Topicm0755446da5b43ac9_1528449000663_0Topic

The exponential functionexponential functionexponential function and its properties. Transforming plots of the exponential function.

Levelm0755446da5b43ac9_1528449084556_0Level

Third

Core curriculumm0755446da5b43ac9_1528449076687_0Core curriculum

V. Functions
The student:

1) identifies functions as clear assignment using word description, table, graph, formula (including formula on different sets);

2) calculates values of the functionfunctionfunction given by the algebraic formula;

3) reads and interprets values of functions determined by tables, graphs, formulas etc., also in cases of using the same source of information a few times or a few sources at once;

4) reads from the graph of the functionfunctionfunction: the domain, the range, roots, monotonic intervals, intervals in which the function takes values not greater (not smaller) or smaller (not greater) than a given number, greatest and smallest values of the functionfunctionfunction (if they exist) in the closed interval and arguments for which the function takes greatest and smallest values;

12) based on the plot of the functionfunctionfunction y = f(x) draws plots of functions y = f(x - a), y = f(x)+b, y = - f(x), y = f(-x);

14) uses exponential and logarithmic, including their plots, to describe and interpret concepts related to practical applications

Timingm0755446da5b43ac9_1528449068082_0Timing

45 minutes

General objectivem0755446da5b43ac9_1528449523725_0General objective

Using mathematical objects, interpreting mathematical concepts.

Specific objectivesm0755446da5b43ac9_1528449552113_0Specific objectives

1. Identifying properties of the exponential functionexponential functionexponential function.

2. Drawing plots of functions y = f(x - a), y = f(x)+b, y = - f(x), y = f(-x) based on the plot of the functionfunctionfunction y = f(x).

3. Communicating in English, developing basic mathematical, computer and scientific competences, developing learning skills.

Learning outcomesm0755446da5b43ac9_1528450430307_0Learning outcomes

The student:

- Identifies properties of the exponential functionfunctionfunction.

- Draws plots of functions y = f(x - a), y = f(x)+b, y = - f(x), y = f(-x) based on the plot of the functionfunctionfunction y = f(x).

Methodsm0755446da5b43ac9_1528449534267_0Methods

1. Situational analysis

2. Expert stations

Forms of workm0755446da5b43ac9_1528449514617_0Forms of work

1. Individual work

2. Group work

Lesson stages

Introductionm0755446da5b43ac9_1528450127855_0Introduction

Six students create three expert groups and prepare information about one of the following subjects before the class.

I. The exponential functions – general formula, plot.

II. Properties of the exponential functionexponential functionexponential function.

III. Transformations of the plot of the functionfunctionfunction

Procedurem0755446da5b43ac9_1528446435040_0Procedure

Students – experts present prepared information one by one. After the presentation, they answer questions from other students and clarify doubts.

Information that should be included in presentations of the expert groups.

I EXPERT GROUP
The exponential functions – general formula, plot.

The general formula of the exponential functionexponential functionexponential function $f\left(x\right)=a^x$, where $x\in R$, a is a set positive number, different than 1.

[Illustration 1]

[Illustration 2]

II EXPERT GROUP
Properties of the exponential functionexponential functionexponential function:

- the domain of the function is the set of all real numbers,
- the range of the functionfunctionfunction is the interval (0,+∞),
- the asymptote of the function is the line y=0 - there are no roots,
- it is monotonic and if a>1, then the function f is increasing and if 0<a<1, then the functionfunctionfunction is decreasing,
- it is injective, so each value is taken by only one argument,
- the plot of the function crosses the axis Y in the point (0,1).

III EXPERT GROUP
Transformation of the plot of the function:

- By transforming the plot of the functionfunctionfunction y = f(x) by p units along the X axis in accordance with the direction of the axis, we obtain the plot of the function y = f(x - p).
- By transforming the plot of the function y = f(x) by q units along the Y axis in accordance with the direction of the axis, we obtain the plot of the function y = f(x)+q.
- By transforming the plot of the functionfunctionfunction y = f(x) in axial symmetry with respect to the X axis, we obtain the plot of the function y = - f(x).
- By transforming the plot of the function y = f(x) in axial symmetry with respect to the Y axis, we obtain the plot of the function y = f(-x).
Students work individually, using computers. Their task is to analyse how the plot of the exponential functionexponential functionexponential function changes in discusses transformations.

