Download Pre-calculus Honors Summer Review

Pre-calculus Honors Summer Review
Solve all problems on a separate sheet of paper. Highlight or box all answers
Simplify:
18. Find the distance between the points
(-3,5) and (4, 7). Find the midpoint.
19. Find the center and radius of the circle:
x2 + y2 - 4x + 10y + 13 = 0
20. Find an equation of a line in standard
form with a y-intercept of 2 that is
parallel to the line 2x – 3y = 0
21. Find an equation in standard form of the
line through (4, 5) perpendicular to the
line through (1, -4) and (7, 3)
22. Find the function values of f(-5), f(0),
(3ab c)
1.
2. √
√
Factor completely:
3.
+2
−3
4. 27
+
5. 3
+ 12
− 15
and f(5)
Perform the indicated operation:
6.
∙
7.
x
8.
2

x 4
9.
3 x if x  0

Given f ( x )   x  1 if 0  x  2

2
 x  2  if x  2
÷
1
x2
x y

y x
2
x  y2
Find the domain
23. f(x) =
Solve:
10. x2 – 6x + 1 = 0
x
1
11.

0
2x  7 x  2
12. |3x + 2| = 4
24. f(x) =
25. Explain how the graph of g is obtained
from the graph of f. f(x) = |x|,
g(x) = -|x+1|
Solve the inequality.
13. – | − 2| − 14 ≤ −25
14. −2| + 3| − 4 > −2
15. 2x2 – 3x ≥ 2
16. Rationalize the denominator:
√
√
√
17. A pasture is twice as long as it is wide.
Its area is 115,200 square feet. How wide is
the pasture?
Determine if the equations are even, odd
or neither
26. g(x) = x6 – 2x2 + 3
27. g(x) = x3 – 5x
Sketch the graph of the given polynomial
functions. Label all intercepts and
approximate max and min values
28. P(x) = - x3 + 9x2 – 24x + 16
29. P(x) = x3 – x2 – 8x + 12
30. P(x) = x4 – x3 + x2 – 3x – 6
For
31.
32.
33.
the given polynomials find all zeros
P(x) = x3 – 27
P(x) = x4 – 16
P(x) = x3 + 6x2 + 14x + 12
Find a polynomial with integer coefficients
That satisfies the given conditions.
34. P of lowest degree and zeros of -3i and
-2.
35. Q has a degree 3, and zeros of -2 and –i
Find the intercepts and asymptotes, and
then sketch a graph of the rational
function.
36. R(x) =
51.
52.
√
53.
√6 − √−18 (2√3 − √−9
√
State the domain:
54. f(x) = log2(x – 4)
55. f(x) =
x
 log( x  1)
2
56. f(x) = ln(2x + 3) – (x/8)
37. R(x) =
Find all solutions. Write unique and
dependent solutions as ordered pairs or
ordered triples.
38. R(x) =
57. 
x  y2
x  y  2
Rewrite each expression as a single
logarithm:
39. log(2x – 6) – 3log(x – 3)
40. log5(x2 – 1) – log5(x – 1)
41. log(2x) + ½ log4 – log(4x)
42. log(2x + 1) + ½ [log(x – 4) – log(x4 – x2-1)]
Solve each logarithmic equation.
43. log23 + log2x = log25 + log2(x – 2)
44. log5x + log5(x + 1) = log520
45. log x  log  x  3   1
Solve each exponential equation
(Use calculator)
46. e3-5x = 16
47. e2x+1 = 200
48. 101-x = 4
Simplify each of the following.
49.
50.
 x 2  y 2  18
58. 
x  y  0
2 x  2 y  2 z  2

59.  x  y  2 z  1
x  y  z  2

x  y  z  2

60. 2 x  3 y  2 z  4
4 x  y  3 z  1

Pre-Calculus Summer Review Answers
1.
23. D: {x| x ≠ 1, x ≠ –1}
3c 7
4a 3
24. D: { x | x ≥ 3}
2. a 2 3 b 2
3. x 2  x  3 x  1
4.  3a  b   9a 2  3ab  b2 
5. 3 x 
22. f (–5) = –15; f (0) = 1; f (5) = 9
1
2
 x  5 x  1
25. reflected over the x-axis; horizontal shift
left 1
26. even function
27. odd function
28. end behavior ↑ left and ↓ right
6. x  2
Zeros of 1 and 4
7. 1
y-int (0, 16)
8.
2x  2
( x  2)( x  2)
9.
1
xy
approximate local max (4, 0)
29. end behavior ↓ left and ↑ right
Zeros of –3 and 2
10. x  3  2 2
11. x  2  11
2
12. x  , x  2
3
13. x  9 or x  13
14. no solution
1
15. x   or x  2
2
16. x 
approximate local min (2, –4)
17  8 2
7
y-int (0, 12)
approximate local max (–1.3, 18.5)
approximate local min(2, 0)
30. end behavior ↑ left and ↑ right
Zeros of –1 and 2
y-int (0, –6)
approximate local min (1, –8)
31. x  3, x 
3  3i 3
2
32. x  2, x  2i
17. the width is 240 ft
33. x  2, x  2  i 2
1 
18. d  53, M  , 6 
2 
34. P  x   x3  2 x 2  9 x  18
19. c   2,  5 and r  4
20. 2 x  3 y  6
21. 6 x  7 y  59
35. P  x   x3  2 x 2  x  2
36. y-int: (0, 2)
x-int: (2, 0)
horizontal asym: y = 0
slant asmy: none
vertical asym: x = –2, x = 1
37. y-int: (0, –4)
50.
1 2
 i
5 5
4
51. 2  i
3
1 2 2
52.  
i
3
3
x-int: (–3, 0)
53. 3 2  9i 6
horizontal asym: y = 4
54. D : {x | x > 4}
slant asmy: none
55. D: {x | x > –1}
vertical asym: x = 3
3

56. D:  x | x   
2

 1
38. y-int:  0, 
 4
57. (4, 2) (1, –1)
x-int: (1, 0)
58. (3, 3) (–3, –3)
horizontal asym: none
59. no solution
slant asmy: y = x
60. (1, 0, 1)
vertical asym: x = ± 2
39. log
2
( x  3)2
40. log 5  x  1
41. 0
3
42. log
2x 1 x  4
x4  x2  1
43. x = 5
44. x = 4
45. x = 5
46. x = 0.05
47. x ≈ 2.15
48. x ≈ 0.40
49. 
5 12
 i
13 13