Learning Functions with Sesame Street By Tom Caron

Learning Functions with Sesame Street By: Tom Caron, Stephen Joseph, Marianne Mousigian, and Leslie Savage

~*Exponential Function*~ Exponential Equation: y=abx Where: a does not equal b, b>0, and b does not equal 1. Domain: set of all real numbers Range: Set of all positive real numbers Intercepts: Y intercept is 1 Asymptotes: the x axis

Exponential Growth and Decay • The graph on the left shows an example of exponential growth with an equation of y=5 x • The graph on the right shows an example of exponential decay with an equation of y=. 7 x

~* Real-life Situation *~ • Cookie Monster is baking cookies because he only has cookie left in is cookie jar. He decides to bake 1 extra large cookie. While adding the eggs he accidentally tipped over the yeast into the bowl. He didn’t realize this until he took the cookie out of the oven and saw how big the cookie was. If the cookie goes into the oven with a diameter of 6 inches by 3 inches every 20 minutes how large will the cookie be when Cookie takes it out of the oven in an hour and a half?

Process • For this situation we have come up with this equation to graph. • 6 x 3 ^ 6. 5

This graph models the real life situation.

Solution

~* Linear Function *~ Linear Equation in Slope Intercept Form y= mx+b Where: m= slope b= y-intercept (or initial condition) Domain: set of real numbers Range: set of real numbers Constant increase situation: positive slope (m) Constant decrease situation: negative slope (-m)

Extra Information *Graphs with same slope (m) in y= mx+b form are parallel lines. * If 2 non-vertical lines are parallel, then they have the same slope. Standard form: Ax+By= C Point-slope form: y-y 1 = m(x-x 1) where m is the slope * The graph of Ax+By= C where A and B are not 0, is a line.

Horizontal and Vertical Lines • The graph to the direct left models the equation X=3. It is an example of a vertical line. The slope is undefined. • The graph to the direct left models the equation y=3. It is an example of a vertical line. The slope = 0.

Process for Oblique lines • Question: Graph the equation 6 x-3 y=12 Answer: First find the x-intercept 6 x-3(0)=12 x=2 Then find the y-intercept 6(0)-3 y=12 y=-4 On the next page you will find the points (2, 0) and (0, -4) plotted on a graph. A line has been drawn that contains these 2 points.

• Oblique lines: (Ex: the line y = x) When in Ax+ By= C form, the slope equals (– A/B when A and B are not equal to 0) The graph to the right models the equation 6 x-3 y=12. The line passes through the origin an xintercept of 2 and a yintercept of – 4.

~* Real-life Situation *~ Zoe’s house is 5, 280 feet away from Elmo’s house. Elmo is walking Zoe home from his house. They are currently 100 feet from Elmo’s house and are moving at a rate of 10 feet per 7 seconds. If they continue to walk at this rate, how much longer in minutes will it take them to reach Zoe’s house?

This graph models the real life situation. As the graph shows this equation has a negative slope because the further they walk the closer they get to their destination.

Process • With this we plot the intercepts (5280, 0) and (0, 5280) and connect the line. We then use linear regression in to find the equation which is: y = -1 x+5280 Here we now have to take the total distance and subtract that from the distance already traveled. 5280 – 100 ft = 2180 ft Now we see that for every 10 ft traveled it takes 7 seconds. So we divide the distance left by the rate which is 10. 5180/10 = 518

Process (continued) Now multiply by the time it takes to travel this which is 7 seconds per ten feet. 518 (7) = 3626 seconds Now it asks for this time in minutes so you have to convert this value into minutes by dividing it by 60. 3626/60 = 60. 433 60. 4 minutes more minutes until they reach Zoe’s house from Elmo’s.

Solution At the constant speed of 10 ft per 7 seconds, it will take 60. 4 more minutes until they reach Zoe’s house from Elmo’s.

~*Power Function*~ Equation: f(x)=xn where n is a positive integer Domain: set of all real numbers Range: when n is positive = all positive real numbers when n is negative = set of all real numbers Asymtotes:

Extra Information Symmetry: When n is even: Reflectional symmetry to the y-axis when n is odd: Rotational Symmetry of 180 degrees around the origin Postulates: Product of powers – x^m times x^n = x^(m+t) Power of Power – (x^m)^n = x^(mn) Power of Product – (xy)^n = x^n times y^n Quotient of powers – For x not equaling 0 (x^m)/(x^n) = x^(m-n)

Simple Power Functions • Graph to the left represents the equation f(x)=x, or the identity function. • The Graph to the left represents the equation f(x)=x 2, or the squaring function.

The graph at the left models the function f(x)=x 3, or the cubing function.

~*Real Life Situation*~ Big Bird is opening his first savings account. If he puts his life savings of 14 cents in the bank, at a rate of 9% compounded quarterly, how long will it take him to accumulate a whole dollar?

Process First we create the power function expression. Which is modeled after: A = P(1+(r/n)nt When P is the principal amount deposited, r is the rate, n is the # of times interest is calculated in one year, and t is the years The equation is : 0. 14(1+(. 09/4))4 t

This graph models the real life situation.

Solution Using the data table to find the value of x when y was equal to or greater than 1. 00 When y is 1. 03 , x is 6. 75 so it took six years and nine months to accumulate one dollar. Big Bird thanks you with his whole heart!!

