Topicm58d58bf4bdca77d5_1528449000663_0Topic

Quadratic monomialquadratic monomialQuadratic monomial and its characteristics

Levelm58d58bf4bdca77d5_1528449084556_0Level

Third

Core curriculumm58d58bf4bdca77d5_1528449076687_0Core curriculum

V. Functions. Basic scope. The student:

7) sketches the graph of thequadratic functionquadratic functionquadratic function given by a formula;

8) interprets the coefficients occurring in the quadratic function formula in the standard, vertex and factored form (if any);

9) determines the quadratic function formula based on the information about this function or its graph.

Timingm58d58bf4bdca77d5_1528449068082_0Timing

45 minutes

General objectivem58d58bf4bdca77d5_1528449523725_0General objective

Interpreting and handling information presented in the form of graphs and formulas.

Specific objectivesm58d58bf4bdca77d5_1528449552113_0Specific objectives

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

2. Preparing a graph of the function f(x) = axIndeks górny 2².

3. Determining the properties of the function f(x) = axIndeks górny 2² based on its graph.

Learning outcomesm58d58bf4bdca77d5_1528450430307_0Learning outcomes

The student:

- prepares  a graph of the function f(x) = axIndeks górny 2²,

- determines the properties of the function f(x) = axIndeks górny 2² based on its graph.

Methodsm58d58bf4bdca77d5_1528449534267_0Methods

1. Mind map.

2. Situational analysis.

Forms of workm58d58bf4bdca77d5_1528449514617_0Forms of work

1. Individual work.

2. Work in small groups.

Lesson stages

Introductionm58d58bf4bdca77d5_1528450127855_0Introduction

Students revise at home what are: definition of functionsfunctionfunctions, ways of describing functions, construction and properties of a monomial.

They use the collected information at the beginning of the lesson to create relevant mind maps.

Procedurem58d58bf4bdca77d5_1528446435040_0Procedure

The students draw a graph of the function f(x) = xIndeks górny 2² (with a given domain) and discuss its properties.

Task 1

The function f(x) = xIndeks górny 2² is determined for x ∈ {-4,-3, -2, -1, 0, 1, 2, 3, 4}.

a) Plot a graph of the function.
b) Specify a range of the function.
c) In which quadrants of the coordination system is the graph placed?
d) Has the function an x‑intercept? If yes, provide it.

Task 2

[Geogebra applet]

The students observe how the graph of the function f(x) = axIndeks górny 2² changes depending on the coefficient a and its sign. They read out the values of the function for $x=\frac{1}{2},x=-\frac{1}{2},x=\frac{5}{2},x=-\frac{5}{2}$.

After the exercise they share their observations and draw conclusions.

The teacher points out elements of the graph of the functionfunctionfunction f(x) = axIndeks górny 2² and discusses its properties.

Conclusion:

The function f(x) = axIndeks górny 2² (a ≠ 0 and x ∈ R) is called quadratic monomial. Its graph is a curve with equation y = axIndeks górny 2². This curve is called a parabola. It is an axisymmetric figure. The symmetry axis is a straight line with the equation x = 0. It intersects with the parabola in a point called a vertex. It also divides the parabola into two parts called arms.m58d58bf4bdca77d5_1527752256679_0The function f(x) = axIndeks górny 2² (a ≠ 0 and x ∈ R) is called quadratic monomial. Its graph is a curve with equation y = axIndeks górny 2². This curve is called a parabola. It is an axisymmetric figure. The symmetry axis is a straight line with the equation x = 0. It intersects with the parabola in a point called a vertex. It also divides the parabola into two parts called arms.

[Illustration 1]

Task 3

The students work in two groups. The first group makes a graph of the function f(x) = 2xIndeks górny 2², the second group makes a graph of the function f(x) = -2xIndeks górny 2².

Then the students read out from the graph:
- the coordinates of the vertex,
- the range of the function,
- the intervals where the function decreases,
- the intervals where the function increases,
- the equation of the symmetry axis of the function,
- the minimum value of the function.m58d58bf4bdca77d5_1527752263647_0Then the students read out from the graph:
- the coordinates of the vertex,
- the range of the function,
- the intervals where the function decreases,
- the intervals where the function increases,
- the equation of the symmetry axis of the function,
- the minimum value of the function.

The summary of the group work should be to determine the properties of the function f(x) = axIndeks górny 2² depending on the sign of the coefficient a.

Conclusions:

- The domain of the function f(x) = axIndeks górny 2² with x ∈ R are the real numbers.
- The function has only one x‑intercept.
- The function is not an injective function.

Other properties of the function depend on the sign of the coefficient a.

The properties of the function f(x) = axIndeks górny 2² for:

[Table 1]

The value of the coefficient a decides about the span of the parabolaparabolaparabola arms – the larger value of |a|, the smaller the span.

An extra task:

Given is the function f(x) = xIndeks górny 2², x∈ R. Prove, that for every natural number n:

a) f(n + 1) - f(n) is a natural odd number,

b) f(n + 5) - f(n + 3) is a number divisible by 4.

Lesson summarym58d58bf4bdca77d5_1528450119332_0Lesson summary

Students perform consolidating exercises. Then they summarize together the activities, formulating conclusions to be remembered:

- The curve with the equation y = axIndeks górny 2² is the graph of the function f(x) = axIndeks górny 2² (a ≠ 0) known as a monomial.

- This curve is called a parabola. It is an axisymmetric figure. The symmetry axis is a straight line with the equation x = 0. It intersects the parabola in the point called a vertex.

- The parabola, which is a graph of the function f(x) = axIndeks górny 2² (a ≠ 0), has a vertex in a point with the coordinates (0,0). This point divides the parabola into two parts called arms.

Selected words and expressions used in the lesson plan

functionfunctionfunction

domain of a functiondomain of a functiondomain of a function

quadratic functionquadratic functionquadratic function

range of a functionrange of a functionrange of a function

quadratic monomialquadratic monomialquadratic monomial

increasing functionincreasing functionincreasing function

decreasing functiondecreasing functiondecreasing function

parabolaparabolaparabola

vertex of a parabolavertex of a parabolavertex of a parabola

arms of a parabolaarms of a parabolaarms of a parabola