FOM 12 7.1- Exponential Functions

Exponential Functions are based on the concept of repeated multiplication.

For example folding a sheet of paper: Number of Folds Number of Layers

(n) (/)

0

1

2

3

4

5

How many layers would there be with 10 folds?

How many layers would there be with 12 folds?

Exponential Function: A function of the form y = a(b )x where a * 0, b > 0, and b * 1.

(variable is in the exponent)

Graphs of Exponential Functions:

y = zx

-

Observations:

7 Exponential growth starts slow but eventually increases very quickly 7 Exponential functions have a horizontal asymptote (a line for which the graph will get closer and

closer to but never actually touch) 7 No x-intercepts 7 Only one y-intercept 7 End Behaviour: QII to QI

7 Domain: x E Ill 7 Range: y > 0

Example 2: y = 3 (2) x

=-L�:- L.-- ... ' I i I

I

-:- ··1·-; : : I , - t· -···- ·-1 r .1 I .-. ·t- I -----T .. L.

r· :·�-!---+-+-"-· --r _.., --t , l.

!· • � -·-· ' -•·

j __ : J ·-·

• I �- T. . . J .

>!_ : I I .. 1· -, -

r I

I •·i I ' '...�---

** The number in front of the base is they-intercept **

Example 3: y = (½)x

i J.

i - : - : .. . �-' ' I j I t- : i -.. -J- - ---: -r-

·-:· -

1.''

I I

** When the base is O < b < 1 we get Exponential Decay and b > 1 is Exponential Growth** - -

Assignment: Pg. 439 #1 - 3

FOM 12 7.2 – Relating the Characteristics of an Exponential Function to its Equation Testing for an Exponential Relationship: # Is each step being multiplied by the same number?

x y 1 5 2 10 3 20 4 40 5 80

Number of x-intercepts: Remember from 7.1 an Exponential Function will never touch the x-axis therefore will never have an x-intercept. Finding the y-intercept (remember 𝑥 = 0 when a graph crosses the y-axis)

𝑦 = 2' 𝑦 = 3(2') 𝑦 = 10(2')

• The leading coefficient is the _____________________________

Is the function Increasing or Decreasing?

𝑦 = 3' 𝑦 = ,12-'

𝑦 = 0.5(2') 𝑦 = 4,13-'

X Y X Y X Y X Y

Example 1: Which exponential function matches each graph below?

𝒚 = 𝟑(𝟎.𝟐)𝒙 𝒚 = 𝟒(𝟑)𝒙 𝒚 = 𝟒(𝟎.𝟓)𝒙 𝒚 = 𝟐(𝟒)𝒙

Assignment: Pg. 448 # 1 – 4, 5 – 10 (odd letters), 12, 13

Increasing if ____________

Decreasing if _______________

Lesson Notes 7-3 Modeling Data Using Exponential Functions

An exponential function of the form f(x) = a(b)x, with a > 0, b > 0, and b ≠ 1, models growth

when b > 1. The y-values increase from left to right along the x-axis.

An exponential function models decay when a > 0 and 0 < b < 1. The y-values decrease from

left to right along the x-axis.

An exponential regression function can be determined the same way as a line of best fit was

determined last chapter.

Example 1: The population of Manitoba is given for the years from 1951 to 2011.

Year 1951 1961 1971 1981 1991 2001 2011

Pop 776.5 921.7 998.9 1035.5 1109.6 1151.4 1250.6

a) Construct a scatter plot to display the data.

b) Use exponential regression to define a function that models the data.

c) Assume the growth rate continues. Estimate the population in 2020.

d) Estimate when the population will reach 1 500 000.

*population in the table above is measured in thousands of people

Example 2: The population of Alberta is given for the years from 2007 to 2011.

Year 2007 2008 2009 2010 2011

Population 3512.7 3591.8 3671.7 3720.9 3779.4

a) Construct a scatter plot to display the data.

b) Use exponential regression to define a function that models the data.

c) Assume the growth rate continues. Estimate the population of Alberta in 2020.

d) Estimate when the population will reach 4 200 000.