~*Quadratic Function*~ Equation: ax 2+bx+c=0 : standard from of a quadratic Domain: Set of all real numbers Range: Determined by examining the graph of the function

~*Real Life Example*~ Bert and Ernie are in a jam. They need to know the total area of their pool and surrounding walkway. Rubber ducky is refusing to return Bert’s paper clips and bathe with Ernie until he is told the answer. Their pool is a rectangle with a length of 50 + 2 w meters and a width of 20 + 2 w meters. “w” is the width of their walkway which is 5 meters.

Process • To solve we draw a diagram then expand the binomial: = (50 + 2 w)(20 + 2 w) = (50 + 2 w) * 20 + (50 + 2 w) * 2 w = 1000 + 40 w + 100 w + 4 w 2 = 4 w 2 + 140 w + 1000 • Now we use the graph to find the y value when x is equal to 5.

The graph shows the curve like shape that is a characteristic of all Quadratic Expressions.

Solution 4 w 2 + 140 w + 1000 when (w = 5 ) 4(5)2 + 140(5) + 1000 100 + 700 + 1000 1800 m 2 When x is equal to five y is 1800 m 2

Solution • Rubber ducky was so happy that we solved this that he returned Bert’s paper clips and bathed with Ernie instantly. Thank you so much!

~*Polynomial Equations*~ • Expression: Anxn + An-1 xn-1 + An-2 xn-2 …+ A 1 x 1 +A 0 • Domain: all real numbers • Range: all real numbers • Intercepts: N/A • Asymptote: none

Extra Information • Leading Coefficient is An • Degree is the largest exponent of x which is n • The number n the degree of the polynomial • This is sometimes called the nth degree polynomial

~*Real Life Situation*~ Grover has a problem he just can’t figure out. Can you help him? … Great! His cousin , Clover, back in Bun Town has decided to attend college. She is planning to attend the fall after her senior year but it costs $5, 000. So she is saving during the next four years of high school. She plans to save $2, 000. 00 her Freshmen year, $1, 450. 00 her Sophomore year, only $750. 00 her Junior year, and $500. 00 her Senior year. At the end of each year she will put the money in a savings account with an annual yield of 7%. Will Clover have enough money to attend college?

Process Since Clover will deposit her Freshmen savings at the end of the year it will collect interest for three years. Her sophomore savings will collect for two years, her junior savings will save for only one year, and her senior savings will not collect interest at all. With this we write the following equation. 2000(1. 07)^3 + 1450(1. 07)^2 + 750(1. 07)^1 + 500

The graph above models the real life situation.

Process (continued) 2000(1. 07)^3 + 1450(1. 07)^2 + 750(1. 07)^1 + 500 2450. 086 + 1660. 105 + 802. 5 +500 4110. 191 + 1302. 5 5412. 691 ~ $5412. 70

Answer The polynomial solves out to be approximately $5412. 70. This is enough for Clover to attend college. Grover is ecstatic to tell his cousin that she will be able to attend and wants to thank you for your hard work.

• Here is Grover and Clover and her cat Cinnamon (left) and Grover saying “thank you”(bottom).

~* Logarithmic Functions *~ Common Logs: log to the base of ten written as logbx=y which equals by=x iff. B>0 and not 1. Natural Logs: logs with bases of e Written as ln for short: ln 1=0 since e 0=1 Logs with bases different then ten: log to the base specified. Written as: Log 22=y or 2 y=2

~* Logarithmic Functions *~ Properties of logbx=y • Domain: all positive real • Range: all real • X-intercept is always one • Asymptote: y-axis Properties of common Log: • Domain: all positive real • Range: real numbers • Asymptote: x-axis and y-axis • x and y intercepts are always one Properties of Natural Log: • Domain: all positive real • Range: all real • Asymptotes: y-axis

Real Life Situation • Grouch is mad because it is too loud for him. He closes his garbage lid but can still hear everything. He petitioned to make a noise ordinance but needs substantial evidence that it is to loud for him. He has measured the noise to be a whooping 1000 watts/ m 2. He is recruiting us to help him on his mission that cannot fail. The comfort zone of hearing is between 20 decibels and 25 decibels. Can Grouch get the ordinance passed on Sesame Street?

Process • We recall the formula: D=10(log N + 12) which converts the number of decibels D from the sound intensity N measured in watts/m 2. So we plug the value of 1000 watts/m 2 into the equation as N. And solve

Process (continued) D=10(log N + 12) D=10(log 1000 + 12) D=10(log 1012) D=10(3. 00518) D=30. 0518 So there is noise up to 30 decibels on Sesame Street.

Solution D=30. 0518 This over the Grouches comfort zone of 2025 decibels. Thanks to you the Grouch got his way and in turn became less grouchy, well for a day anyways. And surprisingly he thanks you. Good work!

How to Tell Them Apart • Linear Functions are a straight line when graphed. • Exponential Functions, either growth of decay, have a curve to them when graphed where there is no x-intercept and the y-intercept is one. • Polynomial Functions have the longest formula of all and has the minimum of two intercepts. • Quadratic Functions have the unmistacabel bell shape to them. They have a maximum of two xintercepts. • Logarithmic Functions have no y-intercept and the x-intercept is always one.