Chapter 4 Exponential and Logarithmic Graphs

8/12/2019 Chapter 4 Exponential and Logarithmic Graphs

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Exponential and Logarithmic Graphs

A. Graphs of Exponential Functions with any Base

(including e):

For graphs of the formf(x) = ax, where aR

+\{1}: (including e)

The maximal domain isR. The range isR+. Thex-axis is the horizontal asymptote. [i.e.y= 0] They-intercept is 1. They are all increasing functions.

Transformations

Dilation:f(x) =Aakx, whereA, kR+.

The graph off(x) = axis dilated by factorAfrom thex-axis and by factor

from they-axis


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Reflection: Iff(x) = - axis reflected in thex-axis. The graph is a decreasing function instead ofan increasing function. They-intercept changes to (0,

1) and the range becomesR

.

Iff(x) = ax

. All key features stay the same but the graph is a decreasing function instead of an

increasing function.

Translation: For all of the graphs of the formf(x) = ax+ b+B, where b, BR, and aR

+\{1},

the maximal domain isR the range is (B, ) the horizontal asymptote isy=Band they are all increasing functions btranslates the graph horizontally,Btranslates the graph vertically.


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Example

Sketch the graph off(x) = 2 ex 1

, showing intercepts and asymptotes, and stating the domain

and the range showing all working.

[Complete Ex. 4A and 4C]


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B. Logarithmic Graphs to any Base(including e):

For graphs of the formf(x) = loga(x), where aR+\{1}: (including e)

The maximal domain isR+; that is, there are no negative values ofx. The range isR. The vertical asymptote is they-axis so there are noy-intercepts. The graph crosses thex-axis at (1, 0) because loga(1) = 0. They are all increasing functions.

Transformations

Dilation:

The functionf(x) =Aloga(x) dilates the graph off(x) = loga(x) by a factor ofAfrom thex-axis.

The vertical asymptote,x-intercept, domain and range remain the same. AsAincreases, thegraph becomes steeper.


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The functionf(x) = loga(kx) dilates the graph off(x) = loga(x) by a factor from they-axis. the

vertical asymptote, domain and range stay the same, but thex-intercept is . As kincreases, the

graph becomes steeper and thex-intercept becomes smaller.

Reflection:

Iff(x) = - loga(x) is reflected in thex-axis. All key features remain the same but the graph is adecreasing function instead of an increasing function.

Iff(x) = loga(-x) is reflected in they-axis. The vertical asymptote and the range remain the same

but thex-intercept and the domain change.


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Translation:

For all of the graphs of the formf(x) =Aloga(x- b) +B, where b, BR, and aR+\{1} and e.

the maximal domain is (b, the range is R the vertical asymptote isx= band they are all increasing functions btranslates the graph horizontally,Btranslates the graph vertically.

Example

Sketch the graph off(x) = 2 loge(3 x) 2, showing intercepts and asymptotes, and stating the

domain, range and transformations. Give exact values or round to 3 decimal places.

[Complete Ex. 4B and 4D]


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E. Finding equations for graphs of Exponential and

Logarithmic functions:

If we know points on the curve, we can substitute the values into the most suitable general

equation:

For an exponential graph the general equation isy=Ae(x+ b)+B. For a logarithmic graph the general equation isy=Aloge(x+ b) +B.

If there are two unknownstwo pieces of information are necessary.

Examples

1.The equation of the graph shown is of the formf(x) = aex+ b. Find the values of aand band

hence find the equation of the function.

2. The equation of the graph shown is of the formy=Aloge(x+ b) +B. Find the values ofA, bandBandhence find the equation.


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F. Addition of Ordinates:

A graph of the sum of two functions can be drawn by sketching the two functions on the sameset of axes and then adding they-values for each value ofx.

If h(x) =f(x) +g(x), domain h(x) = domainf(x) domaing(x). Suitable points at which to add ordinates are:

1. the end points of the graph2. the points of intersection of the two graphs3. thex-intercepts of the two graphs.

The technique can be used for the difference of the two functions, if it is rewritten as a sum: h(x)=f(x) g(x) =f(x) + [

g(x)].

Examples

1. State the domain off(x) = 3x2+ loge(x).

2.Given the graphs off(x) andg(x), sketch the graph of h(x) =f(x) +g(x).


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Further Graphs:

Use a CAS calculator to graph the product of two functions or composite functions.

Obtain the equation of any asymptote of the function by considering asymptotic behaviour of the

individual functions.

Even though you are using the CAS calculator to graph these functions you need to clearly label

on the graph:

Asymptotes with equations axis intercepts as coordinates

Example

Sketch the graph ofy=x2exusing a CAS calculator. Show all axis intercepts and any asymptotes.


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G. Exponential and Logarithmic Functions with Absolute

Values:

The modulus, or absolute value, function is defined as

For y= f|x|, the graph of y= f(x), where x 0, is reflected in the y-axis.

The rule of the composite functiony=f|x|, wheref(x) = loge(x),x> 0, can be written as:

The rule of the composite functiony=f|x|, wheref(x) = ex, can be written as:


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For y= |f(x)|, negative y-values of y= f(x) are reflected in the x-axis.

To obtain the graph ofy= |f(x|, negativey-values ofy=f(x) are reflected in thex-axis. The rule

of the composite functiony=f|x|, wheref(x) = loge(x),x> 0, can be written as:

The rule of the composite functiony= |f(x)|, wheref(x) = ex k, kR

+, can be written as:

Examples

1.For the functiony= 2 loge|x+ 2| 3:

a sketch the graph ofy= 2 loge|x+ 2| 3, showing any asymptotes

bcalculate all axis intercepts both in exact form and correct to 2 decimal places

state the domain and range.


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2.aSketch the graph ofy= |ex 1| showing all axis intercepts and asymptotes.bState the domain and range.


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H. Exponential and Logarithmic Modelling using Graphs:

Graphs of these functions can be used to illustrate the model and make predictions for future

changes.

In most cases when modelling real life situations, the domain is restricted to [0, ) because t=0 when the model begins.

Example

The population of wombats in Snubnose Gully is increasing according to the equation:

W= 100e0.03t

where Wis the number of wombats tyears after 1 January 1998.

aFind the initial size of the population.

bFind the population 2 years and 10 years after the number of wombats was first recorded. Giveanswers to the nearest whole wombat.

cGraph Wagainst tfor 0 t 30.


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dFind the expected size of the population in the year 2020.

eFind the year in which the wombat population reaches 250.

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Chapter 4 Exponential and Logarithmic Graphs