Group WBs: Semester 1 Review Study Notes

Group WBs: Semester 1 Review

1. Inverse of Functions

  • Find the inverse of each function:
    • 1) Function: f(x) = -2x - 4
    • Inverse Steps:
      • Solve for x:
      • x = -2y - 4
      • y = -1/2(x + 4)
    • Inverse Function: f⁻¹(x) = -1/2(x + 4)
    • 2) Function: f(x) = -2/3 * (-x - 2)
    • Inverse Steps:
      • Solve for x:
      • Multiply both sides by -3/2;
      • y = -3/2(x + 2)
    • Inverse Function: Anti-correlation function
    • 3) Function: g(x) = -4x + 16
    • Inverse Steps:
      • Solve for x:
      • x = -4y + 16
      • conveys direct inverse depending on slope which leads to 16 -4x
    • Inverse Function: g⁻¹(x) = (16 - x)/4
    • 4) Function: g(x) = -(-x - 2)
    • Inverse Function:
      • Creates overlap with original as it satisfies inverse conditions

2. Rate of Change

  • Find the rate of change for specific intervals of x:
    • 5) From x = 1 to x = 4
    • Data Points:
      • x: 0, 1, 2, 3, 4, 5
      • f(x): 1, 5, 10, 8, 3, 14
    • Calculation:
      • Rate of Change = (f(4) - f(1)) / (4 - 1) = (3 - 5) / (4 - 1) = -2/3
    • 6) From x = 3 to x = 5
    • Data Points:
      • X: 3 -> 5
      • f(3): 8, f(5): 14
    • Calculation:
      • Rate of Change = (f(5) - f(3)) / (5 - 3) = (14 - 8) / (5 - 3) = 6/2 = 3
    • 7) From x = -3 to x = -1
    • 18) From x = 10 to x = 20

3. Quadratic Formula and Solutions

  • Recall the quadratic formula:
    • The quadratic formula is used to find the solutions of ax² + bx + c = 0. This can be expressed as
    • x=−b±b2−4ac2ax = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}

4. Graphing Equations

  • Graph each equation:
    • 10) Equation: y = |x + 2| + 4
    • 11) Equation: y = 2x - 4

5. Solving Equations

  • Solve each equation:
    • 12) Equation: |106 - 11| = 31
    • Solve absolute equation systematically.
    • 13) Equation: |-8 - m = 13
  • Solve inequalities and graph solutions on number line:
    • 14) Inequality: 6 - 6n > 18
    • 15) Inequality: |5n + 5| ≤ 45

6. Factoring and Quadratic Solutions

  • Solve by factoring:
    • 16) n² + 7n + 10 = 0
    • Factoring gives (n + 5)(n + 2) = 0; Solutions: n = -5, n = -2
    • 17) p² + 2p - 24 = 0
    • Factoring gives (p - 4)(p + 6) = 0; Solutions: p = 4, p = -6
    • 18) 2x² + 7x + 6 = 0
    • Factoring gives (2x + 3)(x + 2) = 0; Solutions: x = -3/2, x = -2

7. Quadratic Formula Applications

  • Use the quadratic formula:
    • 20) n² - 2n - 11 = 0
    • Using the quadratic formula, find solutions.
    • 22) 3n² - 8n + 7 = 0
    • 19) 2n² - 7n - 15 = 0

8. Determining the Discriminant

  • Find the discriminant:
    • 24) 4x² + 7x + 5 = 0
    • Discriminant D = b² - 4ac = (7)² - 4(4)(5) = 49 - 80 = -31 → No real roots
    • 25) -2x² + 8x - 8 = 0
    • D = 8² - 4(-2)(-8) = 64 - 64 = 0 → One real root

9. Equations by Taking Square Roots

  • Solve each equation:
    • 28) 3x² - 4 = -15
    • Rearranged form gives exact roots of function, express final variable terms.
    • 29) 6k² + 8 = -19

10. Circle Equations

  • Standard form equations of circles:
    • 38) x² + y² + 28x - 6y + 201 = 0
    • Rearranged to standard form becomes (x + 14)² + (y - 3)² = r².
    • Center: (-14, 3)
    • Radius: r
    • 39) x² + y² + 28x + 32y + 443 = 0
    • Use completing the square to derive center and radius.