LAGUARDIA COMMUNITY COLLEGE
MEC DEPARTMENT
MAT 117 FINAL REVIEW
1.
Solve for x.
(a)
2.
x
- 3(x – 1) = 5x – 7
2
(b) 5x -
2
(x + 5) = 7x –(x-1)
3
Solve for a.
(a) 3a – (a – 1) ≤ 4a + 15
(b) 4 – 2(3-a) > 5a - 2
3.
Solve for x.
4.
3
2x
−1 =
5
3
Solve for x and express results using interval notation.
5.
(a) 2 x − 5 = 3
(b)
(a) 2 x − 3 - 5 ≥ 6
(b) x − 5 - 3 < 6
Find the equation of the line satisfying the specified conditions.
(a)
(b)
(c)
(d)
(e)
with slope of –3 and passing through (-2,4)
passing through (-3,4) and (-5,6)
slope of 0 and passing through (-2,5)
x-intercept of 3 and y-intercept of 2
passing through (-3,5) with an undefined slope.
6.
Find the equation of the line through the point (-1,4) and parallel to the yaxis.
7.
Find the equation of the line through the point (-2, 6) and parallel to the xaxis.
8.
Let the quadratic function f be defined as f(x) = 4x - x 2 . Compute the
following functional values.
(a) f(-2)
(b) f(0)
(c) f(0.001)
(d) For what values of x is f(x) = 4?
9.
Compute the difference quotient
f ( x + h) − f ( x )
for each function below.
h
(b) f(x) = 3x 2 - x - 6
(a) f(x) = 5 –2x
1
10.
Compute the quotient
f (b) − f (a )
for the following function
b−a
(a) f(x) = -7x – 3 ; a =-2, and b=3
11.
(b) f(x) = x 2 -2x +1 ; a=-1, b =0
Solve the following systems of equations.
(a) 3x – 2y = 4
2x + 3y = 7
(b) 3x + 2y = 12
y = 5x -1
12.
A person invested $20,000 into two accounts that pays 2.5% and 3%
simple interest annually, respectively. Find the amount invested into the
two accounts if the total interest earned after one year is $540
13.
A baker purchased 12 lb of wheat flour and 15 lb of rye flour for a total
cost of $18.30. A second purchase same prices, included 15 lb of wheat
flour and 10 lb of rye flour. The cost of the second purchase was $16.75.
Find the cost per pound of the wheat flour and the rye flour.
14.
Simplify the following expression:
(a) 2(3x 2 - x – 1) – (x 2 - 2x + 5)
15.
(b) (2x – 5) 2
Find the quotient and remainder (if any):
(a) (x 3 + x 2 − 5 x − 6) ÷ ( x + 2)
16.
(b) (x 3 −4 x 2 − 6 x + 8) ÷ ( x − 5)
Factor completely
(a) x 2 - 3x – 18
(b) 12x 2 -17x +6
(c) 8a 3 - 27
(d) 5b 2 - 45
6x 2 + x -2 = 0
17.
Solve the quadratic equation by factoring:
18.
Solve the quadratic equation: 3x 2 - 2x = 0
19.
Solve the quadratic equation by using the quadratic formula: 2x 2 -x – 5 =0
20.
Describe the domain for each of the following functions.
x−3
1
x+3
(a) f(x) = 2
(b) f(x) = −
x
x−2
x − x−6
21.
Graph the quadratic function indicating the intercepts and vertex. Describe
the domain and range.
f(x) = x 2 + 2x – 8
2
22.
A ball is thrown vertically upwards from the top of a 96 foot tower with an
initial velocity of 48 feet per second. Use the function s(t) = -16t 2 +v 0 t +
s 0 where v 0 is the initial velocity and s 0 the initial height.
(a) After how many seconds does the ball attains its maximum height?
(b) What is the maximum height the ball attains?
(c) After how many seconds does the ball return to the ground?
23.
A rectangular field along a river is to be fenced with 800 feet of fencing.
Find the dimension of the field which will produce a maximum area.
24.
A company has found that the revenue, in dollars, from sales of its product
is a function of the unit price p, in dollars, that it charges. If the revenue R
is
R(p) = −
1 2
p + 1800 p
2
(a) What unit price p should be charged to maximize revenue?
(b) What is the maximum revenue?
25.
The marginal cost C (in dollars) of manufacturing x (in thousand) cell
phones is given by
C(x) = 4x2 – 200x + 6000
(a) How many cell phones should be manufactured to minimize the
marginal cost?
(b) What is the minimum marginal cost?
26.
Perform the operation and simplify the rational expressions.
(a)
27.
(b)
x 2 − 7 x + 12
x 2 − 2x − 3
÷
x 2 − 16
x 2 − 2 x − 24
Solve for x and check for any extraneous roots.
(a)
28.
x
x−2
+ 2
2
x −9
x + 3x
x
11
= 2
+2
x + 4 x − 16
(b)
3
2x
+
=2
x −1 x +1
Solve for x and check for any extraneous roots.
(a)
x −1 - 3 = 5
(b)
5x − 1 = 1 + x + 2
3
(c)
x+2 = x
29.
Solve for x.
(a) f(x) = 3x - 4
(b) f(x) =
2x + 3
x −1
(c) f(x) = x − 2
Once you solve for x compute each one for y = - 4.
30.
Find the exponential function of the form f(x) =ab x for which f(-1)=10 and
f(0)=5.
31.
Find the exponential function of the form f(x) =ab x for which f(-1) = 15 and
f(1) = 53 .
32.
Solve for x:
(a) 4 x = 8
= 81
(b) 3
x
(c) 5 x = 25
(d) 3
x −1
33.
Evaluate (a) log 4 2 + log 4 8
(b) log 5 250 - log 5 2
34.
Solve the equation by using logarithm:
35.
Solve for x and check
2
36.
= 27
2 x +1 = 3 2 x +3
(a) log 2 x = 3
(b) log 2 ( x − 1) = 3
(c) log x 16 = 2
(d) log x ( x + 12) = 2
(e) log 2 x + log 2 ( x − 2) = 3
(f) log 2 x − log 2 ( x + 2) = 2
Suppose that $20000 is deposited into an account that earns 6% interest
annually. Assume the interest is compounded monthly. Let A(t) represent
the balance (in thousands of dollars) in the account after t years (or any
fraction thereof).
(a) Find a formula for A(t)
(b) Find the balance after 10 years.
(c) After many years will the balance in the account doubled?
37.
A radioactive substance has a half-life of 200 years. If 500 mg of the
substance was initially present, how much of the substance remains after
640 years?
4
38.
A particle decays at a rate of 0.45% annually. Assume and initial amount
of 300 mg.
(a) How much of the particle remains after 60 years?
(b) What is the half-life of the particle?
39.
A cable 90 feet long is attached to the top of a radio transmission tower
making an angle of 48 o with the ground. How high is the tower?
40.
A 6-foot tall man standing 200 feet from a tower observed the angle of
elevation to the top of tower to be 67 o . How high is the tower?
41.
Find the exact value of the following.
 3π 
(a) cos  
 4 
42.
 2π 
(b) sin  −

