Exploring Exponential Functions

 

Name: Monica Sustaita

 

Title of lesson: Exploring Exponential Functions

 

Date of lesson: Wednesday, Week 4

 

Length of lesson: 1-2 days

 

Description of the class:

                     Name of course:  Algebra I

                     Grade level: 8th or 9th grade

                     Honors or regular: Regular

 

Source of the lesson:

            Exponential functions and the Number e

http://www.langara.bc.ca/~acooper/mathlabs/exp&log/index.htm#

 

TEKS addressed:

(a) Basic understandings

(4) Relationship between equations and functions. Equations arise as a way of asking and answering questions involving functional relationships. Students work in many situations to set up equations and use a variety of methods to solve these equations.

(5) Tools for algebraic thinking. Techniques for working with functions and equations are essential in understanding underlying relationships. Students use a variety of representations (concrete, numerical, algorithmic, graphical), tools, and technology, including, but not limited to, powerful and accessible hand-held calculators and computers with graphing capabilities and model mathematical situations to solve meaningful problems.

(d) Quadratic and other nonlinear functions: knowledge and skills and performance descriptions.

(3) The student understands there are situations modeled by functions that are neither linear nor quadratic and models the situations. Following are performance descriptions.

(A) The student uses patterns to generate the laws of exponents and applies them in problem-solving situations.

(B) The student analyzes data and represents situations involving inverse variation using concrete models, tables, graphs, or algebraic methods.

I.       Overview
In this lesson, the students will become familiar with exponential functions and its properties.  They will also be introduced to the number e and inverses of exponential functions and the natural log function.  This is important to learn because many things in the real world are represented by an exponential function like growth, decay and interest.  It is also the first step in understanding exponential functions that will be used later in their mathematics journey.
 
II.  Performance or learner outcomes

            Students will be able to:           

á        Describe an exponential graph  and give its properties

á        Describe over what interval the function is defined on.

á        Determine what the number e is

á        Describe the inverse function and over what interval this function is defined on.

III. Resources, materials and supplies needed

á        Computer (either one for the whole class, a few for each group or for every individual)

á        The website and Graph Explorer

á        Transparency/large papers

 

IV. Supplementary materials, handouts.

á        Handout-copy of tutorials

 

 

Five-E Organization

Teacher Does                     Probing Questions                      Student Does       

Engage:

Shows the students a real life situation of money being invested and the growth of it over time due to interest.

Why and how did the money grow so fast? 

 

Can this be explained using function?

     

See a general curve of the graph and see that it grows really fast, really quick.

                                                   

Explore:

Give the students a computer (if available) and handout.

 

Activity 1:

Click the 'New Function' button, enter the expression b^x, and color the graph to contrast with the axes.
With b=0 you should see a graph along the x axis for x>0, but not for x<0.

 

 

Now use the animation button to set the b-value to be steadily increasing and describe and explain the result.

 

 

 

Set b=2.

 

 

Graph also b^(x+1).

 

 

 

Then graph c*b^x and watch the graph scale up until it passes b^x (at c=1) and continues on up to match b^(x+1).

 

Describe in words a graphical interpretation of the identity b*b^x=b^(x+1).

 

Do the same with (b^x)/b and (b^x)^a

 

     

 

 

 

 

 

 

 

 

 

 

Why is this?

 

 

 

 

What happens when b=1?

 

 

 

 

Predict the values at x=-2,-1,0,1,and 2.

 

 

 

 

What does this do the graph?

 

 

At what value of c does this happen?

 

Go to the website and follow along in the handout.

 

 

 

 

 

 

 

 

Because a negative exponent is the reciprocal and 0 does not have one because it is undefined.

 

ItÍs a line at 1.

 

 

 

 

Make predictions.

 

Note that the ordinates (y-values) are as expected for x= -2,-1,0,1,and 2.

 

Note how it is shifted one unit to the left of the graph of y=b^x..

 

At c=b.

 

 

 

 

    

Explain:


On paper, have students write properties of exponents.

 

 

 

As a group, come up with 4 or 5 characteristics of exponential.

                                               

Extend:

Create a linear function graph which slope 1 and y-intercept 1 (and a color distinct from your b^x graph).

 

With b set equal to 2 compare the two graphs.

Change the b-value to 4 and again compare the slopes.

Compare again at b=3,
and at b=2.5. Centre your view on the point (0,1) and zoom in to make the crossing clearer.

Now reduce the step size and increase b gradually until the graphs coincide. Zoom in again and repeat the process to try and get as close an estimate as you can to the b-value which makes the two graphs tangent at x=0.

 

If you are careful you may be able to estimate the b value which gives tangency correct to two decimal places.

Explain e. Try graphing this function and zoom in on the point where it appears to cross the x axis.

Elaborate:

Recall b^x note that at b=0 the function is only defined for x>0, and for b<0 there is no graph at all.

Vary b>0.

 

 

 

For any base b between 0 and 1, or greater than 1, the function giving y=b^x is invertible, i.e. and inverse exists.

