Exponential Graphs
Exponential Equations
Logarithms
Talk the Talk
Modeling
100
Is this graph represent exponential GROWTH or DECAY?
EXPONENTIAL GROWTH
The outputs (y-values) are "growing" or getting larger in the negative direction. The outputs will approach negative infinity.
100
Does this exponential function model GROWTH or DECAY?
y=0.75(5/6)^x+3
DECAY
The common ratio (r) in this equation is less than 1 which indicated decay (the outputs are getting smaller by a factor of 5/6 for every increase of 1 in the input.
100
What is the OUTPUT of a LOGARITHMIC function?
an EXPONENT
Logarithms are the inverse of exponential functions and used when solving exponential equations where the exponent is unknown. The output of a logarithm is an exponent.
100
What do you call of 2 value in the exponential equation?
f(x)=2∙5^x
INITIAL VALUE
When the base of this function is raised to an input of 0, the result is 1. So your are left with the coefficient being multiplied by 1. This means the coefficient is the INITIAL VALUE or what you started with before the function began to grow or decay.
100
A credit card company offers you a $500 limit at an interest rate of ONLY 2% per month. You think this is a ripper deal so you sign up and immediately go out an buy a Supreme sweatshirt for $500. You love your sweatshirt and wear it almost everyday for a year. You feel bad for your roommate who bought a similar sweatshirt at Target for only $15, but his doesn't say Supreme on it...lame. After a year the credit card company starts calling and demanding their money...how much do they claim you owe them after 12 months?
yapprox$634.12
y=500(1.02)^12
200
Write the END BEHAVIOR of this graph.
x->oo f(x)->oo
x->-oo f(x)->3
200
Write the exponential equation for the table below.
200
Estimate the value of log3 (90) WITHOUT a calculator.
xapprox4.1
Since 34=81 and 35=243, we can see x must be between 4 and 5. Since 90 is much closer to 81 than 243, we can estimate the value to be slightly over 4.
200
What are the restrictions on the BASE of an EXPONENTIAL function?
The base can be any positive number except 1.
0<base, base ne1
200
A large pepperoni pizza costs 5.700 Omani rials at Pizza Hut. If pizza has been increasing in price by 3% each year, how much did a pizza cost when Mr. Becker was in Algebra 2? (Way back in 1997.)
Aapprox2.975 __OMR
5.7=A(1.03)^22
300
Write the equation of the exponential graph.
y=4/9(3/2)^x
300
Solve for the exponential equation WITHOUT using logarithms or a calculator.
2^(3x)=64
x=2
Since 26=64 we can rewrite the equation as 23x=26. That means 3x=6, so x=2.
300
Write the logarithm in EXPONENTIAL form.
7^-2=1/49
300
The percentage at which the output of an exponential functions increases or decreases by for each increase of one in the input.
GROWTH RATE
300
The current population in Muscat is 797,000 people. In 2003 the population was 635,000. Assuming exponential growth, at what rate is Muscat's population growing per year?
rapprox0.0143 or Rapprox1.43%
797,000=635,000(r)^16
400
Identify the TRANSFORMATIONS applied to f(x) in order to create g(x). Be specific.
VERTICAL SHIFT (TRANSLATION) down 4
HORIZONTAL SHIFT (TRANSLATION) 4 right
REFLECTED over the y-axis
DILATED by a factor of 3
400
Solve the exponential equation WITHOUT using logarithms or your calculator.
x=9
Since 27x+3=(33)(x+3) we can rewrite the equation as 34x=33x+9. That means 4x=3x+9, so x=9.
400
y=log_3((x-5)/2)-4
The inverse of an exponential function is a logarithm. So rewrite the original function in logarithmic form by switching x and y and then isolating y.
400
What does the 1 represent in the exponential equation?
y=A(1+R)^x
100% of the INITIAL VALUE
This is what you started with before the function began growing or decaying at a particular rate.
400
The cost of a new Ferrari Spider is $272,700. You want to sell the car before the price drops below $200,000. If the car depreciates at a rate of 15% per year, how long before you must sell it?
tapprox1.91 __years
200,000=272700(0.85)^t
500
The blue graph is of f(x)=2x. Find the equation of the green graph g(x) which has been transformed.
g(x)=2^((x-4))-3
500
f(-1)=27 and f(2)=1 are coordinates that can be found using an exponential equation. Find the equation that created these points.
f(x)=9(1/3)^x
Set up a system of equations. Isolate r or A and then use substitution to solve.
500
Solve for x WITHOUT a calculator.
x=-2 and 4
First write the equation in exponential form. Then use Algebra to isolate x.
2^3=(x^2-6x)/(1-x)
500
Explain why the COMMON RATIO of an exponential function can never be negative.
Thinking in terms of (1+R)...if the decay rate (R) is greater than one, it means you lost MORE than what you originally started with...not logical.
Thinking in terms of rx...if you raise a negative base to an even power, the result is positive. If you raise the same base to an odd power, the result is negative. This would create a discrete graph like below...
500
Create a function based on the date below. Use that to determine when will (did) the number of iPhone units sold exceed 50 million?
xapprox5.50
...so almost exactly between 2012-2013
yapprox1.114(1.998)^x
Perform an exponential regression to find the equation. Then find the x-value that produces a y-value of 50. This can be done with the "intersect" tool or using logarithms.