1.3.2 notes

October 07, 2014
Agenda
• Warm Up
• Go over homework
• Key Concepts/Notes
• Example #1 & #2
• Extra Practice
• Homework
Warm Up
1.3.2: Creating and Graphing Exponential Equations
pg. 79
Read the scenario and answer the questions that
follow.
One form of the element beryllium, beryllium-11, has a half-life of
about 14 seconds and decays to the element boron. A chemist
starts out with 128 grams of beryllium-11. She monitors the element
for 70 seconds.
1.
What is the equation that models the amount of beryllium-11
over time?
2.
How many grams of beryllium-11 does the chemist have left
after 70 seconds?
October 07, 2014
1.3.2: Creating and Graphing Exponential Equations
1.
What is the equation that models the amount of beryllium-11
over time?
> y = abx, where y is the final value, a is the initial value, b is
the rate of growth or decay, and x is the time.
y = unknown
a = 128 grams
b = 0.5
> Time = 70 seconds, but this needs to be converted to time
periods before substituting the value for x.
• Convert 70 seconds into 14-second time
periods.
1 time period = 14 seconds
x=5
>
Substitute all the variables into the
equation.
y = abx
y = 128(0.5)5
2. How many grams of beryllium-11 does the chemist have left
after 70 seconds?
Apply the order of operations to the equation from the end
of problem 1.
y = 128(0.5)5
y = 4 grams
1.3.2: Creating and Graphing Exponential Equations
October 07, 2014
Agenda
• Warm Up
• Go over homework
• Key Concepts/Notes
• Example #1 & #2
• Extra Practice
• Homework
pg. 75 #7-10
October 07, 2014
October 07, 2014
1.3.2 Creating & Graphing Exponential Equations
I
•
•
•
pg. 78
can identify the following:
Exponential Equation
Exponential Growth & Decay
Independent & Dependent Variables
I can create and graph an exponential equation of two
variables
I can interpret and describe parts of a graph
pg. 65
I will be able to...
October 07, 2014
y = a(b)x
or
Key Concepts
pg. 80
• The general form of an exponential equation is
y = a • bx, where a is the initial value, b is the rate of decay or
growth, and x is the time. The final output value will be y.
• Since the equation has an exponent, the value increases or
decreases rapidly.
• The base, b, must always be greater than 0 (b > 0).
October 07, 2014
Key Concepts, continued
pg. 80
• Look for words such as double, triple, half, quarter—such words
give the number of the base. For example, if an experiment
begins with 1 bacterium that doubles (splits itself in two) every
hour, determining how many bacteria will be present after x
hours is solved with the following equation: y = (1)2x, where 1 is
the starting value, 2 is the rate, x is the number of hours, and y
is the final value.
pg. 80
y = a(1 r)t
October 07, 2014
Guided Practice
pg. 81
Example 1
If a pendulum swings to 90% of its height on each swing and starts
out at a height of 60 cm, what is the equation that models this
scenario? What is its graph?
1. Read the problem statement and then reread the scenario,
identifying the know quantities.
2. Substitute the known quantities into the general form of the
exponential equation y = abx, where a is the initial value, b is the
rate of decay, x is time (in this case swinging), and y is the final
value.
Guided Practice
pg. 81
Example 1, continued
If a pendulum swings to 90% of its height on each swing and starts
out at a height of 60 cm, what is the equation that models this
scenario? What is its graph?
3. Create a table of values.
4. Set up the coordinate plane. (in student textbook)
5. Plot the points on the coordinate plane and connect the points
with a line (curve).
October 07, 2014
pg. 81
October 07, 2014
October 07, 2014
Guided Practice
pg. 82
Example 2
The bacteria Streptococcus lactis doubles every 26 minutes in
milk. If a container of milk contains 4 bacteria, write an equation
that models this scenario and then graph the equation.
Guided Practice: Example 2, continued
• Read the problem statement and then
reread the scenario, identifying the known
quantities.
Initial bacteria count = 4
Base = 2
Time period = 26 minutes
Guided Practice: Example 2, continued
• Substitute the known quantities into the general
form of the exponential equation
y = abx, where a is the initial value, b is the base,
x is time (in this case, 1 time period is 26
minutes), and y is the final value.
