Download PreCalc Summer Packet

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Precalculus Name ____________________________________ Resources: Math Dictionary: h
ttp://www.mathwords.com/a_to_z.htm Video Tutorials: There are a selection of videos for each lesson on Mrs. Jamison’s Precalculus Website (h
ttps://sites.google.com/site/mrsjamisonswebsite/pre­calculus ) or you can search your own on Youtube or Khan Academy. Real Numbers & Algebraic Expressions (P.1) 1. Given the set of numbers listed below, list all numbers that are n
atural numbers , whole numbers , i ntegers , r ational numbers , and i rrational numbers . Remember that numbers may fall in more than one category. A. Natural: _____ {
}
− 9, − 45 , 0, 0.25, √3, 9.2, √100 B. Whole: _____, _____ C. Integers: _____, _____, _____ D. Rational: _____, _____, _____, _____, _____, _____ E. Irrational: _____ 2. Evaluate each algebraic expression for the given value of the variable or variables. Show your work. A. 5x + 7; x = 4 B. 6(x + 5) − 13; x =− 7 1−(x−2)2
C. ; x =− 1 1+(x−2)2
D. 1 2
−b±√b −4ac ; 2a
a = 4, b =− 20, c = 25 Summer Packet ( NO CALCULATOR)
Precalculus 3. Simplify each expression. Show your work. A. 4(2y − 6) + 3(5y + 10) B. 4(5y − 3) − (6y + 3) Exponents & Scientific Notation (P.2) 4. Simplify each exponential expression, then evaluate w
ithout using a calculator . B. − 24 A. (− 2)4 D. (53 )0 8
E. 35
3
C. 22 · 23 F.
2−4 · 22 5. SImplify each exponential expression. Write your final answer using only positive exponents. A. x−2 y C. x−10 · x5 B. x0 y 5 D. (x−2 )−3 14
E. xx−7
F.
(− 3x2 y 5 )2 C.
√− 36 Radicals & Rational Exponents (P.3) 6. Evaluate each expression or indicate that the root is imaginary. A. √49 B.
(− 3)2 √
7. Simplify each expression. Assume all variables are positive. 2 Summer Packet ( NO CALCULATOR)
A. √50 B.
D. √
1
81
Precalculus √45x2 E. √
49
16
8. Add or subtract terms whenever possible. A. 7 3 + 3 √ √
C.
√150x4 F. √3x
B.
√2x2 · √6x √8 + 3√2 − 2√32 9. Rationalize the denominator. Give your answers in simplified radical form. 2
A. 3
√
1
B. 8
√
5
C. 2+ 3
√
Polynomials (P.4) 10. Perform the indicated operations. Write the resulting polynomial in standard form. Standard form begins with the highest power, decreasing for each additional term. 3 A. (− 6x3 + 5x2 − 8x + 9) + (17x3 + 2x2 − 4x − 13) B. (17x3 − 5x2 + 4x − 3) − (5x3 − 9x2 + 11) Summer Packet ( NO CALCULATOR)
Precalculus 11. Find each product. Write the resulting polynomial in standard form. A. (3x + 5)(2x − 4) B. (4x − 3)(4x + 3) C. (2x + 5)2 D. (7x2 + 1)(x2 − 2x + 3) Factoring Polynomials (P.5) 12. Factor out the greatest common factor. A. 3x2 + 6x B. 9x4 − 18x3 + 27x2 13. Factor by grouping. A. x3 − 2x2 + 5x − 10 B. x3 − x2 + 2x − 2 14. Factor each trinomial. A. x2 + 5x + 6 B. x2 − 2x − 15 C. x2 − 8x + 15 D. 3x2 − 25x − 28 15. Factor the difference of squares. a2 − b2 = (a + b) (a − b) A. x2 − 100 4 B. 64x2 − 81 Summer Packet ( NO CALCULATOR)
Precalculus 16. Factor any perfect square trinomials, or state that the polynomial is prime. a2 + 2ab + b2 = (a + b)2 , a2 − 2ab + b2 = (a − b)2 A. x2 + 2x + 1 B. x2 − 14x + 49 C. 4x2 + 4x + 1 D. 9x2 − 6x + 1 17. Factor completely. A. 4x2 − 4x − 24 C. x3 + 2x2 − 9x − 18 B. 2x4 − 162 D. x3 − 4x Rational Expressions (P.6) 18. Factor and simplify each rational expression below. −
A. x2 6x+9
−
3x 9
2
B. x +12x+36
x2 36
−
