EQ: How do I identify exponential growth? Bellwork:

Transcription

1 EQ: How do I identify exponential growth? Bellwork: 1. Bethany's grandmother has been sending her money for her birthday every year since she turned 1. When she was one, her grandmother sent her $5. Every year she sends Bethany twice as much money as she did the previous year. How much money will Bethany receive for her 5th birthday? Explain how you came up with your answer.

2 Investigation You have been assigned the task of making ballots for the upcoming student elections. Because the school is trying to save paper, you want to see how many ballots can be cut out of a single piece of paper. Using the scissors and paper provided make a table that displays many ballots you can make with and number of cuts (n). Start by recording the number of ballots created for 1, 2, 3, 4, and 5 cuts. Then try to create a rule that shows how many ballots will be created with n cuts. To simplify the process, each cut should cut the piece(s) of paper exactly in half. How many ballots could you make if you could make 20 cuts? 30? How many tomes would you have to cut a piece of paper in order to create enough ballots for all 800 students at our school? You have 15 minutes.

3 So what did your data look like? Is the relationship between cuts and total ballots linear? How do you know? Number of Ballots Cuts n Ballots ??? If the relationship isn't linear, what is it? Add row or column to your table that shows how you got from the number of cuts to the number of ballots. That is to say, what mathematical operation did you perform? Cuts n Ballots ??? My Thinking 1x2 1x2x2 1x2x2x2 1x2x2x2x2 1x2x2x2x2 x2 1 n number of 2's

4 The relationship between the number of cuts and the total number of ballots made is known as an exponential relationship. In an exponential relationship, a fixed change in the independent variable results in the dependent variable being multiplied by a fixed amount. In our ballot example, every time the cuts (I.V.) increased by 1, the number of ballots (D.V.) was multiplied by 2. Exponential relationships are easy to recognize in tabular form. In an exponential relationship, there is no constant rate of change, and the dependent variable starts out by growing or shrinking at a slow rate, but then increases rapidly. x y The table above shows the equation 3 x. Notice how the relationship starts out growing slowly, but then shoots up quickly.

5 Exponential relationships are pretty easy to identify in equations as well, but there are 3 different ways to write exponential equations: Expanded Form: When we write a number as a product of multiple factors. 6 x 6 x 6 x 6 x 6 x 6 x 6 Exponential Form: When we write a number using a base raised to an exponent or "power". 6 7 The number above is read as "six to the seventh power." the number six is the base and the number 7 is the exponent or "power." Standard Form: Standard form is what we get when we evaluate an expression in expanded or exponential form. 6 7 = 279,936 so 279,936 would be considered standard form. Work with your partner and think of one situation in which we might wnat to use each of the 3 different forms.

6 EQ: How do I operate with exponents? Bellwork: Write each number in exponential notation: 1. 5 x 5 x 5 x 5 x 5 x 5 x x 8 x 8 x 8 x x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 Write each number in expanded form: Write each number in standard form:

7 What do exponential relationships look like in graphic form? Sketch a graph of your data from the ballot investigation. What does the graph look like? Total Ballots Produced Ballot Making Investigation Number of Cuts

8 But that graph really only gives us a small picture of an exponential relationship. Lets go back to our table of data: Cuts n Ballots ??? My Thinking 1x2 1x2x2 1x2x2x2 1x2x2x2x2 1x2x2x2x2 x2 1 n number of 2's Was anyone able to write a rule that defines the relationship? B = 2 n Total ballots is equal to 2 to the power of n, or 2 to the nth power. If we were to write this using our standard variables x and y, it would be: y = 2 x Using a graphing calculator, sketch a graph of y = 2 x Zoom out to see what the graph looks like for many values. Try graphing y = 3 x or y = 4 x

9 Operating with Exponents Now that we know what exponents look like, its important for us to know what to do when we see them in equations and expressions. When we operate with exponents, we perform the four basic mathematical operations (+, -,, ). Consider the following expression: Find the value of the expression in standard notation. 8 4 = 32 That wasn't so hard was it? But how would we express the product in exponential notation? First let's look at how we would write the expression in expanded form. Write each of the factors from the expression above in expanded form. (2 2 2) (2 2) Do we really need the parentheses in the expression above? What happens if you remove them? Now write the expression again in exponential form: 2 5

10 So we've decided that: = 2 5 What relationship do you notice between the exponents? Rule #1 a x a y = a (x+y) Practice: Simplify the following expressions and leave your answer in exponential form

11 So what about division? consider the following expression: Again, we can start by writing the expression in expanded form: ( ) (4 4 4) What is another way of writing a division problem? That's right! We can write it as a fraction. ( ) (4 4 4) Then we can use what we know about the multiplicative identity. Remember, the multiplicative identity is the value which we can multiply by any other value without changing its identity. In this case (and most others) the multiplicative identity is 1. So in our fraction above, we can cancel 3 pairs of 4's because 4/4 is 1. This leaves us with: = 4 4 So, = 4 4 What is the relationship between the exponents in the equation above?

12 Rule #2 a x a y = a (x-y) Practice:

13 So now your asking, we've done multiplication and division, what could possibly be left? Well, there is addition and subtraction, but unfortunately there's not much we can do to add or subtract exponential relationships. But what if we took one exponential relationship and took it to a higher power? Consider the following expression: (3 3 ) 4 Just like before, we can write the expression in expanded form. You try it this time. See of you can write a rule that tells us what to do when we raise one exponent to another power. (3 3 3) (3 3 3) (3 3 3) (3 3 3) = 3 12 (3 3 ) 4 = 3 12

14 Rule # 3 (a x ) y = a (x y) Practice: 1. (5 2 ) 5 2. (3 6 ) 2 3. (2 7 ) 8 4. (1 4 ) 6 5. ( ) 3

15 There are also a few special cases. Consider the following expression: 5 0 This one's a little harder to figure out. I find it easiest to do this one in table form. 5 3 = 5 x 5 x 5 x = 5 x 5 x = 5 x = 1 Rule #4 a 0 = 1 Practice: I'm not going to insult your intelligence by making you practice this one.

16 Another special case: What happens when I have a negative exponent? Consider the following expression: 5-3 Lets go back to our table from the last rule and continue it. 5 3 = 5 x 5 x 5 x = 5 x 5 x = 5 x = = = = If we rewrite the division as fractions, we get: 5-1 = = = 1 5 3

17 Rule #5 a -x = 1 a x Practice

18 *****CAUTION***** Negative bases can be tricky. It's important to follow the order of operations to the letter. Consider the following expression: -3 2 It's tempting to evaluate the expression like this: -3 2 = (-3) (-3) = 9 But the order of operations says differently. PEMDAS tells us to do exponents before we affix negative signs. Why? Well, a negative sign is like multiplying a value by -1. Multiplication comes after exponents in the order of operations. So = (-1) (3) (3) = -9 However, if we want to include the negative sign with the number 3, all we have to do is use some cleverly placed parentheses. (-3) 2 = 9

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