 3 
(a) Convert the radian measure
2π
to degrees
3
(b) Compute the sin ( 23π ) and cos ( 23π )
(c) Compute the cot ( 23π )
43.
Let the trigonometric function f be defined as f(x) = 4sin (3x)
(a) What is the amplitude?
(b) What is the period?
(c) Sketch two periods of f.
44.
Find the values of a and b;
12
b
380
a
5
 5π 
(c) tan  
 6 
ANSWERS
1. (a) 4/3 (b) -13/5
2. (a) a ≥ -7 (b) a<0
3. (a) 4,1 (b)12/5, 3/5
4. (a) (- ∞ ,-4]
∪ [7, ∞ ) (b) (-4,14)
5. (a) y=-3x-2 (b) x+y = 1 (c) y =5 (d) 2x+3y = 6 (e) x=-3 6.x = -1
7. y = 6
8. (a) -12 (b) 0 (c) 0.003999 (d) 2
9. (a) –2 (b) 6x + 3h -1
11. (a) (2,1)
(b)
10. (a) –7 (b) –3
(1413 , 1357 )
12. $12,000 @ 2.5%; $8,000 @ 3%
13. wheat $.65 ; rye $.70
15. (a) x
2
−x − 3
16. (c) (2a-3)(4a
19.
14. (a) 5x
(b) x
2
2
+x −1 +
3
x−5
2
−7
(b) 4x
2
−20 x + 25
16. (a) (x-6)(x+3) (b) (3x-2)(4x-3)
+6a + 9) (d) 5(b-3)(b+3)
17. -2/3,1/2
1 ± 41
4
18. 0, 3/2
20. (a) x ≠ 3,-2
21.
(b) x ≥ -3, x ≠ 0,2
22. (a) 3/2 second (b) 132 feet (c) 4.37 sec
(− 1,−9 )
Domain: (− ∞, ∞ )
Range: [− 9, ∞ )
Vertex:
23. 400 ft by 200 feet
24. (a)1800
25. (a) 25,000
26. (a)
(b) $3500
27. (a) -7,3 (b) -5
29. (a) x = (y+4)/3
x=0
(b) $1,620,000
2x 2 − 5x + 6
x( x − 3)( x + 3)
28. (a) 65 (b) x=2; x= 1/4 (extraneous)
(c) x = 2; x = -1 (extraneous)
(b) x = (y = 3)/(y – 2)
x = 1/6
(c) x = y² + 2
x = 18
6
1

2
32. (a) x = 32
x
1

5
30. f(x) =5 
33. (a) 2
(b) x = 16
(c)
x=± 2
(b) 3
35. (a) x = 8
x
31. f(x) = 3 
(d) x = 16
34.
(b) x = 9
(c) x = 4
36. (a) A(t) = 20000(1.005)
12 t
(d) x = 4
(b) $36,387.93
3 log 3 − log 2
≈ -1.73
log 2 − 2 log 3
(e) x = 4
(f) no solution
(c) 11.6 years
37. 54.4 mg
38. (a) 229mg (b) 154 years
39. 67 feet
40. 477 feet
41. (a) -
2
3
3
(b) (c) 2
2
3
42. (a) 120
43.
o
(a) 4
(b)
(b)
sin
( 23π ) =
3
1
; cos ( 23π ) = −
2
2
2π
3
(c)
44.
a
≈ 9.45; b ≈ 7.39
Developed by R. Meangru Revised on 6/7/2013
Adapted to MAT 117 by J. Perez 11/23/2016
7
(c) -
3
3