(U2) Lesson 5 - The Twin Powers - JAMES TAYLOR | Library

Today's Learning Goal:
Use properties of negative exponents to examine key features of continuously increasing or decreasing exponential functions.
Explain the end behavior of exponential functions on the portion of the graph where the domain values are negative.
Use properties of exponents to rewrite exponential expressions that involve negative exponents
Reminders:
Discrete functions have distinct and separate values and are used for things that can be counted. (countable)
Continuous functions can have any value within a specific interval and values are connected. (measureable)
Domain of a function: The numbers that can be used as possible inputs
Today's Materials:
Laptop
Pencil
Binder
Please complete the Jump Start (activator). This is independent it should be silent.

Required

0

a^-1 = 1/a
This is true based on the negative exponent property rule, we investigated last class.

Prove it by simplifying this expression. a^3/a^4

Required

0

What observations do you make about Carlos’s graph?

How about especially when the x-values are negative?

(List what you notice about the graph, include some observations about when the x values are negative)

Required

0

Clarita: "Sometimes it helps to make a table and look for patterns in the numbers. I’m going to try that."

Complete Clarita’s table. Think about any patterns that you see.

Use fractions instead of decimals throughout the task to make the patterns easier to see.

Required

0

Compare the graphs of f(x) and g(x).

What do you notice?

Required

0

Clarita: "I think we should make a table to compare the two functions. I’ve added a column to my previous table."

Complete Clarita’s table and make sure that the table matches the graphs of the functions.

Use fractions (not decimals).

Carlos: "I remember this about negative exponents: 
Negative exponents always have a "twin": an equivalent expression that is written with a positive exponent."

(The y-values of inputs with negative exponents are the reciprocals of the y-values with the opposite positive exponents.)

Clarita: "I think we could play around with these negative exponents and use them with other bases"

Try applying the multiplication rule for exponents,
 to these expressions.  When you have finished, use another method to verify that your answers are correct. 

"Show your work" when showing use of the rule and the other method to verify your answer as correct. 

Required

1

Try applying the division rule for exponents,
 to these expressions.  When you have finished, use another method to verify that your answers are correct. 

"Show your work" when showing use of the rule and the other method to verify your answer as correct. 

Required

1

Try applying the power of a power rule for exponents,
 to these expressions.  When you have finished, use another method to verify that your answers are correct. 

"Show your work" when showing use of the rule and the other method to verify your answer as correct. 

Required

1

Required

0

Exit Ticket:
Explain why raising a positive number to a negative power doesn’t result in a negative answer.

Independent Practice:

Required

1

Write the radical expression in rational form

Required

1

Write the expression in radical form

Required

1

Fill the table for using the function.

Independent Practice: 
Linear vs Exponential

Required

1

Complete the table below for the following job offers. Then answer questions.

After graduation, you are offered two jobs. Cedar Grove Associates, offered to start you at $30,000 with a 6% increase per year. Maple Grove Associates offered to start you at $40,000 with a $1200 raise per year.

Compare the two jobs offered by completing the table below.

Required

1

Think about the type of function Cedar Grove models. Write the explicit equation.

Use c(x) for the total money earned at cedar grove and x for the number of years.

Required

1

Think about the type of function Maple Grove models. Write the explicit equation.

use m(x) for maple grove.

Required

1

If you plan on moving to a different state in 5 years which company would be the better option for you to choose.

Required

1

If your plans change and you don't move, which company would be the better option to choose as a long term career?

Required

1

A company is struggling financially at the moment.
A new employee is shown the salary schedule and is
given the equation y = (25,000)(0.99) 15 .

What does 25,000 refer to in this situation?

Required

1

A company is struggling financially at the moment.
A new employee is shown the salary schedule and is
given the equation y = (25,000)(0.99) 15 .

What does the 0.99 refer to in this situation?

Required

1

A person is keeping track of the height
of a tree in their backyard.
The height is modeled by the equation
y = 4.5x + 18 , where x represents the
number of months and y represents
the height, in inches.

What does 4.5 refer to in this situation?

