Topicmbc2b6f0c5f25d13d_1528449000663_0Topic

Number of x‑intercepts of the quadratic function

Levelmbc2b6f0c5f25d13d_1528449084556_0Level

Third

Core curriculummbc2b6f0c5f25d13d_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 a 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.

Timingmbc2b6f0c5f25d13d_1528449068082_0Timing

45 minutes

General objectivembc2b6f0c5f25d13d_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 objectivesmbc2b6f0c5f25d13d_1528449552113_0Specific objectives

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

2. Finding the number of x‑intercepts of a quadratic function using its graph.

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

Learning outcomesmbc2b6f0c5f25d13d_1528450430307_0Learning outcomes

The student:

- finds the number of x‑intercepts of a quadratic function using its graph,

- determines the number of x‑intercepts of a quadratic function using its equation.

Methodsmbc2b6f0c5f25d13d_1528449534267_0Methods

1. Situational analysis.

2. Sentence completion.

Forms of workmbc2b6f0c5f25d13d_1528449514617_0Forms of work

1. Individual work.

2. Work in small groups.

Lesson stages

Introductionmbc2b6f0c5f25d13d_1528450127855_0Introduction

The students sort out the knowledge about the quadratic function using the method of the sentence completion, initialized by the teacher. They remind the definition and the graphical interpretation of the x‑intercept of the function.

Procedurembc2b6f0c5f25d13d_1528446435040_0Procedure

The teacher informs the students that during the lesson they will determine the number of x‑intercepts of the quadratic function described in a graphical and algebraic way.

The students work in groups.

Task for a group

Plot the graph of the functions f and g. Find the number of x‑intercepts for each function using their graphs.mbc2b6f0c5f25d13d_1527752256679_0Plot the graph of the functions f and g. Find the number of x‑intercepts for each function using their graphs.

Group 1: f(x) = (x - 2)Indeks górny 2² - 3, g(x) = -(x + 2)Indeks górny 2² + 2

Group 2: f(x) = -(x - 2)Indeks górny 2² - 4, g(x) = -(x + 1)Indeks górny 2² + 3

Group 3: f(x) = 3(x + 2)Indeks górny 2², g(x) = -2(x - 3)Indeks górny 2²

Discussion – how many x‑intercepts can the quadratic function have?

Conclusion:

The quadratic function can have two x‑intercepts, one x‑intercept or any of them.

The students are thinking together if a number of x‑intercepts of the quadratic function could be determined without plotting its graph. They check their hypotheses by analysing the Slideshow. They draw a conclusion.

[Slideshow]

Analyse carefully the Slideshow presenting the position of the graph of the quadratic function $y=a(x-p{)}^2+q$: depending on the coefficients a and q.

Note in each case what sign has the product $a·q$. What do you notice? Draw a conclusion.

Conclusion:

The quadratic function $y=a(x-p{)}^2+q$:

- has no x‑intercept if $a·q>0$,
- has one x‑intercept if
- has two x‑intercepts if $a·q<0$.

The students solve the exercises individually using the conclusion.

Task 1

Determine the number of x‑intercepts for each function:
$y=-\frac{3}{8}(x+9{)}^2+2$,
$y=\frac{3}{8}(x-5{)}^2+5$,
$y=-\frac{3}{8}(x+9{)}^2$.

Task 2

The parabola, which is the graph of the quadratic function, has the vertex at the point (-3,2). The parabola is open upwards. How many x‑intercepts has this function?mbc2b6f0c5f25d13d_1527752263647_0The parabola, which is the graph of the quadratic function, has the vertex at the point (-3,2). The parabola is open upwards. How many x‑intercepts has this function?

Task 3

Determine the number of the solutions of the equation.

a. $2+(x+1{)}^2=0$

b. $9-(4-x{)}^2=0$

c. $-4x^2+6=0$

Task 4

The quadratic function f has two different x‑intercepts xIndeks dolny 1₁ and xIndeks dolny 2₂. The symmetry axis of the parabola, which is the graph of this function, is the line x = 0.

Prove that

a. $2·x_1·x_2<0$

b. $x_1+x_2<2$

An extra task:

Determine all values of the coefficient b for which the function $y=x^2+bx-4$ has exactly one x‑intercept.

Lesson summarymbc2b6f0c5f25d13d_1528450119332_0Lesson summary

The students perform consolidating exercises. Then they summarize together the lesson, formulating conclusions to be remembered:

The quadratic function $y=a(x-p{)}^2+q$:

- has no x‑intercept if $a·q>0$,
- has one x‑intercept if $a·q=0$,
- has two x‑intercepts if $a·q<0$.

Selected words and expressions used in the lesson plan

standard formstandard formstandard form

numerical coefficientsnumerical coefficientsnumerical coefficients

vertex formvertex formvertex form

arms of the parabolaarms of the parabolaarms of the parabola

vertex of the parabolavertex of the parabolavertex of the parabola

x‑intercepts of a functionx‑intercepts of a functionx‑intercepts of a function

coordinates of the vertex of the parabolacoordinates of the vertex of the parabolacoordinates of the vertex of the parabola