Since the base is repeating in units other than 1, use the
equation
, where t = 26.
Guided Practice: Example 2, continued
a
=4
b=2
October 07, 2014
Guided Practice: Example 2, continued
• Create a table of values.
x
y
0
4
26
8
52
16
78
32
104
64
Guided Practice: Example 2, continued
• Set up the coordinate plane.
Determine the labels by reading the problem again. The
independent variable is the number of time periods. The time
periods are in number of minutes. Therefore, “Minutes” will be
the x-axis label. The y-axis label will be the “Number of
bacteria.” The number of bacteria is the dependent variable
because it depends on the number of minutes that have
passed.
Guided Practice: Example 2, continued
minutes and the table of
values. The table of values showed 4 time periods. One time
period = 26 minutes and so 4 time periods = 4(26) = 104
minutes. This means the x-axis scale needs to go from 0 to 104.
Use increments of 26 for easy plotting of the points. For the yaxis, start with 0 and go to 65 in increments of 5. This will make
plotting numbers like 32 a little easier than if you chose
increments of 10. See the graph that follows.
The x-axis needs a scale that reflects the time period of 26
October 07, 2014
Guided Practice: Example 2, continued
Guided Practice: Example 2, continued
• Plot the points on the coordinate plane and
connect the points with a line (curve).
When the points do not lie on a grid line, use estimation to
approximate where the point should be plotted. Add an arrow
to the right end of the line to show that the curve continues in
that direction toward infinity, as shown in the graph that
follows.
October 07, 2014
Guided Practice: Example 2, continued
Key Concepts
pg. 87
Introducing the Compound Interest Formula:
• The general form of the compounding interest formula is
where A is the initial value, r is the interest rate, n is the number of
times the investment is compounded in a year, and t is the number of
years the investment is left in the account to grow.
October 07, 2014
pg. 82
Guided Practice
Example 3
An investment of $500 is compounded monthly at a rate of 3%. What
is the equation that models this situation? Graph the equation.
Exit Slip
Graph the following exponential equation.
X
Y
October 07, 2014
1.3.2 Creating & Graphing Exponential Equations
I
•
•
•
pg. 78
can identify the following:
Exponential Equation
Exponential Growth & Decay
Independent & Dependent Variables
I can create and graph an exponential equation of two
variables
I can interpret and describe parts of a graph
pg. 85 #1-5
October 07, 2014
Agenda
• Warm Up
• Go over homework
• Example #3
• Extra Practice (if necessary)
• Problem Based Task
• Homework
Exit Slip
Graph the following exponential equation.
X
Y
October 07, 2014
Hands free
pencils down
pg. 86
Warm-Up
Exponential Equations
Determine if the following are exponential growth or decay.
3. If Downers Grove's population is growing at a rate of 3.5%
each year. Determine how many people there will be in 7 years
if there are currently 42,378 people.
Hands free
pencils down
Warm-Up
Exponential Equations
pg. 86
Determine if the following are exponential growth or decay.
3. If Downers Grove's population is growing at a rate of 3.5%
each year. Determine how many people there will be in 7 years
if there are currently 42,378 people.
October 07, 2014
Agenda
• Warm Up
• Go over homework
• Example #3
• Extra Practice (if necessary)
• Problem Based Task
• Homework
pg. 85 #1-5
October 07, 2014
pg. 85 #1-5
pg. 85 #1-5
October 07, 2014
1.3.2 Creating & Graphing Exponential Equations
I
•
•
•
pg. 78
can identify the following:
Exponential Equation
Exponential Growth & Decay
Independent & Dependent Variables
I can create and graph an exponential equation of two
variables
I can interpret and describe parts of a graph
Problem Based Task 1.3.2
pg. 83
October 07, 2014
Problem Based Tasks - Sample Response
October 07, 2014
Problem Based Tasks - Sample Response
Problem Based Tasks - Sample Response
October 07, 2014
Exit Slip
1.3.2 Creating & Graphing Exponential Equations
I
•
•
•
pg. 78
can identify the following:
Exponential Equation
Exponential Growth & Decay
Independent & Dependent Variables
I can create and graph an exponential equation of two
variables
I can interpret and describe parts of a graph
October 07, 2014
pg. 85 #6-10
pg. 85 #6-10
October 07, 2014

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