Topicm9e7c415affae5ce8_1528449000663_0Topic

Exponential function and its properties

Levelm9e7c415affae5ce8_1528449084556_0Level

Third

Core curriculumm9e7c415affae5ce8_1528449076687_0Core curriculum

V. Functions. Student:
4. reads the domaindomaindomain, the set of valuesset of valuesset of values, roots, monotonicity intervals, intervals in which the function has greater (not less) or less (not greater) values than the given number, the least and greatest values of the function (if exist) in a given closed interval and arguments at which the function is minimized or maximized;
14. uses exponential and logarithmic functions, including their graphs, to describe and interpret issues related to practical applications.

Timingm9e7c415affae5ce8_1528449068082_0Timing

45 minutes

General objectivem9e7c415affae5ce8_1528449523725_0General objective

To interpret, analyse and manipulate information presented in the text, both mathematical and popular science, as well as in the form of graphs, diagrams, tables.

Specific objectivesm9e7c415affae5ce8_1528449552113_0Specific objectives

1. To communicate in English, develop mathematics and basic scientific and technical and IT competences, improve learning skills.

2. To draw a graph of the function $f\left(x\right)=a^x$.

3. To determine the properties of the function $f\left(x\right)=a^x$ based on its graph.

Learning outcomesm9e7c415affae5ce8_1528450430307_0Learning outcomes

The student:

- draws a graph of the function $f\left(x\right)=a^x$.

- determines the properties of the function $f\left(x\right)=a^x$ based on its graph.

Methodsm9e7c415affae5ce8_1528449534267_0Methods

1. Mind map.

2. Case study.

Forms of workm9e7c415affae5ce8_1528449514617_0Forms of work

1. Individual work.

2. Group work.

Lesson stages

Introductionm9e7c415affae5ce8_1528450127855_0Introduction

Students work in small groups. Their task is to systematize previously acquired knowledge about functions and the ways of defining them.
They present their ideas in the form of a mind map.

Procedurem9e7c415affae5ce8_1528446435040_0Procedure

The teacher informs students that the aim of the lesson is to get to know the graph and properties of the function $f\left(x\right)=a^x$, where $a>0$ and $a\ne 1$.

Students work in two groups and together prepare graphs of $f\left(x\right)=a^x$ function.

Task - Group 1
Using the table, draw the graph of the following function.

$f\left(x\right)=3^x,x\in R$

[Table 1]

Task - Group 1
Using the table, draw the graph of the following function.

$f\left(x\right)=(\frac{1}{3}{)}^x,x\in R$

[Table 2]

Students present the results of their work and discuss the set of valuesset of valuesset of values of function whose graph they drew, and define where zeros of the function are, or where it is monotonic.

The teacher informs students that the function $f\left(x\right)=a^x$, where $a>0$ and $a\ne 1$ is called the exponential functionexponential functionexponential function. He / She gives students a definition of this function.

Definition
The function $f\left(x\right)=a^x$, where $a>0$ and $a\ne 1$, is called the exponential function.m9e7c415affae5ce8_1527752263647_0The function $f\left(x\right)=a^x$, where $a>0$ and $a\ne 1$, is called the exponential function.

Students work independently and analyze the applet presenting graphs of exponential functions. Based on the graphs, they determine their properties. They draw their own conclusions.

Task

Analyze the material contained in the description. Discuss the properties of the function $f\left(x\right)=a^x$ depending on the value of $a(a>1$ or $a\in (0,1)).$

[Geogebra applet]

Students should note that:

- The graph of the function $f\left(x\right)=a^x$, where $a>0$ and $a\ne 1$ is in the first and second quadrant of the coordinate system.

- It intersects the Y axis at the point of coordinates (0,1).

- The domaindomaindomain of this function is all real numbers.

- The set of valuesset of valuesset of values is the interval $(0,+\infty )$.

- The function is increasing for $a>0$, decreasing for $0<a<1$.

- The asymptote of the graph of this function is the X axis.

Students use the acquired knowledge to solve the following tasks.

Task
Draw graphs of the functions $f\left(x\right)=2^x$ , $x\in R$ and $g\left(x\right)=(\frac{1}{2}{)}^x$, $x\in R$ in one coordinate system. How are the graphs of these functions located relative to each other? Formulate the appropriate conclusion.

Conclusion
The graph of function $f\left(x\right)=2^x$ is symmetrical to the graph of function $g\left(x\right)=(\frac{1}{2}{)}^x$ with respect to the axis Y.m9e7c415affae5ce8_1527752256679_0The graph of function $f\left(x\right)=2^x$ is symmetrical to the graph of function $g\left(x\right)=(\frac{1}{2}{)}^x$ with respect to the axis Y.

Task
Draw graphs of the functions $f\left(x\right)=4^x$ and $g\left(x\right)=-4^x$ in one coordinate system. How are the graphs of these functions located relative to each other? Formulate the appropriate conclusions.

Conclusion
The graph of function $f\left(x\right)=4^x$ is symmetrical to the function graph $g\left(x\right)=-4^x$ with respect to them9e7c415affae5ce8_1527712094602_0The graph of function $f\left(x\right)=4^x$ is symmetrical to the function graph $g\left(x\right)=-4^x$ with respect to theaxism9e7c415affae5ce8_1527752256679_0axisX.m9e7c415affae5ce8_1527712094602_0X.

Task
What is the largest and the smallest value of the function $f\left(x\right)=-(\frac{1}{5}{)}^x$ in the range of ?

Task
The graph of the exponential functionexponential functionexponential function $f\left(x\right)=a^x$ includes the point $K(-2,9)$. Check whether the $M(\frac{1}{2},\frac{\sqrt{3}}{3})$ point also belongs to the graph of this function.

An extra task
Solve this inequality graphically $(\frac{1}{2}{)}^x>8$.

Lesson summarym9e7c415affae5ce8_1528450119332_0Lesson summary

Students do revision exercises and together summarize the lesson formulating the rule to remember.
The function $f\left(x\right)=a^x$, where $a>0$ and $a\ne 1$ specified for $x\in R$ is called the exponential functionexponential functionexponential function.

Selected words and expressions used in the lesson plan

asymptote of the function graphasymptote of the function graphasymptote of the function graph

decreasing functiondecreasing functiondecreasing function

domaindomaindomain

exponential functionexponential functionexponential function

increasing functionincreasing functionincreasing function

monotonicity of the functionmonotonicity of the functionmonotonicity of the function

set of valuesset of valuesset of values

zero of the functionzero of the functionzero of the function