Topicm2ce0b418d57b5e66_1528449000663_0Topic

Minimum and maximum value of the quadratic function in a closed interval

Levelm2ce0b418d57b5e66_1528449084556_0Level

Third

Core curriculumm2ce0b418d57b5e66_1528449076687_0Core curriculum

V. Functions. The student:

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

Timingm2ce0b418d57b5e66_1528449068082_0Timing

45 minutes

General objectivem2ce0b418d57b5e66_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 objectivesm2ce0b418d57b5e66_1528449552113_0Specific objectives

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

2. Determining the minimum and maximum value of the quadratic functionquadratic functionquadratic function in a closed interval.

Learning outcomesm2ce0b418d57b5e66_1528450430307_0Learning outcomes

The student:

- determines the minimum and maximum value of the quadratic function in a closed interval,

- determines arguments for which the function takes on maximum and minimum values.

Methodsm2ce0b418d57b5e66_1528449534267_0Methods

1. Diamond ranking.

2. Situational analysis.

Forms of workm2ce0b418d57b5e66_1528449514617_0Forms of work

1. Individual work.

2. Work in small groups.

Lesson stages

Introductionm2ce0b418d57b5e66_1528450127855_0Introduction

The students work in groups.

They sort out the properties of the quadratic function known to them using the diamond ranking method.

Procedurem2ce0b418d57b5e66_1528446435040_0Procedure

The teacher informs the student that the goal of the lesson is the determination of the minimum and maximum value of the quadratic functionquadratic functionquadratic function in a closed intervalclosed intervalclosed interval.

Discussion – when does the quadratic function reach its minimum/maximum value and what formula can describe it.

The conclusion which the students should draw.

The quadratic functionquadratic functionquadratic function y= axIndeks górny 2² + bx + c, where a ≠ 0, x ∈ R:

- for a > 0 reaches at the point x = $\frac{-b}{2a}$ , the minimum value of $\frac{-∆}{4a}$,

[Illustration 1]

- for a < 0 reaches at the point x = $\frac{-b}{2a}$ , the maximum value of $\frac{-∆}{4a}$.

[Illustration 2]

The students work in groups and discuss how to determine the minimum and maximum value of the quadratic function in a closed interval. They analyse the material contained in the applet, draw a conclusion and create a corresponding algorithm.

Task
[Geogebra applet]

Conclusion:

The minimum/maximum value of the quadratic function in a closed interval is at the one of the edges of this interval or in the vertex of the parabola, which is the graph of this function (only if the vertex belongs to this interval).m2ce0b418d57b5e66_1527752256679_0The minimum/maximum value of the quadratic function in a closed interval is at the one of the edges of this interval or in the vertex of the parabola, which is the graph of this function (only if the vertex belongs to this interval).

The algorithm for determining the minimum/maximum value of the quadratic functionf(x) = axIndeks górny 2² + bx + c in the closed intervalclosed intervalclosed interval <k, l>.

[Illustration 3]

Students use the elaborated flowchartflowchartflowchart in exercises.

Task
Find a minimum and maximum value of the function:
a) f(x) = -(x - 3)Indeks górny 2² +4 in the interval <-2, 3>,
b) f(x) = -3xIndeks górny 2² - 12x - 6 in the interval <-4, 1>,
c) f(x) = 2xIndeks górny 2² + 4x +7 in the interval <-3, 0>.

Task
Find the minimum value of the expression xIndeks górny 2² + yIndeks górny 2², when x + y = 4.

Task
The sum of the length of the side of a triangle and the height perpendicular to this side is equal to 10. What length should have the side of the triangle, so that the triangle area is the largest? Calculate this area.m2ce0b418d57b5e66_1527752263647_0The sum of the length of the side of a triangle and the height perpendicular to this side is equal to 10. What length should have the side of the triangle, so that the triangle area is the largest? Calculate this area.

An extra task
For what value of does the function f(x) = $\frac{1}{x^2-x+{\displaystyle \frac{1}{2}}}$ take on the maximum value? Find this value.

Lesson summarym2ce0b418d57b5e66_1528450119332_0Lesson summary

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

Conclusion:

The minimum / maximum value of the quadratic function in a closed interval is at the one of the edges of this interval or in the vertex of the parabola, which is the graph of this function (only if the vertex belongs to this interval).m2ce0b418d57b5e66_1527752256679_0The minimum / maximum value of the quadratic function in a closed interval is at the one of the edges of this interval or in the vertex of the parabola, which is the graph of this function (only if the vertex belongs to this interval).

Selected words and expressions used in the lesson plan

quadratic functionquadratic functionquadratic function

range of functionrange of functionrange of function

closed intervalclosed intervalclosed interval

vertex of the parabolavertex of the parabolavertex of the parabola

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

minimum value of a function in a closed intervalminimum value of a function in a closed intervalminimum value of a function in a closed interval

maximum value of a function in a closed intervalmaximum value of a function in a closed intervalmaximum value of a function in a closed interval

flowchartflowchartflowchart