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In math, people often invent ways to extend concepts to areas that might not make sense at first. Pretty much everyone can understand what means, because they understand that it represents multiplying by the number 2 a total of 3 times. Yet, what does or mean? Does it make sense to talk about multiplying by a number a negative amount of times? Lets explore.
Exercise #1: We can think of powers of 2 as representing multiplication of the number 1 repeatedly. (a) Fill in the pattern for powers that are not negative. (b) If positive exponents indicated multiplying What does this lead you to fill in for the number 1 by 2 repeatedly, then negative exponents should indicate ______________________.
We want the pattern of positive, integer powers to extend to zero exponents and negative, integer exponents. We can now define zero and negative exponents as follows.
Zero & Negative Exponents
1.) Zero Exponents:
as long as
2.) Negative Exponents:
Exercise #2: Which of the following is not equivalent to?
[1] [3]
[2] [4]
Exercise #3: If , then which of the following is the value of Show the work that leads to your answer. Remember exponents always come before multiplication.
[1] [3]
[2] [4]
Exercise #4: Using all your known exponent rules, simplify each of the following completely.
(a) (b) (c) (d)
So, we now see that the subtraction rule for exponents is consistent with negative and zero exponents. For now, we just want to be comfortable that negative exponents indicate division and positive exponents indicate multiplication. Exercise #5: Consider the exponential function. Find each of the following without the use of your calculator. (a) (b) (c)
7.) The number of people, n, ho know a rumor can be modeled using the equation , where is the number of days since Monday. (a) Explain why. What does this (b) What is the value of? What does this represent in terms of the situation modeled? represent in terms of the situation modeled?
_____ 8.) Consider the function. When the value of x is increased by , the output is: [1] multiplied by 3 [3] multiplied by [2] divided by 3 [4] divided by
Review Section :
_____ 9.)
_____ 10.)
Homework Answers
1.) a) b) 1 c)
d) 4 e)1 f)
6.) a) b) c)
7.) a) represents the fact that 20 people knew the rumor on Monday. b) represents the fact that 5 people knew the rumor two days before Monday (Saturday).
8.) 2
9.) 1
10.) 3
Exercise #2: Consider the function
(a) Evaluate. What point does this indicate (b) Without the use of your calculator, on the graph of g? determine the values of and
(c) Using your graphing calculator, sketch a graph (d) Why is this exponential function always of this function using the WINDOW decreasing while the one in Exercise # and. Mark the y-intercept. is always increasing?
Exercise #3: For each of the following exponential functions, give its y-intercept and tell whether it is increase or decreasing.
(a) (b) (c)
The equations of exponential functions are relatively easy to determine, if you understand this lesson so far. See what you can do in the next exercise.
Exercise #4: Find the equation of the exponential function, in form, for the function given in the table below. Show or explain your thinking.
4.) Graphing a basic exponential can be challenging because of how quickly they grow (or decay). In this exercise, we will graph one of the most basic.
(a) Evaluate each of the following and state the coordinate point that occurs on the graph of based on the calculation.
(b) Evaluate each of the following. Remember your facts about negative exponents and give the point on the graph of.
(c) Using the points you found in (a) and (b), graph this function for the domain interval.
5.) Classify each of the following exponential functions as either increasing or decreasing and give the value of their y-intercepts.
(a) (b) (c)
_____ 6.) Which of the following could be the equation of the exponential function shown graphed below?
[1] [3]
[2] [4]
Review Section :
_____ 7.)
