Rational Functions
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
Is the following function growth or decay?
f(x) = 2(1.07)x
Growth
Write the following in logarithmic form:
ex=9
ln(9)=x
Graph the following exponential function, give its domain and range, as well as any asymptotes
f(x) = 4-x-5
Shift down 5, Reflection over y-axis
Domain: (-inf, inf)
Range: (-5,inf)
Asymptote: y=-5
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
Solve the following:
2x= 32
x=5
Evaluate
log4(64)
3
Because 43=64
Graph the following log function, state the domain and range, as well as any asymptotes
f(x) = log3(x-2)+7
Shift right 2, up 7
Domain: (2,inf)
Range: (-inf, inf)
Vertical asymptote: x=2
Simplify:
(6x^2+24x+24)/(3x+6)
2(x+2)
Solve the following:
4 2x-1=27
2x=7
x=ln7/ln2
Expand
log(x2y3/z-2w4)
log(x2y3/z-2w4)=2log(x) + 3log(y) +2log(z) -4log(w)
Graph the following rational function:
f(x) = x/(x2-9)
Vertical asymptotes at x=3,-3
Horizontal asymptote at y=0
Domain: x cannot equal 3,-3
Range: (-inf,inf)
Subtract.
-5/(x2-3x-4) - 1/(4-x)
1/(x+1)
Give the equation of an exponential function that passes through the given points:
(1,6) & (3,54)
f(x) = 2(3)x
Plug in both points and solve for a & b
6 = a(b)1
54 = a(b)3
Write the following in terms of the natural log:
log4(x)
ln(x)/ln(4)
This is the change of base formula
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
Solve.
-4/(2x+5) = 6/(x+1) + 1/(2x+5)(x+1)
x=-35/16
A bacterial culture starts with 50 bacteria and grows at a rate of 7.34% per hour.
a) Write an exponential equation for the number of bacteria present at time t, measured in
hours.
b) Predict how many bacteria will be present after 10 hours?
c) How long will it take for the number of bacteria to double?
a) A(t) = 50(1.0734)t
b) A(10) = 50(1.0734)10=101.53
c) 2=(1.0734)t
t=ln(2)/ln(1.0734)
Solve the following:
log2(x+1) + log2(x) = log2(2)
log2(x2+x) = log2(2)
x2+x=2
x2+x-2=0
(x+2)(x-1)
x=1, (x=-2 is extraneous)
Graph
f(x) = ln(ex)
(hint: this can be done very simply)
y=x (straight line)