 

Explain its inverse function is called the logarithm with base b. To avoid using subscripts we will write this as log(b,x) but in standard notation with b as a subscript after the word 'log'.

 

Enter the function log(b,x). Vary b and note that this relation is preserved.


Graph the functions b^(logb,x) and  log(b,b^x) and explain how they are related.

 

 

 

 

 

 

 

Is the slope of y=2^x more or less than 1 at x=0?

 

Which slope is more?

 

 

 

Which slope is greater in each of the cases?

 

 

 

 

 

 

 

 

 

 

 

 

Approximately, what is this number?

 

 

 

Why again?

So when is there always a graph?  What is the domain?

Are there any b-values for which the graph fails the Horizontal Line Test?

 

 

 

 

 

 

 

 

How does this graph relate to that of b^x?

 

Is there any difference?  Why?

 

Continue to follow along in the next activity.

 

 

 

 

Less.

 

 

4^x.

 

 

 

 

 

 

 

 

 

 

 

They come up with a number that has a limiting value of (1+x)^(1/x) as x approaches 0.

 

Between 2 and 3 or 2.718ƒ

 

 

 

For b>0 there is always a graph.  The domain is all real numbers.

 

B=1 because itÍs a straight line.

 

 

 

 

 

 

 

 

ItÍs the reflection of b^x about the line y=x. (inverse of b^x)

   

  Evaluate:

Have each group extent the properties of exponents to include the number e and log functions.

 

Have each group present and formulate a class consensus.  Hang these properties up on the paper.    

 

 

 

 

Percent effort each team member contributed to this lesson plan:

___%___       ____Name of group member_____________________

 

___%___       ____Name of group member_____________________


                                                                                      Name________________________            

http://www.langara.bc.ca/~acooper/mathlabs/exp&log/index.htm#

 

Exponential Functions and the Number e

Activity#1 - Some First Examples

Click the 'New Function' button, enter the expression b^x, and colour the graph to contrast with the axes.
With b=0 you should see a graph along the x axis for x>0, but not for x<0. Why is this? ________________________________________________________________________________________________________________________________________________

Now use the animation button to set the b-value to be steadily increasing and describe and explain the result.
What happens when b=1? __________________________________________________

Set b=2 and note that the ordinates (y-values) are as expected for x= -2,-1,0,1,and 2. Predictions:        x=-2   y=______                                                                                                                          x=-1   y=______                                                                                                                     x=0    y=______                                                                                                                     x=1    y=______                                                                                                                        x=2    y=______                                                                                

Graph also b^(x+1) and note how it is shifted one unit to the left of the graph of y=b^x.

Then graph c*b^x and watch the graph scale up until it passes b^x (at c=1) and continues on up to match b^(x+1). At what value of c does this happen? ______________________

Describe in words a graphical interpretation of the identity b*b^x=b^(x+1).
 _______________________________________________________________________________________________________________________________________________________________________________________________________________________

Activity#2 - Slopes at x=0

Create a linear function graph which slope 1 and y-intercept 1 (and a color distinct from your b^x graph).

With b set equal to 2, by comparing the two graphs decide whether the slope of y=2^x is more or less than 1 at x=0. ____________________

Change the b-value to 4 and again compare the slopes. ___________________________

Compare again at b=3 and at b=2.5. ______________________   ___________________

Centre your view on the point (0,1) and zoom in to make the crossing clearer.

Now reduce the step size and increase b gradually until the graphs coincide.

Zoom in again and repeat the process to try and get as close an estimate as you can to the b-value which makes the two graphs tangent at x=0.


(You may find it easier to check whether the b^x graph actually crosses below 1+x for some x>0 by zooming in on a point slightly to the right of x=0, and if you think you have gone too far then you can check whether b^x goes below 1+x on the left by zooming in a bit to the left of x=0)

If you are careful you may be able to estimate the b value which gives tangency correct to two decimal places.

Your value should then agree to that accuracy with the number e that we obtained as the limiting value of (1+x)^(1/x) as x approaches 0. Try graphing this function and zoom in on the point where it appears to cross the x axis.

________________________________________________________________________________________________________________________________________________

Activity#3 - Inverse Functions

For the function b^x note that at b=0 the function is only defined for x>0, and for b<0 there is no graph at all (actually there are values of x for which (-1)^x makes sense, but the y values jump back and forth between -1 and +1 so there is no way of drawing a continuous graph through all of them and the graph is not visible).

Graph at ________________  Domain:________________________________________

As you vary b>0 are there any b-values for which the graph fails to pass the Horizontal Line Test? _______________________________________________________________

For any base b between 0 and 1, or greater than 1, the function giving y=b^x is invertible and its inverse function is called the logarithm with base b. To avoid using subscripts we will write this as log(b,x) but in standard notation with b as a subscript after the word 'log'.

Enter the function log(b,x) and note how its graph is related to that of b^x. ________________________________________________________________________________________________________________________________________________

Vary b and note that this relation is preserved.

Graph the functions b^(logb,x) and  log(b,b^x) and explain how they are related.
Is there any difference? If so explain what is it and why it occurs.

________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________