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MATH 90

Final Exam

NAME___________________________

May 8, 2020

Show all work on this paper, if possible. You may use extra paper, but then be sure to clearly label the

problems. Turn in all of your work. Scientific calculator is required. Test is 100 pts.

1.

Sketch a graph of each function. Indicate all asymptotes and intercepts. (6 pts ea)

b) f (x) = log2 (x – 1 )

a) f (x) = 2 - ex

Asymptote (x=1)

Asymptote (y=2)

Y Intercepts =

(0,1)

X Intercepts =

(ln(2),0)

Exponential functions have a horizontal

asymptote. The equation of the horizontal

asymptote is y=2

X

Y

-2

1.86

-1

1.63

0

1

1

-0.71

2

-5.38

To find the intersection with the X-axis, 0 is

replaced by Y is solved for X. While to find the

intersection with the Y-axis, 0 is replaced by X

and resolved for Y.

Intersection(s) with x-axis: (𝐥𝐧(𝟐), 𝟎)

Intersection(s) with axis and: (𝟎, 𝟏)

X Intercepts =

(2,0)

To make the graphing, we must use the vertical

asymptote and a few selected points. An asymptote is

a line that approaches a curve, but never touches it.

The vertical asymptote is x=1, while the horizontal

asymptote does not possess

x

Y

2

0

3

1

5

2

To find the intersection with the X-axis, 0 is replaced by

Y, and resolved for X. To find the intersection with the

Y-axis, 0 is replaced by X and resolved for Y.

Intersection(s) with x-axis: (2,0)

Intersection(s) with axis and: None

2.

Use properties of logarithms to expand as much as possible. (5 pts)

log (

𝑥3

)

𝑦𝑧 2

log (

𝑥3

) = log 𝑥 3 − log(𝑦𝑧 2 )

𝑦𝑧 2

= 3 log 𝑥 − [log 𝑦 + log 𝑧 2 ]

= 3 log 𝑥 − [log 𝑦 + 2 log 𝑧]

𝒙𝟑

𝐥𝐨𝐠 (𝒚𝒛𝟐 ) = 𝟑 𝐥𝐨𝐠 𝒙 − 𝐥𝐨𝐠 𝒚 − 𝟐 𝐥𝐨𝐠 𝒛

3.

Use properties of logarithms to condense into a single logarithm. (5 pts)

1...