a{x}^2+bx+c$, where a, b and c are real numbers and a is a number different than zero. The plot of this function is a parabola,
- The discriminant of the quadratic polynomial is a number marked by the symbol $∆$, such that $∆=b^2-4ac$.m81161e1b19e9adb2_1527752256679_0- The root of the function is such an argument for which the function takes the value 0. Graphically, it is the point of intersection of the plot with the axis X,
- The quadratic function is a function defined by the formula $f\left(x\right)=a{x}^2+bx+c$, where a, b and c are real numbers and a is a number different than zero. The plot of this function is a parabola,
- The discriminant of the quadratic polynomial is a number marked by the symbol $∆$, such that $∆=b^2-4ac$.

Procedurem81161e1b19e9adb2_1528446435040_0Procedure

Students work using the JIGSAW method.

The teacher divides students into 3 persons groups. Each member of the group gets different task from
the tasks below. After solving the tasks, students gather in groups that were doing the same task. They discuss the solutions and clarify any doubts. Then, they return to the initial groups and present the solutions to other members. They discuss the relation between the value of the discriminant of the quadratic polynomialdiscriminant of the quadratic polynomialdiscriminant of the quadratic polynomial and the number of roots.

Task 1

1. Draw plots of functions:

a) $f\left(x\right)=(x+1{)}^2-4$

b) $f\left(x\right)=x^2+x-6$

(Hint: write the formula in the vertex form.)

2. Give the number of its roots.

3. Calculate the discriminant of the quadratic polynomialdiscriminant of the quadratic polynomialdiscriminant of the quadratic polynomial.

Task 2

1. Draw plots of functions:

a) $f\left(x\right)=(x-1{)}^2$

b) $f\left(x\right)=x^2+6x+9$

(Hint: write the formula in the vertex form.)

2. Give the number of its roots.

3. Calculate the discriminant of the quadratic polynomial.

Task 3

1. Draw plots of functions:

a) $f\left(x\right)=(x+2{)}^2+3$

b) $f\left(x\right)=x^2+2x+3$

(Hint: write the formula in the vertex form.)

2. Give the number of its roots.

3. Calculate the discriminant of the quadratic polynomial.

The teacher evaluates students’ work and clarifies doubts.

After having completed the exercise, students write proper conclusions.

Conclusions:

The quadratic function defined by the formula $f\left(x\right)=a{x}^2+bx+c$:
- has two real roots xIndeks dolny 1₁ and xIndeks dolny 2₂ if and only if the discriminant ∆ is positive. Then we can write the formula of the function in the factored form:

where: $x_1=\frac{-b-\surd ∆}{2a}$, $x_2=\frac{-b+\surd ∆}{2a}$,
- has exactly one root x if and only if the discriminant ∆ is equal to zero. Then we can write the formula of the function in the factored form:

where: $x_0=\frac{-b}{2a}$,
- has no roots only and only if the discriminant ∆ is negative. Then we cannot write the formula of the function f in the factored form. 

Students work individually, using computers. Their task is to apply learnt information while identifying the number of roots of the quadratic function.

[Geogebra applet]

An extra task:

There is the quadratic functionquadratic functionquadratic function $f\left(x\right)=-\frac{1}{2}(x+6)(x-26).$ Give the vertex form of this function.

Lesson summarym81161e1b19e9adb2_1528450119332_0Lesson summary

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

- the root of the function is such an argument for which the function takes the value 0. Graphically it is the point of intersection of the plot with the axis X,
- the quadratic function is a function defined by the formula $f\left(x\right)=a{x}^2+bx+c$, where a, b and c are real numbers and a is a number different than zero. The plot of this function is a parabola,
- the discriminant of the quadratic polynomial is a number marked by the symbol ∆, such that $∆=b^2-4ac$,m81161e1b19e9adb2_1527752256679_0- the root of the function is such an argument for which the function takes the value 0. Graphically it is the point of intersection of the plot with the axis X,
- the quadratic function is a function defined by the formula $f\left(x\right)=a{x}^2+bx+c$, where a, b and c are real numbers and a is a number different than zero. The plot of this function is a parabola,
- the discriminant of the quadratic polynomial is a number marked by the symbol ∆, such that $∆=b^2-4ac$,

The quadratic function defined by the formula $f\left(x\right)=a{x}^2+bx+c$:
- has two real roots xIndeks dolny 1₁ and xIndeks dolny 2₂ if and only if the discriminant ∆ is positive. Then we can write the formula of the function in the factored form:

where: $x_1=\frac{-b-\surd ∆}{2a}$, $x_2=\frac{-b+\surd ∆}{2a}$,

- has exactly one root x if and only if the discriminant ∆ is equal to zero. Then we can write the formula of the function in the factored form:

where: $x_0=\frac{-b}{2a}$,

- has no roots only and only if the discriminant ∆ is negative. Then we cannot write the formula of the function f in the factored form. 

Selected words and expressions used in the lesson plan

discriminant of the quadratic polynomialdiscriminant of the quadratic polynomialdiscriminant of the quadratic polynomial

factored form of the quadratic functionfactored form of the quadratic functionfactored form of the quadratic function

negative discriminantnegative discriminantnegative discriminant

one rootone rootone root

parabolaparabolaparabola

positive discriminantpositive discriminantpositive discriminant

quadratic functionquadratic functionquadratic function

root of the functionroot of the functionroot of the function

two different rootstwo different rootstwo different roots