19. Factor and simplify, then multiply or divide each rational expression. 2
A. x x−2 9
5 ·
x2 −3x
−12
x2 +x
2
B. x2x−25
−2
2
÷ x x+10x+25
2 +4x 5 −
Summer Packet ( NO CALCULATOR)
Precalculus 20. Add or subtract the rational expressions below. Remember to get a common denominator, if necessary. Simplify the result, if possible. 4x+1
A. 6x+5
+ 8x+9
6x+5 2x
C. x+4
+
x−1
x+5
x2 +3x
−
x2 +x
4
+
4
x+3
B. x2 +x−12
D. x2 +6x+9
x2 −12
−12
Linear Equations (P.7) 21. Solve and check each linear equation. A. 3(x − 2) + 7 = 2(x + 5) B. 3(x − 4) − 4(x − 3) = x − (x − 2) 22. Solve and check each linear equation. x
x 3
A. 4 = 2 + −3
x+3
B. 6
=
3
8
+
x−5
4
Quadratic Equations (P.8) 23. Solve each quadratic by factoring. 6 A. x2 − 3x − 10 = 0 B. x2 = 8x − 15 C. 6x2 + 11x − 10 = 0 D. 3x2 + 12x = 0 Summer Packet ( NO CALCULATOR)
Precalculus 24. Solve each quadratic by the square root method. A. 3x2 = 27 B. 5x2 + 1 = 51 C. (x + 2)2 = 9 D. (3x − 4)2 = 8 25. Solve each quadratic by completing the square. A. x2 + 6x = 7 B. x2 + 4x + 1 = 0 b2 −4ac
26. Solve each quadratic using the Quadratic Formula, x = −b±√2a
. A. x2 + 5x + 3 = 0 B. 3x2 = 6x − 1 Linear Inequalities (P.9) 27. Solve each linear inequality and graph the solution set on a number line. 7 Summer Packet ( NO CALCULATOR)
Precalculus A. 8x − 2 ≥ 14 B. 5x + 11 < 26 C. 3 ≤ 3 − 4x < 19 Graphs & Graphing Utilities (1.1) 28. Complete the table for each equation, then use the ordered pairs to graph the equation. 8 A. y = x2 − 2 x y ­3 ­2 ­1 0 1 2 3 Summer Packet ( NO CALCULATOR)
B. y = ∣x + 1∣ − 2 9 Precalculus x y ­3 ­2 ­1 0 1 2 3 Summer Packet ( NO CALCULATOR)
Precalculus Lines & Slope (1.2) 29. Calculate the slope between each pair of points. Then, indicate whether the line containing the points is vertical, horizontal, or oblique. A. (− 2, 1), (2, 2) B. (4, − 2), (3, − 2) C. (5, 3), (5, − 7) 30. Write equations in P
oint­Slope Form a
nd S
lope­Intercept Form to satisfy the given conditions. P
oint­Slope Form: y − y 1 = m(x − x1 ) ; Slope­Intercept Form: y = mx + b . Conditions Point­Slope Form Slope­Intercept Form A. Slope: − 5 , passing through (− 4, − 2) B. Slope: − 23 , passing through the origin C. Passing through (3, 5) and (8, 15) D. x ­intercept: 4 , y ­intercept: − 2 E. Passing through (− 2, − 7) , p
arallel to y =− 5x + 4 F. Passing through (− 4, 2) , perpendicular to y = 13 x + 7 10 Summer Packet ( NO CALCULATOR)
Precalculus Distance & Midpoint Formulas; Circles (1.3) 31. Find the distance between the points below. Simplify radicals, when necessary. Distance Formula: d =
√(x − x ) + (y − y ) 2
1
2
2
1
2
B. (− 2, − 6), (3, − 4) A. (2, 3), (3, 5) 32. Find the midpoint of each line segment with the given endpoints. Midpoint Formula: M P =
(
) x1 + x2 y 1 + y 2
, 2 2
B. (− 3, − 4), (6, − 8) A. (10, 4), (2, 6) 33. Write the standard form of the equation for a circle given the center and radius. 2
2
Standard Form of Circle Equation: (x − h) + (y − k ) = r2 , where (h, k) is the center and the radius is r . A. Center: (3, 2) , r = 5 B. Center: (− 5, 0), r = √10 34. State the center and radius of the circle with the given equation. 2
2
A. (x − 3) + (y + 1) = 4 35. A circle has a diameter with endpoints at (− 4, − 3) and (− 10, 5) . A. Find its center. B. Find its radius. C. Write the standard form of the equation for the circle. 11 2
B. x2 + (y − 7) = 36 Summer Packet ( NO CALCULATOR)
36. 12 Precalculus