Rules of exponents Rule 1 a0= 1 Rule 2 𝑎 −𝑛 = 1 𝑎 𝑛 2^3 8 2^2 4 2^1 2 2^0 1 2^-1 ½ 2^-2 1/4 Rule 1 a0= 1 Rule 2 𝑎 −𝑛 = 1 𝑎 𝑛 Example 2 −3 = 1 2 3

Practice with graphing f(x) =1(2)^x -3 1(2)-3 1/8 -2 1(2)-2 ¼ -1 1(2)-1 ½ 1(2)0 1 1(2)1 2 1(2)2 4 3 1(2)3 8

𝑓 𝑥 =80 1 2 𝑥 x f(x) -1 1 2 3 4 5 6 7

Chapter 5.1 and 5.2 Exponential functions I can distinguish between situations that can be modeled with linear functions and with exponential functions. I can show that exponential growth eventually exceeds linear growth. I can graph exponential functions and determine end behavior and other characteristics. I can write exponential functions from a table, graph, or context.

Compare the graph of an exponential increasing function to a linear increasing function.

Linear or exponential what do you think? Amount a person gets paid hourly Growth of the internet over time Value of a car over time The amount of people infected by the flu bug over time The amount of a drug in your body over time Price for renting a car over the number of miles. Distance traveled in a SUV over time. Interest on a loan over time

Utah’s population since 1950.

f(x) = mx + b m = slope b = y intercept f(x) = a(b)x a =y intercept b is the common multiplier or the base.

With exponential functions the slope is different depending on the interval you are interested in. What is the slope of this curve over the following intervals. 0≤𝑥 ≤1 1≤𝑥≤2 2≤𝑥≤3 0≤𝑥≤3

What do you notice about the 2 functions from their graphs?

What do you notice about these two functions from their graphs? One increases more quickly than the other.

What is it about this function that makes it decrease? The multiplier is less than 1

The population of a town is 100 people in the year 2000 and increases by 8 percent each year. Which graph represents the increase in population as given in the scenario above? The Blue graph it increases 8 percent over the previous year. The black graph add the same amount each year.

A man invests 50 dollars in a stock that earns 10 percent interest compounded yearly. Which function represents the scenario described? The blue function it increases by 10 percent over the previous year. What is the difference between the 2 functions? One increases at the same rate, the other increases exponentially.

A man takes 20 milligrams of ibuprofen A man takes 20 milligrams of ibuprofen. The amount of ibuprofen in the blood stream decreases by 20 percent each hour. Explain the following graph and function to your neighbor.

A man buys a 100 dollar toy. Each year the value of the toy decreases 10 percent? After how many years will the toy be worth 60 dollars?

Secret of getting rich is compound interest Secret of getting rich is compound interest. I am making interest on my interest. t in years 100 (1.10)t P(t) 100 1 100(1.10) 110 2 100(1.10)(1.10) 121 My extra dollar is interest on the interest 3 100(1.10)(1.10)(1.10) 133.1

Simple Interest Can you write an explicit equation for the problem? Can you write a linear function for the problem?

Compound interest Suppose that Raul deposits $1000 into an account that earns 5% compound interest each year. Complete the table to show Raul’s account balance after each year. Can you write a geometric sequence for the given table? Can you write an exponential function for the given table?

Graphs for Nico and Raul Explain the differences between Nico’s graph(black) and Raul’s graph (blue).

