IM Alg1.5.20 Lesson: Changes over Equal Intervals
For each given expression, write an equivalent expression with as few terms as possible.
Here is a graph of where .
Here is an expression we can use to find the difference in the values of when the input changes from to . Does this expression have the same value as what you found in the previous questions? Show your reasoning.
How do the values of change whenever increases by 4? Explain or show how you know.
Write an expression that shows the change in the values of when the input value changes from to .
Show or explain how that expression has a value of 8.
Here is a table that shows some input and output values of an exponential function . The equation defines the function. How does change every time increases by 1? Show or explain your reasoning.
Choose two new input values that are consecutive whole numbers and find their output values. Record them in the table. How do the output values change for those two input values?
Complete the table with the output when the input is x and when it is x+1. Look at the change in output values as the increases by 1. Does it still agree with your findings earlier? Show your reasoning.
Pause here for a class discussion. Then, work with your group on the next few questions. Choose two -values where one is 3 more than the other (for example, 1 and 4). How do the output values of change as increases by 3? (Each group member should choose a different pair of numbers and study the outputs.)
Complete this table with the output when the input is x and when it is x+3.
Look at the change in output values as increases by 3. Does it agree with your group's findings in the previous question? Show your reasoning.
For integer inputs, we can think of multiplication as repeated addition and exponentiation as repeated multiplication: We could continue this process with a new operation called tetration. It uses the symbol , and is defined as repeated exponentiation: . Compute and .
If , what is the relationship between and ?