Topicm797c96537fbf8726_1528449000663_0Topic

Determining x‑intercepts of the quadratic functionquadratic functionquadratic function

Levelm797c96537fbf8726_1528449084556_0Level

Third

Core curriculumm797c96537fbf8726_1528449076687_0Core curriculum

V. Functions. The student:

4) reads out from a graph of a function: domain, range, x‑intersections, intervals of monotonicity, intervals in which the function takes on values greater (not smaller) or smaller (not larger) than the given number, maximum and minimum values of the function (if any) in a given closed interval and arguments for which the function takes on maximum and minimum values.

Timingm797c96537fbf8726_1528449068082_0Timing

45 minutes

General objectivem797c96537fbf8726_1528449523725_0General objective

Interpreting and handling information presented in the text, both mathematical and popular science, as well as in the form of graphs, diagrams, tables.

Specific objectivesm797c96537fbf8726_1528449552113_0Specific objectives

1. Communicating in English, developing mathematical and basic scientific‑technical and IT competence, forming of learning skills.

2. Understanding the relation between the number of x‑intercepts of the quadratic functionquadratic functionquadratic function and the sign of ∆.

3. Determining x‑intercepts of the quadratic functionx‑intercepts of the quadratic functionx‑intercepts of the quadratic function using its equation.

Learning outcomesm797c96537fbf8726_1528450430307_0Learning outcomes

The student:

- determines the number of x‑intercepts depending on the sign of ∆,

- determines an x‑intercept using the equation of a function.

Methodsm797c96537fbf8726_1528449534267_0Methods

1. Sentence completion.

2. A trash and a suitcase.

Forms of workm797c96537fbf8726_1528449514617_0Forms of work

1. Individual work.

2. Work in small groups.

Lesson stages

Introductionm797c96537fbf8726_1528450127855_0Introduction

The students sort out the knowledge about the quadratic function (x‑intercept, finding x‑intercepts using a graph) using the method of the sentence completion.

Procedurem797c96537fbf8726_1528446435040_0Procedure

The teacher informs the student that during the lesson they will determine x‑intercepts of the quadratic function.

Discussion – how to find x‑intercepts of the quadratic functionx‑intercepts of the quadratic functionx‑intercepts of the quadratic function when the formula of the function is known.

Conclusion:

To find x‑intercepts of the quadratic function $f\left(x\right)=a{x}^2+bx+c$, where $a\ne \; 0$ you need to solve the equation $f\left(x\right)=0$.m797c96537fbf8726_1527752263647_0To find x‑intercepts of the quadratic function $f\left(x\right)=a{x}^2+bx+c$, where $a\ne \; 0$ you need to solve the equation $f\left(x\right)=0$.

The students work in small groups and determine x‑intercepts using the conclusion.

Task
Find x‑intercepts of the function:

• $f\left(x\right)=-\sqrt{2}{x}^2$

• $g\left(x\right)=x^2-16$

• $h\left(x\right)=9-4x^2$

• $m\left(x\right)=x^2-2x-3$

While discussing solutions the students should note that the function should be written in the factored form. Perhaps for some groups it was very difficult.

The teacher proposes to analyse the Slideshow in which the general formula for x‑intercepts of the quadratic functionx‑intercepts of the quadratic functionx‑intercepts of the quadratic function is presented and to write down a conclusion.

[Slideshow]

Conclusion:
The quadratic function $y=a{x}^2+bx+c$ , where $a\ne 0$and$∆=b^2-4ac$:
- has only one x‑intercept, $x_0=\frac{-b}{2a}$, if and only if$∆=0$,
- has two x‑intercepts $x_1=\frac{-b-\sqrt{∆}}{2a}$ and $x_2=\frac{-b+\sqrt{∆}}{2a}$, if and only if $∆>0$, - has no x‑intercept, if and only if $∆<0$.

The students work in groups and solve the exercises using the conclusion.

Task
Evaluate based on the value of the discriminantdiscriminantdiscriminant, how many x‑intercepts each of given functions has:

• $f\left(x\right)=0,5x^2+8x+32$

• $h\left(x\right)=9x^2+4-2x$

• $m\left(x\right)=x^2-2x+1$

Task
Determine x‑intercepts (if they exist) for each function:

• $g\left(x\right)=2x^2-6x+5$

• $f\left(x\right)=-2x^2+4+2x$

• $h\left(x\right)=3x^2+18x+12$

Task
The quadratic function $f\left(x\right)=2x^2+8x+6$ has two different x‑intercepts.
Prove that each of them belongs to the interval $(-4,0)$.

Task
The number $(-3)$ is one of the intercepts of the function $f\left(x\right)=2x^2+bx-3$. Find b and the second x‑intercept.m797c96537fbf8726_1527752256679_0The number $(-3)$ is one of the intercepts of the function $f\left(x\right)=2x^2+bx-3$. Find b and the second x‑intercept.

An extra task
The height of a cylinder is 3 cm greater than its diameter. The total surface area is equal to $120\pi$ cmIndeks górny 2². Find the height of the cylinder.

Lesson summarym797c96537fbf8726_1528450119332_0Lesson summary

Students perform consolidating exercises. Using a trash and a suitcase method they choose the most important competences and information to be remembered. Then they summarize together the lesson, formulating conclusions to be remembered:

The quadratic functionquadratic functionquadratic function $y=a{x}^2+bx+c$ , where $a\ne 0$ and $∆=b^2-4ac$:
- has only one x‑intercept, $x_0=\frac{-b}{2a}$, if and only if$∆=0$,
- has two x‑intercepts $x_1=\frac{-b-\sqrt{∆}}{2a}$ and $x_2=\frac{-b+\sqrt{∆}}{2a}$, if and only if $∆>0$,
- has no x‑intercept, if and only if $∆<0$.

Selected words and expressions used in the lesson plan

quadratic functionquadratic functionquadratic function

discriminantdiscriminantdiscriminant

numerical coefficientsnumerical coefficientsnumerical coefficients

vertex form of the quadratic functionvertex form of the quadratic functionvertex form of the quadratic function

x‑intercepts of the quadratic functionx‑intercepts of the quadratic functionx‑intercepts of the quadratic function

number of x‑interceptsnumber of x‑interceptsnumber of x‑intercepts

factoringfactoringfactoring