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Exponents and Exponential Functions. Chapter 7. Complete the following chart:. Why is ? . Decreasing the exponent by 1 is really dividing by the base (which in this case is 2). What do you think will be?. Remember: . What would the next two be? .
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Exponents and Exponential Functions Chapter 7
Why is ? • Decreasing the exponent by 1 is really dividing by the base (which in this case is 2)
What do you think will be? Remember: What would the next two be? Dividing by a fraction is the same as multiplying by its reciprocal
A closer look at negative exponents: Use the negative sign from the negative exponent to clue you in to make it a fraction!
What if the negative exponent was on the bottom? The negative tells you to move the value to the other side of the fraction bar
Examples 1) 2) 3)
What about parenthesis? The exponent is applied to all numbers and variables in the parenthesis
Multiplying powers with the same base Exponent base Same Base Different Base
Multiplying powers with the same base Add exponents!
Multiplying powers with the same base Add exponents!
examples Remember to be sure the number is in scientific notation!
Raising a Power to a Power Multiply the exponents!
Dividing powers with the same base Subtract exponents!
Function Notation y = 2x + 4 is a linear function Function notation replaces y with f(x) f(x) = 2x + 4 When asked what the function would equal if x was 3, the equation would say: f(3) = 2(3) + 4 = 10 So f(3) = 10Everywhere you saw an x, you replaced it with 3 and solved
Find… • f(12) if f(x) = -3x – 10 • g(-1) if g(x) = 7x + 19 • h(10) if h(x) = x² + 2x - 18
Exponential Functions Cannot equal 1 and must be greater than zero Cannot equal zero
Domain/Range • Domain: The possible x values • Range: The possible y values Domain: x = all real numbers Range: y > 0
The differences of x are constant The product is constant for y Exponential Function!!
Determine if these are exponential functions 1) YES! 2) NO!
Given the domain {-2, -1, 0, 1, 2}, Does the range increase or decrease? Decreases!
Given the domain {-2, -1, 0, 1, 2}, Does the range increase or decrease? INCREASES!
Exponential Growth Growth factor (more than one) Initial amount
Examples: Since 2005, the amount of money spent at restaurants in the US has increased by 7% each year. In 2005, about $360 billion was spent at restaurants. If the trend continues, about how much will be spent at restaurants in 2015? 1 + % for Exponential Growth
Examples: Suppose the population of deer in a region was 3500 in the year 2000. Since then, the population has grown by 3.5% annually. What will the approximate population be in the year 2020?