Math 3C Midterm 2

Rational Functions

100

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

200

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

300

Simplify:

(6x^2+24x+24)/(3x+6)

2(x+2)

400

Subtract. 

-5/(x²-3x-4)  -   1/(4-x)

1/(x+1)

500

Solve. 

-4/(2x+5) = 6/(x+1) + 1/(2x+5)(x+1)

x=-35/16

Exponential

100

Is the following function growth or decay?

f(x) = 2(1.07)^x

Growth

200

Solve the following:

2^x= 32

x=5

300

Solve the following:

4 2^x-1=27

2^x=7

x=ln7/ln2

400

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

500

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)

Log

100

Write the following in logarithmic form:

e^x=9

ln(9)=x

200

Evaluate 

log₄(64)

3

Because 4³=64

300

Expand 

log(x²y³/z^-2w⁴)

log(x²y³/z^-2w⁴)=2log(x) + 3log(y) +2log(z) -4log(w)

400

Write the following in terms of the natural log:

log₄(x)

ln(x)/ln(4)

This is the change of base formula

500

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)

Graphing

100

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

200

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

300

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)

400

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

500

Graph

f(x) = ln(e^x)

(hint: this can be done very simply)

y=x (straight line)