Where do we find exponential functions? Increasing Percent increase Decreasing Percent decrease Population growth Humans, animals, bacteria Spread of disease Technology People with cell phones People with internet Recipients of spam email Viral you tube Computer virus Deflation Cars, music equipment, machinery Amount of drug in bloodstream Population

Population increase and population decrease A town has 1000 people. Each year the population increases by 5 percent. Write a function for this situation. f(x)=a(b)x f(x) = 1000(1 +.05)x A town has 1000 people each year the population decreases by 5 percent. Write a function for this situation f(x)= a(b)x f(x) = 1000(1-.05)x

At this moment, the population of Downtown is 20,000, and the population of Uptown is 6000. But over many years, people have been moving away from Downtown at a rate of 1.5% every year. At the same time, Uptown’s population has been growing at a rate of 1.8% each year. Time Population downtown Population Uptown YEars people 20,000(1-.015)t 6000(1+.018)t 20000 6000 1 20000(.985) 6000(1.018) 2 20000(.985)(.985) 3 20000(.985)(.985)(.985

At this moment, the population of Downtown is 20,000, and the population of Uptown is 6000. But over many years, people have been moving away from Downtown at a rate of 1.5% every year. At the same time, Uptown’s population has been growing at a rate of 1.8% each year. Graph the equation and Answer problems 1-4 page 308 and 309.

Analyzing a graph of an exponential function. Asymptotes and intercepts. Y=4(2)x An asymptote is a line that the graph of function keeps getting closer to but never touches. The y asymptote is ____ The y intercept is ____ The x intercept is ____

Analyzing a graph of an exponential function. Asymptotes and intercepts. Y=4(2)x + 2 An asymptote is a line that the graph of function keeps getting closer to but never touches. The y asymptote is ___ The y intercept is ____ The x intercept is ____

Analyzing a graph of an exponential function. Asymptotes and intercepts. Y=4(2)x - 2 An asymptote is a line that the graph of function keeps getting closer to but never touches. Asymptote is _________ Y intercept is _______ X intercept is ________

Analyzing a graph of an exponential function. Asymptotes and intercepts. A man stuffs 500 dollars in his mattress. He also puts 500 dollars in the bank with 5% interest compounded yearly. Could you write an equation to show how this man’s money grows over time. f(x) = 500(1.05)x + 500

Analyzing a graph of a function with domain and range Analyzing a graph of a function with domain and range. What can the inputs and outputs be? Domain in orange is all the inputs of a graph. It can be written −∞ ≤𝑥 ≤ ∞ Range is all of the outputs on a graph. It can be written 𝑦>0

Analyzing a graph of a function with domain and range Analyzing a graph of a function with domain and range. What can the inputs and outputs be? Write down the domain and range for your neighbor.

Word problems sometimes have added limitations to the domain and range. A man has a 100 savings bond that earns 5% interest compounded yearly. The Bond expires after 20 years. Think of what X could be. Write an equation and think of what y could be.

5.3 I can determine an average rate of change over an interval on an exponential function. I can translate exponential and linear graphs, both vertically and horizontally. I can show a translation through function notation, and through coordinate notation, and through graphing.

I can determine the average rate of change of an exponential equation over a given interval. A town has 100 people. Each year the population increases by 5 percent. f(x) = 100(1 +.05)x Years x Dollars y 100 2 110.25 12 179.59 14 197,99 Find the rate of change over the given intervals 0≤𝑥≤2 ______ 12≤𝑥≤14 ______ 0≤𝑥≤14 _______

Determine the average rate of change of the exponential function over the following intervals. 𝑓 𝑥 =100 (.5) 𝑥 0≤𝑥≤1 1≤𝑥≤2 2≤𝑥≤3 0≤𝑥≤3 x F(x) 100 1 50 2 25 3 12.5

Introduction to vertical translation x f(x) g(x) h(x) 1 2 I get 2 dollars an hour f(x) =2x I get 2 dollars an hour plus a 4 dollar bonus g(x) = 2x+4 or g(x)= f(x)+4 I get 2 dollars an hour and my wages are garnished I lose 6 dollars of my paycheck. h(x) = 2x -6 or h(x)= f(x)-6

Vertical translations What happens to the graph and the table when I add 2 to the function f(x) f(x) = 2x g(x) = f(x) + 2 or g(x) = 2x +2 x f(x) g(x) -1 -2 2 1 4 6

