Summer Review Sheets for Precalculus

Summer Review Sheets for Precalculus--1
I. Linear Equations and Inequalities
Solve for the variable:
1.
3(4x – 6) – (2x + 7) = 10 – 5(4 – 3x)
2.
2.6y – 3.4 = 3.6 – 4.4y
3.
4x  5
 19
3
4.
k + 6 < -3
5,
2(7  x)  x  17
6.
6(2n  4)  5(3n  4)  10
7.
x
x
 2  11 
5
2
8.
5 x  6  16
9.
10  3x  4
10.
 5 6  y  4  6
For #11 – 14, solve for the indicated variable:
11. Given P = 2L + 2W, solve for L.
12. Given V=r2h, solve for h.
Given F 
Gm1 m2
:
r2
13. Solve for m1 .
14. Solve for r.
Summer Review Sheets for Precalculus--2
II. Simplifying Radicals, Coordinate Geometry, and Trigonometry
Simplify:
1.
4.
7.
50
48
6
28
4 2
2. 2 108
6
3.
5. 7 3  8 6
6.
8. 5 50
9.
2
5
3 1

6 3

6 3

10. Find the distance between the given points. Write the result in simplified radical
form.
a) K = (1, 5)
L = (-3, 7)
b) R = (5, 6)
Q = (9, 10)
11. Find the midpoint of the segment AB in each example below.
a) A = (5, 4)
B = (7, -10)
b) A = (-5, -7)
B = (-1, -3)
12. Point A has coordinates (-4, 3), and the midpoint of AB is the point (1, -1). What
are the coordinates of point B?
13. Find each labeled length or angle measure; round to the nearest tenth.
y
13
13
x
20
14
75
50
x
350
x
6
14. Find the length of the base in an isosceles triangle, if each leg = 16 cm and the vertex
angle measures 40º.
15. Find the value of x, in simplified radical form, in each right triangle below:
3
49
3
x
15
x
x
60
8
Summer Review Sheets for Precalculus—3
III. Multiply and Factor
Multiply:
1. (2x + 5)(x – 3) =
2. (5x – 6)2 =
3. (2x + 3y)(2x – 3y) =
4. (x + 1)(x + 2)(x + 3) =
Factor completely:
5. x2 – 7x + 12
6. 2x2 + 7x + 3 =
7. 9x2 – 6x + 1 =
8. 4x2 – 9 =
9. 10x2 + 35x + 15 =
10. x3 – 4x2 – 45x =
11. 25x2 – y2 =
12. x4 – 13x2 + 36 =
13. 4x2 + 4x – 35 =
14. 16m2 – 24m + 9 =
For each problem below, find the value of c that makes the expression a trinomial square,
and write the expression as a perfect square.
15. x2 – 4x + c =
16. x2 + 10x + c =
17. x2 – 5x + c =
18. x2 + 9x + c =
Solve. Write answers in simplified radical form.
19. Solve by completing the square: x2 + 8x – 15 = 0
20. Solve by factoring: x2 + 2x – 35 = 0
21. Solve by using the quadratic formula: x2 – 6x + 3 = 0
Solve by any method:
22. 2x2 + 9x – 5 = 0
23. x2 + 6x = -27
24. x3 – 4x2 + 3x = 0
25. 4x2 = 100
Summer Review Sheets for Precalculus—4
IV. Linear Functions and Systems of Equations
1. Write the equation of the line that contains P(7, -3) and has slope =
4
:
7
a) in point-slope form (y – y1 = m(x – x1))
b) in slope-intercept form (y = mx + b)
2. Write the equation of the line through points P(1, 5) and Q(-2, 3) in any form.
3. Write an equation in standard form (Ax + By = C) of the line through (-4, 6) that is
parallel to the line 3x – 4y = 5.
4. Determine the value of the constant k so that the line through A and B is perpendicular
to the line through C and D, where A = (2, 1), B = (6, 3), C = (4, k), and D = (3, 1).
5. Show that ABCD is a parallelogram, if A = (-3, -3), B = (2, -5), C = (5, -1), and D =
(0, 1).
Solve for x and y in each system below:
6.
2x – 3y = 7
3x + y = 5
7.
3x + 2y = 17
2x – 3y = 7
8.
2x – y = 5
-4x + 2y = 7
9.
3x – y = 5
x = 4 – 2y
10.
12x – 7y = -20
5x – 8y = 73
11.
y = 5x – 12
y = -2x + 9
12. Five tables and eight chairs costs $115 altogether. Three tables and five chairs costs
$70. Determine the cost of each table and each chair.
13. A boat can go 10 miles upstream (against the current) in 4 hrs. It makes the return
trip downstream in 2 ½ hrs. How fast is the current?
14. Solve this system of linear equations, using matrices:
2x + y – z = -5
-5x – 3y + 2z = 7
x + 4y – 3z = 0
Summer Review Sheets for Precalculus—5
V. Functions
1. If f(x) = 2x + 1 and g(x) = │x + 2│, find:
a) f(10)
b) g(-5)
(c) f(g(1))
e) g ◦ f(12)
d) g(f(-4))
2. If f(x) = 7x2 + 3 and g(x) = 2x – 9, then g(f(2)) =
3. If f(x) = x2 – 2, find (simplified):
b) f(3x – 5)
a) f(7y)
4. Tell the domain for each function below:
a) f ( x) 
x5
3x  6
b) f ( x)  2 x  9
5. Find the inverse function f-1(x) for each function below:
a) f(x) = 4x + 1
c) f ( x) 
1
x
b) f ( x) 
x
2
3
d) f(x) = 3x
6. Let f(x) = -3x and g(x) = x2 – 4. Find and simplify f(g(x)).
7. Let f ( x)  x 2 and g ( x)  x  2 .
a) Tell the domain of each function.
b) Find f(g(11)) and g(f(11)).
8. Let f(x) = x2 – 4 and g(x) = 3x.
a) What are the zeroes of f?
b) For what values of x does f(x) = g(x)?
9. Refer to the graph of the function f at the right:
a) f(0) =
b) Solve f(x) = 0.
c) Solve f(x) = 4.
d) What is the range of f?
10. a) If x varies inversely as y, and if x = 18 when y = 52, find x when y = 234.
b) The distance d that a body falls toward the earth varies directly as the square of the
time t that it has been falling. If d = 18 when t = 3, find the value of d when t = 5.
Summer Review Sheets for Precalculus—6
VI. Exponents and Logarithms
Simplify. Leave answers with positive exponents only.
1.
x 
4 2
1
4
3
4
4.
x x
7.
 r 1 s 4
 2  2
v m
3



2.
5.
 x 
2y (2y)
11.
 2a 2
 1
 b



3 2
3
3
9.



4
 x9
 2
x
3
12.
8.
 6 x 3b 2 c 5
x 1b 1c 2
10.
1 1
  
 2 3
x 2  x8
x 2
3.
7 y 3  8 y 9
6.
 x 3
 7
x



2
2
 3x 
2 4
13.
9x 6
Evaluate, without a calculator:
1
3
14.
8
17.
log232
2
3
15.
64
18.
log 0.0001
3
16.
16 2
19.
log 6
1
36
Write as a sum or difference of logs:
20.
 C 
log 3 
D 
21.
log M 2 N
22.
x3 y
log
z
Write as a single logarithm:
23.
5 log2x + 2 log2y
24.
3 log a – 2 log c
25.
½ log 4 – 2 log 3

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