[Geogebra applet]

After having completed the exercise, students write down formulas of functions obtain as a result of presented transformations.

- By transforming the plot of the functionfunctionfunction $y=a^x$ by p units along the X axis in accordance with the direction of the axis, we obtain the plot of the function $y=a^{x-p}$.
- By transforming the plot of the function $y=a^x$ p by q units along the Y axis in accordance with the direction of the axis, we obtain the plot of the function $y=a^x+q$.
- By transforming the plot of the function $y=a^x$ in axial symmetry with respect to the X axis, we obtain the plot of the function $y=-a^x$.
- By transforming the plot of the functionfunctionfunction $y=a^x$ in axial symmetry with respect to the Y axis, we obtain the plot of the function $y=a^{-x}$.

The teacher divides students into four groups that approach information stations. Each task group does exercises prepared by the teacher. Experts help students, clarify doubts. The teacher supervises groups’ work. After doing exercises from one part, students go on to the next station.

I GROUP - task
Draw in one coordinate system plots of functions $f\left(x\right)=3^x$ and $f\left(x\right)=(\frac{1}{3}{)}^x$. What can you say about the mutual position of plots of these functions?

II GROUP - task
Draw the plot of the functionfunctionfunction $f\left(x\right)=4^x$ and identify its properties.

III GROUP - task
In the drawing there is the plot of the exponential functionexponential functionexponential function. On separate pieces of paper, draw plots of functions after given transformations. Write formulas of obtained functions.

Transformations:

a. translation by 3 units to the right along the X axis,
b. translation by 2 units up along the Y axis,
c. symmetry with respect to the X axis,
d. symmetry with respect to the Y axis.

[Illlustration 3]

The teacher evaluates students’ work and clarifies doubts.

An extra task:
Draw the plot of the functionfunctionfunction $f\left(x\right)=-2^{(x-3)}+4$.

Lesson summarym0755446da5b43ac9_1528450119332_0Lesson summary

Students do the revision exercises. Then together they sum‑up the classes, by formulating the conclusions to memorise.

The general formula of the exponential functionexponential functionexponential function: $f\left(x\right)=a^x$, where $x\in R$, a is a set positive number, different than 1.

Properties of the exponential function:
- the domain of the function is the set of all real numbers, - the range of the function is the interval (0,+∞),
- the asymptote of the functionfunctionfunction is the line y=0 - there are no roots,
- it is monotonic and if a>1, then the function f is increasing and if 0<a<1, then the function is decreasing,
- it is injective, so each value is taken by only one argument,
- the plot of the functionfunctionfunction crosses the axis Y in the point (0,1).

- By transforming the plot of the function $y=a^x$ by p units along the X axis in accordance with the direction of the axis, we obtain the plot of the function $y=a^{x-p}$.m0755446da5b43ac9_1527752256679_0By transforming the plot of the function $y=a^x$ by p units along the X axis in accordance with the direction of the axis, we obtain the plot of the function $y=a^{x-p}$.
- By transforming the plot of the function $y=a^x$ by q units along the Y axis in accordance with the direction of the axis, we obtain the plot of the function $y=a^x+q$.m0755446da5b43ac9_1527752263647_0By transforming the plot of the function $y=a^x$ by q units along the Y axis in accordance with the direction of the axis, we obtain the plot of the function $y=a^x+q$.
- By transforming the plot of the function $y=a^x$ in axial symmetry with respect to the X axis, we obtain the plot of the functionfunctionfunction $y=-a^x$.
- By transforming the plot of the function $y=a^x$ in axial symmetry with respect to the Y axis, we obtain the plot of the functionfunctionfunction $y=a^{-x}$.

Selected words and expressions used in the lesson plan

asymptote of the plot of the exponential functionasymptote of the plot of the exponential functionasymptote of the plot of the exponential function

exponential functionexponential functionexponential function

functionfunctionfunction

injectivenessinjectivenessinjectiveness

monotonicitymonotonicitymonotonicity

symmetry along the X axissymmetry along the X axissymmetry along the X axis

symmetry along the Y axissymmetry along the Y axissymmetry along the Y axis

translation of the plot along the X axistranslation of the plot along the X axistranslation of the plot along the X axis

translation of the plot of the functiontranslation of the plot of the functiontranslation of the plot of the function