Vertical translations. Fill in the blanks for g(x) in the table Vertical translations. Fill in the blanks for g(x) in the table. Explain how you got them f(x) = 1(2)x g(x) = f(x) + 2 or g(x)=1(2)x + 2 x f(x) g(x) 1 2 4

Horizontal translations with linear functions Horizontal translations with linear functions. Is this translation vertical or horizontal? f(x) = 2x g(x) = 2x+4 or g(x) = 2(x+2) x f(x) g(x) -1 -2 2 4 1 6

Horizontal translations f(x) = 2x g(x) = 2 (x+1) x f(x) g(x) -1 ½ 1 2 4 Notice the output for the two functions is the same, except g(x) comes one place before f(x).

Paddy paper practice.. 1. Graph the original function on grid paper. 2. Trace the graph of the function to paddy paper. 3. Look at the new function and move the paddy paper so the graph on the paddy paper shows the new function.

Coordinate notation is a way of showing the transformation from one function to another 𝑓 𝑥 =2𝑥 𝑡𝑜 𝑔 𝑥 =2𝑥+2 𝑓 𝑥 =2𝑥 𝑡𝑜 𝑔 𝑥 =2(𝑥+3) 𝑓 𝑥 =2 (3) 𝑥 to g x =2 (3) 𝑥+4 𝑓 𝑥 =2 (3) 𝑥 to 𝑔 𝑥 = 2(3) 𝑥 +1 𝑥,𝑦 → 𝑥,𝑦+2 (𝑥,𝑦)→(𝑥−3,𝑦)

Practice with translations and equations. If f(x)=1(2)x Then what is the equation for the translated graph?

Practice with translations and equations. If f(x)=1(2)x Then what is the equation for the translated graph?

Practice with translations and equations. If f(x)=1(2)x Then what is the equation for the translated graph?

I can write an equation given a graph. Y=a(b)x What is the y intercept? Put it in place of a. What am I multiplying by each time? Put it in place of b. F(x)=20(.5)x

Practice writing exponential equations from a graph with your partner.

Practice writing exponential equations from a graph with your partner.

How to write an equation given a table X F(x) -1 20 10 1 5 2 2.5 1. Determine if you are multiplying by the same amount through equal intervals. 2 Find the y intercept. 3. Plug an x and y coordinate into your equation to determine the multiplier. 4. f(x) = a (b)x Fill in for a and b I am seeing a pattern of halves. The y intercept is 10 5 = 10(b)1 5 10 = 10𝑏 1 10 1 2 = b 𝑓 𝑥 =10 ( 1 2 ) 𝑥

How to write an equation given a table X F(x) -1 ¼ ½ 1 2 1. Determine if you are multiplying by the same amount through equal intervals. 2 Find the y intercept. 3. Plug an x and y coordinate into your equation to determine the multiplier. 4. f(x) = a (b)x Fill in for a and b

How to write an equation given a table X F(x) -1 125 100 1 80 2 64 1. Determine if you are multiplying by the same amount through equal intervals. 2 Find the y intercept. 3. Plug an x and y coordinate into your equation to determine the multiplier. 4. f(x) = a (b)x Fill in for a and b

Write an exponential equation given two ordered pairs Write an exponential equation given two ordered pairs. This is pretty much the same as with a table. You can only do this if you know that there is an exponential relationship between the two ordered pairs (0,3),(2,6.75) Find the y intercept. Use the other point to fill in for the equation y=a(b)x Solve for b. Put a and b into the function.

I can solve exponential equations by making a common base. 4x+3 = 23x-6 Use your understanding of exponents to make a common base Set the exponents equal to each other Solve

I can solve exponential equations by making a common base. Use your understanding of exponents to make a common base Set the exponents equal to each other Solve

I can solve exponential equations by making a common base. 3 2𝑥+5 =1 Use your understanding of exponents to make a common base Set the exponents equal to each other Solve

I can solve exponential equations by making a common base. 1 9 2𝑥 = 3 𝑥+2 Use your understanding of exponents to make a common base Set the exponents equal to each other Solve