FOM 12 Exponential Function and Carbon Dating Carbon Dating

The decay of radioactive elements can sometimes be used to date events fort he earth’s past. In a living organism, the ratio of radioactive carbon, carbon-14, to ordinary carbon remains fairly constant. However, when the organism dies, no new carbon is ingested and the proportion of carbon-14 decreases as it decays An estimate of the age, x years, of organic remains can be determined

from the exponential function 𝑷 = 𝟏𝟎𝟎%𝟏𝟐'

𝒙𝟓𝟕𝟑𝟎, where P is the

percentage of carbon-14 remaining.

a) State a suitable domain and range for the exponential function.

b) Graph the function using a calculator window [Xmin = 0, Xmax = 20,000, Xscl = 10,000] and [Ymin = 0, Ymax = 100, Yscl = 50]

c) On May 1, 2012, a family in Whitehorse was digging up their basement when they discovered what were believed to be the remains of a prehistoric bison. If the skeleton of the bison is about 10,000 years old, what percentage, to the nearest whole number, of carbon-14 should the paleontologist find remaining in the bones that were found?

d) A bone fragment with the carved image of a mammoth was discovered in the southern U.S. state of Florida in June 2011. If the carbon dating test indicated that approximately 20.35% of carbon-14 was left, estimate the age of the bone fragment to the nearest 1 000 years

FOM 12 7.4- Characteristics of Logarithmic Functions

with Base 10 and Base e

A Logarithm is an Exponent written in a different way

102 = 100 - ).

103 = 1000 ).

101 = 10 )

log10 (100) = 2

Example 1: What is the value of log10 (20) ?

Example 2: Sketch a graph of the function y = log10 (:X::)

,. """"'.� ;r -1 .... -·...,

�rna:,n:

'Ra.n3e !

....

I 1, I .. I

�· r\l JI .... .... . �-

., D ,_

I

1 •

�

I, l"'

II .. h ,: ' ' I.-

i'il • \ LI p I _I

Introducing "e": e is a constant number named e in honour of Euler who proved that it was irrational.With the possible exception of rr, e is the most important constant in mathematics since it appears inmyriad mathematical contexts involving limits and derivatives. The numerical value of e ise = 2.7 l 828 l 828459045235360287471352662497757 ...

Logarithms with a base of e are called Natural ,Logarithms.

y = loge (x) is equivalent to

Example 3: Sketch a graph of the function y = -4 ln (x)

y = In (x)

I

ii � -·

__...

-

.

1

� ,

•

r ' l

., • ....

Determine:

the x-intercept:

end behaviour:

number of y-intercepts:

domain and range:

....

�·

a)

Comparing Exponential Functions & Logarithmic Functions

�)(�,�� ... iho..l Loa RlTH )..l\C....

Example 4: Which function matches each graph below?

y= 2(o.� Y=sew

y b) 8

4

-8 -4 o-4

·S

4 8

c)

X

y= -2 ln(x)

y d) y

8 8

4 4

-8 -4 0 4 8

-4

,c

Assignment: Pg. 482 #1, 2, 4, 5, 6, 8, 10, 12 (Extension #14, 15)

Lesson Notes 7-5 Modeling Data Using Logarithmic Functions

A logarithmic function may be a good model for a set of data if a scatter plot of the data forms an

increasing or decreasing curve in Quadrant I and/or Quadrant IV.

The general form of the logarithmic regression model is y = (constant) + (multiplier)•lnx.

A logarithmic curve of best fit can be used to predict values that are not recorded or plotted.

Predictions can be made by reading values from the curve of best fit on a scatter plot or by using

the equation of the logarithmic regression function.

Example 1: Determine the equation of the logarithmic regression function that models the

given data, and describe these characteristics of it graph:

x 2 4 6 8 10 12 14 16 18 20

y 16.6 33.1 42.8 49.7 55.0 59.4 63.0 66.2 69.0 71.6

Example 2: Jamie earned $4000 in her job after school. The table shows Jamie’s

balance, to the nearest dollar, over the first 5 years. Use logarithmic regression to

determine when the investment will grow to $6000.

Amount 4343 4168 4000 4525 4914 4716

Time 3 4 5 0 1 2

The location of any intercepts

The end behaviour

The domain and range

Increasing or decreasing?

Example 3: Create a Logarithmic Regression function for the following data.

(Round to the nearest hundredth)

Example 4: (Pg. 488) Lydia is researching the rise in tuition fees for post-secondary education in

Alberta for her school website. She found some data that uses the tuition fees in 1992 as the

benchmark, assigning them a value of 100%. The tuition fees in all other years are compared with

the tuition fees in 1992.

a) Create a scatter plot to visualize the type of relationship (regression)

b) Create a regression function for the data using your calculator

c) Estimate when the tuition fees will be double what they were in 1992.

d) what will the tuition fees be in the year 2010

Assignment: Pg. 494 #1-3, 5-7, 9

x 100 90.26 73.90 60.51 49.54 40.56

Y 0 0.01 0.03 0.05 0.07 0.09

Tuition Fees as a Percent

of Costin 1992 (%) Year

37.8 1979

43.8 1982

54.0 1984

58.8 1986

69.4 1989

100.0 1992

222.7 1999

287.1 2004

305.9 2006