Download ~* Exponential Function *~

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=5x
• The graph on the right shows an example of
exponential decay with an equation of y=.7x
~* 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-y1 = m(x-x1) 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 6x-3y=12
Answer:
First find the x-intercept
6x-3(0)=12
x=2
Then find the y-intercept
6(0)-3y=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
6x-3y=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 = -1x+5280
Here we now have to take the total distance and subtract
that from the distance already traveled.
5280 – 100ft = 2180ft
Now we see that for every 10ft 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 10ft per 7seconds, 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)=x2, or
the squaring function.
The graph at the left models the function
f(x)=x3, 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))4t
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:
ax2+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 + 2w meters and a
width of 20 + 2w meters. “w” is the width of their walkway which
is 5 meters.
Process
• To solve we draw a diagram then expand
the binomial:
= (50 + 2w)(20 + 2w)
= (50 + 2w) * 20 + (50 + 2w) * 2w
= 1000 + 40w + 100w + 4w2
= 4w2 + 140w + 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
4w2 + 140w + 1000 when (w = 5 )
4(5)2 + 140(5) + 1000
100 + 700 + 1000
1800m2
When x is equal to five y is 1800m2
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-1xn-1 + An-2xn-2 …+ A1x1 +A0
• 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 $5,000.00. 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 + 1450(1.07)^2 + 750(1.07)^1 + 500
2450.086 + 1660.105 + 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:
ln1=0 since e0=1
Logs with bases different then ten: log to the
base specified. Written as:
Log22=y or 2y=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/ m2. 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/m2.
So we plug the value of 1000 watts/m2 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.