Pre-Calculus 12H.

Chapter 7: Exponential Functions

Assignment Outline

This unit examines key characteristics and properties of exponential functions. Techniques used to solve

exponential equations will be taught and applied to solving problems.

Date

Topic

Assignment

7.1 Exponential Functions p. 342 #1-4, 5ac, 6, 7

7.2 Transformations of Exponential Functions

p. 354 #1-5, 7, 9, 10, 12

7.3 Exponential Equations p. 364 #1-5, 7aceg, 10, 11

Ch. 7 Test

Exponential Functions

This module will explore the properties of exponential functions of the form y=cx,c>0,c≠1. Transformations

(reflections, stretches, and translations) of functions as they apply to exponential functions will be discussed.

Solving Exponential Equations

Exponential equations, in one variable, will be solved using a common base.

Modelling Exponential Behaviour

Real-world problems involving exponential growth and decay will be modelled and solved using exponential

functions.

Name: ____________________ Pre- Calculus 12 H. Date: _____________

Chapter 7 – Exponential Functions 7.1 – Characteristics of Exponential Functions

An exponential function is a function that can be written in the form ( ) xy f x c

Where c is a constant (c > 0) and x is a variable

INCREASING DECREASING

1c , xcy 1c0 , xcy

Example: Graph each exponential function. Then identify the following:

a) xy 2 b)

x

y

2

1

What is the x intercept? (Let y =0) What is the x intercept? (y =0)

What is the y-intercept? (Let x = 0) What is the y-intercept? (Let x = 0)

Domain: Range: Domain: Range:

Asymptote: Asymptote:

Domain and Range

x & y intercept(s)

Increasing or Decreasing

Equation of the Horizontal Asymptote

x y

x y

Applying Graphing Tools to the Exponential Function xby

1. Give a rough sketch of the following functions (dot the function xy 2 on each):

a) xy 2 b) xy 2 c) xy 4 d) y

1

2

x

a) What happens to )(xf when it becomes )(xf ?

b) What happens to )(xf when it becomes )( xf ?

c) What happens to )(xf as b gets larger than 1?

d) What happens to )(xf when b is between 0 and 1?

Example: Write the function of the following graphs in xcy form.

GROWTH and DECAY

0c Exponential Growth

10 c Exponential Decay

*Half-Life: Length of time for an unstable element to spontaneously decay to one half its original mass.

Example: A radioactive sample of radium (Ra-225) has a half-life of 15 days.

a) Beginning with 1.0 grams, create an exponential graph that models the mass of Ra-225 over time. What

value does the mass remaining approach as time passes? What are the domain and range of this function?

b) Write the exponential decay model that relates the mass of Ra-225 remaining to time, in 15-day

intervals.

c) Estimate how many days it would take for Ra-225 to decay to 30

1 of its original mass.

Exponential functions are used to model populations, carbon date artifacts, help coroners determine time of

death, compute investments, as well as many other applications.

The most common applications: population growth, exponential decay, and compound interest

A good model for most exponential functions is:

n

t

rAA )(0 where

growth of periodn

timet

aygrowth/dec of rater

amount initial A

tat timeamount A

0

Note: the term half-life refers to the amount of time it takes for decay to be at a rate of 0.5

Example: A scientist is doing an experiment on bacterial growth. She places a single bacterium in a petri

dish and it divides to form two new bacteria within one hour. This process will continue as long as there is

sufficient food and space for the bacteria.

a) Write an equation to illustrate this situation. What would the domain be?

b) What is the population size after?

3 hours?

3.5 hours?

4 hours?

24 hours?

48 hours?

c) If the scientist added an antibiotic to the petri dish medium, the population of bacteria decreases at a rate

of 7% per hour. What equation could you use to illustrate the relative size of the population over time?

Approximately how many hours would have to pass so that the population is 50% of what it was to start

with?

Example: There are 40,000 homes in a certain city. Each year 10% of the homes are expected to

disconnect from septic systems and connect to the sewer system. Write an exponential decay model for the

number of homes that still use septic systems. Use the graph of the model to estimate when about 17,200

homes will still not be connected to the sewer system.

Assignment: p. 342 #1-4, 5ac, 6, 7