Miejsca zerowe funkcji kwadratowej. Postać iloczynowa funkcji kwadratowej

Learning objectives

You will learn to draw the plot of the quadratic function based on its formula and read roots from the plot of the function as well as to identify the number of roots based on the discriminant of the quadratic polynomial and calculate them.

Learning effect

• You draw the plot of the quadratic function based on its formula and read roots from the plot of the function as well as identify the number of roots based on the discriminant of the quadratic polynomial and calculate them.

Revise terms: the root of the functionroot of the functionroot of the function, the quadratic functionquadratic functionquadratic function, the discriminant of the quadratic polynomialdiscriminant of the quadratic polynomialdiscriminant of the quadratic polynomial.

• The root of the functionroot of the functionroot 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 functionquadratic functionquadratic 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 parabolaparabolaparabola.

• The discriminant of the quadratic polynomialdiscriminant of the quadratic polynomialdiscriminant of the quadratic polynomial is a number marked by the symbol $∆$, such that $∆=b^2-4ac$.

After having completed the exercise, write proper conclusions. See if they match the following conclusions.

The quadratic functionquadratic functionquadratic 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 rootone root 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.

Task 4

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nagranie abstraktu

nagranie abstraktu

Nagranie dostępne na portalu epodreczniki.pl

nagranie abstraktu

Open the applet and apply learnt information while identifying the number of roots of the quadratic functionquadratic functionquadratic function.

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Aplikacja geogebra: Miejsca zerowe funkcji kwadratowej. Poniżej znajduje się galeria będąca wersją alternatywną dla aplikacji.

Aplikacja geogebra: Miejsca zerowe funkcji kwadratowej. Poniżej znajduje się galeria będąca wersją alternatywną dla aplikacji.

Zasób interaktywny dostępny pod adresem https://zpe.gov.pl/a/D13V0trMU

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Na rysunku przedstawiony jest wykres funkcji kwadratowej f – parabola oraz zapis f(x) równa się minus 2 razy x do kwadratu dodać 4 razy x.

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Na rysunku przedstawiony jest wykres funkcji kwadratowej f – parabola, zapis f(x) równa się minus 2 razy x do kwadratu dodać 4 razy x oraz zapisy przedstawiające obliczenie wartości wyznacznika trójmianu kwadratowego delta równa się b do kwadratu odjąć 4 razy a razy c, poniżej delta równa się 4 do kwadratu odjąć 4 razy minus dwa razy 0, poniżej delta rowna się 16.

Do the revision exercises.

• The root of the functionroot of the functionroot 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 functionquadratic functionquadratic 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 parabolaparabolaparabola.

• The discriminant of the quadratic polynomialdiscriminant of the quadratic polynomialdiscriminant 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 exactlyone rootone root 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.

Exercises

Glossary

Keywords

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

quadratic functionquadratic functionquadratic function

root of the functionroot of the functionroot of the function