Pre-Calculus 11: Final Review

Transcription

1 Pre-Calculus 11 Name: Block: FORMULAS Sequences and Series Pre-Calculus 11: Final Review Arithmetic: = + 1 = + or = Geometric: = = or = Infinite geometric: = Trigonometry sin= cos= tan= Sine Law: = = Cosine Law: = + 2cos cos= Special Triangles: Quadratic Functions and Equations Vertex form: = + Standard form: = ++ = Quadratic formula: = ±

2 Chapter 3 Final Review Quadratic Functions Vertex Form 1. Graph each parabola and state its characteristics. a) = 4 b) = Coordinates of the vertex: Equation of the axis of symmetry: y-intercept: x-intercepts, if any: Domain: Range: Maximum or minimum value: Coordinates of the vertex: Equation of the axis of symmetry: y-intercept: x-intercepts, if any: Domain: Range: Maximum or minimum value: 2. Write an equation for a parabola vertex at (-3, 5) passing through the point (2, -45).

3 3. Determine the vertex of each quadratic function. Use the = method. a) = 4 12 b) = Standard Form 4. Graph each of the following, either by completing the square or using the = method. a) = 8+14 b) = Quadratic Problems 5. The sum of two numbers is 60. Find the numbers if their product is a maximum.

4 Chapter 4 Final Review Quadratic Equations Solving Quadratic Equations 1. Solve +2 3=0 by graphing. Root(s): = 2. Solve by factoring. a) +3 28=0 b) 4 3=0 c) 2 =27 15 d) 2 +5=3

5 e) 16 49=0 f) 12 27=0 3. Solve by taking the square root. a) 5 67=18 b) 2 =81 c) 25 +4=23 4. Solve using the quadratic formula. State your answers as both exact and approximate to two decimal places. a) 2 + 4=0 b) =0

6 5. Find the discriminant and the nature of the roots for the following quadratic equations. (Hint: use 4) a) 2 4= 2 b) 3 =+9 6. The area of a board is 270 cm 2, and the length is 17 cm greater than the width. Build a quadratic equation and solve it to find the dimensions of the board. 7. A springboard diver s height, in metres, above the water, is given by the equation h= , where h is the height in metres, and is the time in seconds. When does the diver hit the water?

7 Chapter 5 Final Review Radical Expression and Equations Radical Expressions 1. Simplify by writing as a mixed radical. (a) 32 (b) 581 (c) Add or subtract. (a) (b) Multiply. Simplify your answers where possible. (a) 2 20 (b) (c) Divide. Rationalize the denominator when necessary and simplify your answer. (a) (b) (c)

8 Radical Equations 5. Solve each equation. Remember to check for extraneous solutions! (a) 10= 32 2 (b) = Chapter 6 Final Review Rational Expressions and Equations Rational Expressions 1. Multiply or divide and simplify completely. Identify the non-permissible values. (a) (b) 2. Add or subtract and simplify completely. Identify the non-permissible values. (a) (b) +

9 (c) Rational Equations 3. Solve each equation. Remember to check for extraneous solutions! (a) = (b) + =1 (c) + =

10 Chapter 7 Final Review Absolute Value and Graphing Rational Functions Absolute Value Functions 1. Graph the following absolute value functions without making a table of values. Also express each as a piecewise function. (a) = +3 2 (b) 2 4 Piecewise function: Piecewise function: 2. Solve for x and check for extraneous roots. Show all of your steps. (a) 23 5 (b) 4 21

11 (c) Graphing Rational Functions 3. Graph the rational functions. Also label each asymptote with its equation. (a) (b)

12 Chapters 8/9 Final Review Systems of Equations and Linear/Quadratic Inequalities Solving Systems of Equations 1. Solve this system graphically Solve each system algebraically by substitution or elimination (your choice). (a) (b) =6

13 Inequalities 3. Graph the following inequalities. (a) < 2 (b) Solve the following inequalities. (a) 616>0 (b)

14 (c) <0 (d) Chapter 1 Final Review Sequences and Series Arithmetic Sequences/Series 1. For the arithmetic sequence 1, 4, 7, 10,... find the following: (a) t8 (b) tn 2. Determine the number of terms in this arithmetic sequence: 29, 24, 19,..., Find the sum of the first ten terms of this arithmetic series:

15 Geometric Sequences/Series 4. For the geometric sequence 1, 3, 9, 27,... find the following: (a) t11 (b) tn 5. Determine the sum of the first 8 terms of the geometric series Calculate the sum of the geometric series: Decide whether each infinite geometric series is convergent or divergent. State the sum of the series, if it exists. (a) (b) 3+12

16 ANSWERS Ch. 3 Final Review 1. a) a) vertex: (0, -4) axis of symm: x = 0 y-int: -4 x-int: -4 and 4 domain: range: 4, max or min: -4 (min) Vertex: (1, 8) Axis of symm: x = 1 y-int: 6 x-int: -1 and 3 domain: range: 8, max or min: 8 (max) a) (2,-16) (-2, 3) 4. a) b) and 30 Ch. 4 Final Review 1. x = 1 and a) 4, -7 b) 0, c) -9, d) -3, e) ± f) ± 3. a) ± 17 b) -7, 11 c) ± 4. a) exact: ± ; approx: and 1.19 b) exact: ± ; approx: -0.12, a) 0; one distinct real root b) -107; no real roots cm by 10 cm 7. 2 seconds after diving

17 Ch. 5 Final Review 1. (a) 4 2 (b) 15 3 (c) (a) (b) (a) 2 10 (b) (c) (a) 3 (b) (c) 5. (a) 14 (b) 10 (8 is extraneous) Ch. 6 Final Review 1. (a) ; 0, 0 (b) ; 3, 1,3 2. (a) ; 2 (b) ; 0 (c) ; 0,1,5 3. (a) 3 (b) 4 (c) 5 Ch. 7 Final Review 1. (a) (b) 5, 3 +1,> 3 2, 2 +6,>2 2. (a) 4, 1 (b) 1 (x = -5 is extraneous) (c) 1, 5 3. (a) (b) horiz. asymptote: y = 1; vert. asymptote: x = -3 horiz. asymptote: y = 0; vert. asymptote: x = 2

18 Ch. 8/9 Final Review 1. (-1, -5) and (0, -2) 2. (a) (1, -1) and (3, 3) (b) (1, -4) 3. (a) (b) 4. (a) <2 or >8 (b) or 4 (c) 3<< (d) 7 2 Ch. 1 Final Review 1. (a) 22 (b) 3n (a) (b) 3 n (a) convergent; (b) divergent (no sum)