Algebra 2 Final Exam Review Notes

Overview

• This video aids in preparing for the Algebra 2 final exam.
• Time stamps are available for specific question types in the video description.
• Refresher for first semester concepts can be found in the midterm exam review video.

Question 1: Inverse Variation

1. Problem Statement: Given y varies inversely with x, y = 3 when x = -8.
   • Find the equation relating x and y:
     • The formula for inverse variation: y = a / x
     • Substitute values:
       • 3 = a / -8
       • a = -24
     • Final Equation: y = -24 / x
   • Find y when x = 4:
     • y = -24 / 4 = -6

Question 2: Joint Variation

2. Problem Statement: Given Z varies jointly with X and Y, Z = -72 when X = 9 and Y = -4.
   • Equation:
     • Z = k * X * Y
     • Substitute values:
       • -72 = k * 9 * -4
       • Solve for k:
         • k = 2
     • Final Equation: Z = 2 * X * Y
   • Find Z when X = 6 and Y = 4:
     • Z = 2 * 6 * 4 = 48

Question 3: Combined Variation

3. Problem Statement: D varies directly with M and inversely with the cube of P.
   • Equation: D = k * M / P^3
   • Joint Variation: R varies jointly with Y and the square root of X:
     • R = k * Y * sqrt(X)*

Question 4: Graphing Functions

4. Function: y = -2 / (x - 1) + 3
   • Vertical Asymptote: x = 1
   • Horizontal Asymptote: y = 3
   • Domain:
     • (-∞, 1) ∪ (1, +∞)
   • Range:
     • (-∞, 3) ∪ (3, +∞)

Question 5: Simplifying Algebraic Expressions

5. Simplification: x^2 - 16 / x^2 - 7x + 12
   • Factor numerator and denominator:
     • Numerator: x^2 - 16 = (x - 4)(x + 4)
     • Denominator: x^2 - 7x + 12 = (x - 3)(x - 4)
   • Cancel common factors:
     • Final answer: (x + 4)/(x - 3)
   • Excluded Values: Analyze for original expression: x ≠ 4, 3

Question 6: Subtracting Fractions

6. Common denominator needed for: 2 / (x - 5) - 3 / (x + 4)
   • Combined expression: (x + 4)*2 - (x - 5)*3 / (x - 5)(x + 4)
   • Expand numerator and simplify.

Question 7: Solving Equations

7. Equation: Solve 3/(2x - 1) = 1/(x - 5).
   • Cross Multiply: 3(x - 5) = 2x - 1
   • Simplify and solve for x, ensuring to check for extraneous solutions.

Question 8: Distance and Midpoint Formula

8. Find distance between points (3, -8) and (-4, 16):
   • Use formula: d = √[(x2 - x1)² + (y2 - y1)²]
   • Midpoint formula: M = ((x1 + x2)/2, (y1 + y2)/2).

Question 9: Probability

• Problem: Probability of drawing a blue marble from a hat of 7 blue, 3 red, and 5 white.
• Compute Probability: P(B) = (# of blue marbles) / (total # of marbles) = 7 / 15.

Additional Concepts and Questions

• Book Ordering (Question 10): 6! = 720 ways to order six different books.
• Combination Lock (Questions 11): Calculate combinations considering restrictions and repetitions.
• Statistics (Question 18): Calculate mean, median, mode, range, standard deviation, and identify outliers.
• Normal Distribution (Question 19): Use mean and standard deviation to find probabilities.
• Triangle Problems (Questions 31-34): Apply trigonometric ratios and inverse functions.
• Area of Triangles (Questions 37-40): Use relevant formulas like Heron's and sine-based area calculations.
• Trigonometric Functions (Questions 41-43): Understand periodicity, domains, and ranges of functions.