Rational Expressions
What is the first step you should ALWAYS due when simplifying rational expressions?
See if there is a factor you can take out of everything! (Might find hidden holes)
For rational expressions, explain how to find vertical and horizontal asymptotes.
Vertical - set the denominator = 0
Horizontal - long run behavior (as x approaches +- infinity)
tricks: same leading power - divide coefficients
greater on top - no horizontal asymptote
greater on bottom - y=0
Which of the following is not an exponential function and why?
(1/10)^x
8^x
(-1/10)^x
(-1/10)^x
Cannot have negative base
How to write the following in Logarithmic form?
3^2 = 9
log{3} 9 =2
What does b^(log{b}(x)) equal?
How about log{b}b?
Be able to prove your answer
x
1
Simplify:
(6x^2+24x+24)/(3x+6)
2(x+2)
Find the vertical and horizontal asymptote(s).
y = (x+1)/(2x+3)
Vertical: x=-3/2
Horizontal: y=1/2
What is the equation for the growth/decay model?
What determines whether it is growth or decay?
Explain how one would apply the formula to the following problem:
The # of Salmon in a pond are decaying exponentially at rate r of 10% per year. If there are 50,000 Salmon initially, how many will be there in 5 years?
P(t) = A(e^r)^t
e is base
r is rate:
r>0 - growth
r<0 - decay
t - time elapsed
A - initial value
Plug in corresponding values
Solve the following:
log {121} 11 = y
y=1/2
Expand:
log(x+5)^1/2
1/2log(x+5)
Simplify:
(x^2-1)/(2x^2+7x+5) divided by (x^2+5x+6)/(4x^2-25)
(x-1)(2x-5)/(x+3)(x+2)
Does the following function have a removable discontinuity?
y = (x^2+3x+2)/(x+5)(x+1)
Yes! at x=-1
Given two points
(1,3) and (2,4.5)
Find the exponential function that passes through them
f(x) = 2(1.5)^x
(plug in both points, solve for one variable and plug into other equation)
What is the range of any log function?
How do you find domain?
x-intercept?
Key point?
Vertical asymptote?
Bonus: Is there a horizontal asymptote for a log function? Why?
(-inf, inf)
Argument>0
Plug in 0 for y
Where y=1
No, the graph spans all y-values as x goes to infinity
Condense the following:
2log3 + log5 - log6
What properties did you use?
log45/6
logA + logC = logAC
and
logA - logC = logA/C
Subtract.
-5/(x^2-3x-4) - 1/(4-x)
1/(x+1)
Find the vertical and horizontal asymptotes.
f(x) = (x+5)(x-5)/x(x^2-6x+5)
hint. what should we do before finding the asymptotes?
Vertical: x=0, x=1
Horizontal: y=0
Given the parent function: y = (1/3)^x
Describe the transformations on the following:
y = -(1/3)^x
y= (1/3)^-x
y= (1/3)^(x+1) + 7
AND what are the horizontal asymptotes?
Reflection over x - y=0
Reflection over y - y=0
left 1, up 7 - y=7
Find the domain and vertical asymptote of the following:
log{4}(x-2) +4 = f(x)
(2,inf)
x=2
Expand:
ln(5x sqrt((x+6)/(x-6)))
ln5 + lnx +1/2ln(x+6) - 1/2ln(x-6)
Solve.
-4/(2x+5) = 6/(x+1) + 1/(2x+5)(x+1)
x=-35/16
Sketch the graph.
y= (3x^2-6)/(x+5)(x-4)
Sketch of graph on slide show (screenshot from desmos)
Sketch the graph and label intercepts, domain/range and the horizontal asymptote.
y= -(1/5)^(-x-7)
Graph on slide show # 2 (screenshot from desmos)
Sketch the graph:
5-log{3}(x+4) = f(x)
Expand:
log{3}(7x^2+21x)/(7x(x-1)(x-2))
log{3}(x+3) - log{3}(x-1) - log{3}(x-2)