Graphing Exponential FunctionsExplain how to tell if an exponential function is growth or decay, and what an exponential growth and decay graph looks like.

Graphing Exponential FunctionsMM2A2. Students will explore exponential functions.b. Investigate and explain characteristics of exponential functions, including domain and range, asymptotes, zeros, intercepts, intervals of increase and decrease, rates of change, and end behavior. (4.4, 4.5, 4.6)c. Graph functions as transformations of . (4.4, 4.5, 4.6)

Graphing Exponential Functions• An exponential function has the form:

• (this is a little different than the book)• If a > 0 and b > 1, then the function is an

exponential growth function, and b is called the growth factor

• If a > 0 and 0 < b < 1, then the function is an exponential decay function, and b is called the decay factor

• What if a < 0?• What if there is a negative in front of the x?

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Graphing Exponential Functions

• An exponential function has the form:

• An asymptote is a line that a graph approaches more and more closely. The asymptote for this function will be y = k

• The graph will be shifted horizontally by h• The end behavior of a function’s graph is the

behavior of the graph as x approaches positive or negative infinity.

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Graphing Exponential Functions

• Minimum steps to graph exponential functions:1. Determine the asymptote by looking at the value

of “k” – sketch it on the graph2. Determine the general shape and end conditions

by looking at the values of “a”, “b”, and the sign on the x.

3. Find the y-intercept by calculating and plotting x = 0

4. Determine the rough shape by calculating and plotting two more points (x = 1 may be good)

5. Smoothly draw the curve

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• Watch videos:• http://www.onlinemathlearning.com/sketch-

exponential-graph.html• Watch 1st video from 3:30 introducing e• Watch 2nd video for negative transformations

Graphing Exponential Functions

• Determine the general shape, intervals of increase/decrease, asymptote, end behavior, domain, range, x and y intercepts, and average rate of change -1 < x < 2 of the following graphs?

523)( xxf 35.1)( xxg

xxi 25.0)(

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xxh 8.05.1)(

22.05.0)( 3 xxj

Rate of Change

• What is rate of change?• How do we calculate it?• We can calculate the “average rate of change”

for non-linear functions• Find the average rate of change of the

following functions for -2 x 2

523)( xxf 35.1)( xxg

12.03)( 5 xxh

Practice

• Page 122, # 1 – 12 all• Page 127, # 1 – 6 all with explanations and 7 –

16 all.

Warm Up

• Find asymptote, determine general shape of curve, find y-intercept, graph, state the domain and range, end conditions, and find average rate of change 0 < x < 1

38.05.1)( xxh

432)( xxg

Given: f(x) = (2x + 7)(2x – 9), Find:

• Vertex – is it a minimum or maximum?• AoS• Domain• Range• x-intercepts• y-intercept• Intervals of increase/decrease• End conditions• Average rate of change from 0 < x < 2

Given the function: f(x) = 2(x - 4)2 + 3, find:

• Vertex – is it a minimum or maximum?• AoS• Domain• Range• x-intercepts• the solutions• y-intercept• Intervals of increase/decrease• End conditions• Average rate of change from 0 < x < 2• Write the function in standard form

Match the function to the graph8

7

6

5

4

3

2

1

-1

-6 -4 -2 2 4 6

q x = 0.5xh x = 0.2xg x = 3xf x = 2x

Given the function, find:

• Domain• Range• y-intercept• Asymptote• Zeros• Intervals of increase/decrease• End conditions• Average rate of change from -2 < x < 2• Is function a growth or decay function?

xxf 3)(

Given the function, find:

• Domain• Range• y-intercept• Asymptote• Zeros• Intervals of increase/decrease• End conditions• Average rate of change from -2 < x < 2• Is function a growth or decay function?

xxf 75.0)(

Given the function, find:

• Domain• Range• y-intercept• Asymptote• Zeros• Intervals of increase/decrease• End conditions• Average rate of change from -2 < x < 2• Is function a growth or decay function?

35.4)( xxf

Given the function, find:

• Domain• Range• y-intercept• Asymptote• Zeros• Intervals of increase/decrease• End conditions• Average rate of change from -2 < x < 2• Is function a growth or decay function?

63.0)( xxf

Compare and Contrast:

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trPA 1

(_#_# startend timesofgrowthStartEnd __#)1(_#_#

nt

n

rPA

1

Practice

• Packet last two sheets - 4 problems