Required

1

A person is keeping track of the height
of a tree in their backyard.
The height is modeled by the equation
y = 4.5x + 18 , where x represents the
number of months and y represents
the height, in inches.

What does the 18 refer to in this situation?

(U2) Lesson 5 - The Twin Powers

a^-1 = 1/a This is true based on the negative exponent property rule, we investigated last class. Prove it by simplifying this expression. a^3/a^4

What observations do you make about Carlos’s graph?How about especially when the x-values are negative? (List what you notice about the graph, include some observations about when the x values are negative)

Clarita: "Sometimes it helps to make a table and look for patterns in the numbers. I’m going to try that."Complete Clarita’s table. Think about any patterns that you see. Use fractions instead of decimals throughout the task to make the patterns easier to see.

Compare the graphs of f(x) and g(x).What do you notice?

Clarita: "I think we should make a table to compare the two functions. I’ve added a column to my previous table."Complete Clarita’s table and make sure that the table matches the graphs of the functions. Use fractions (not decimals).

Exit Ticket:Explain why raising a positive number to a negative power doesn’t result in a negative answer.

Independent Practice:

Write the radical expression in rational form

Write the expression in radical form

Fill the table for using the function.

Independent Practice:

Think about the type of function Cedar Grove models. Write the explicit equation. Use c(x) for the total money earned at cedar grove and x for the number of years.

Think about the type of function Maple Grove models. Write the explicit equation.use m(x) for maple grove.

If you plan on moving to a different state in 5 years which company would be the better option for you to choose.

If your plans change and you don't move, which company would be the better option to choose as a long term career?

A company is struggling financially at the moment.A new employee is shown the salary schedule and isgiven the equation y = (25,000)(0.99) 15 .What does 25,000 refer to in this situation?

A company is struggling financially at the moment.A new employee is shown the salary schedule and isgiven the equation y = (25,000)(0.99) 15 .What does the 0.99 refer to in this situation?

A person is keeping track of the heightof a tree in their backyard.The height is modeled by the equationy = 4.5x + 18 , where x represents thenumber of months and y representsthe height, in inches.What does 4.5 refer to in this situation?

A person is keeping track of the heightof a tree in their backyard.The height is modeled by the equationy = 4.5x + 18 , where x represents thenumber of months and y representsthe height, in inches.What does the 18 refer to in this situation?

a^-1 = 1/a
This is true based on the negative exponent property rule, we investigated last class.

Prove it by simplifying this expression. a^3/a^4

What observations do you make about Carlos’s graph?

How about especially when the x-values are negative?

(List what you notice about the graph, include some observations about when the x values are negative)

Clarita: "Sometimes it helps to make a table and look for patterns in the numbers. I’m going to try that."

Complete Clarita’s table. Think about any patterns that you see.

Use fractions instead of decimals throughout the task to make the patterns easier to see.

Compare the graphs of f(x) and g(x).

What do you notice?

Clarita: "I think we should make a table to compare the two functions. I’ve added a column to my previous table."

Complete Clarita’s table and make sure that the table matches the graphs of the functions.

Use fractions (not decimals).

Exit Ticket:
Explain why raising a positive number to a negative power doesn’t result in a negative answer.

Think about the type of function Cedar Grove models. Write the explicit equation.

Use c(x) for the total money earned at cedar grove and x for the number of years.

Think about the type of function Maple Grove models. Write the explicit equation.

use m(x) for maple grove.

A company is struggling financially at the moment.
A new employee is shown the salary schedule and is
given the equation y = (25,000)(0.99) 15 .

What does 25,000 refer to in this situation?

A company is struggling financially at the moment.
A new employee is shown the salary schedule and is
given the equation y = (25,000)(0.99) 15 .

What does the 0.99 refer to in this situation?

A person is keeping track of the height
of a tree in their backyard.
The height is modeled by the equation
y = 4.5x + 18 , where x represents the
number of months and y represents
the height, in inches.

What does 4.5 refer to in this situation?

A person is keeping track of the height
of a tree in their backyard.
The height is modeled by the equation
y = 4.5x + 18 , where x represents the
number of months and y represents
the height, in inches.

What does the 18 refer to in this situation?