Algebra 2 Sem2 Final REVIEW Notes

Rational Exponents and Radicals

• Simplifying Expressions with Rational Exponents:

  • 7^1 \cdot 7^{1/4} = 7^{1 + 1/4} = 7^{5/4}

• Simplifying Radical Expressions:

  • \sqrt{300} = \sqrt{100 \cdot 3} = \sqrt{100} \cdot \sqrt{3} = 10\sqrt{3}
  • \sqrt[3]{25} = \sqrt[3]{5^2} = 5^{2/3}
  • Cannot be simplified further without additional context or instructions.

• Combining Radical Expressions:

  • 7\sqrt{8} + \sqrt{7} = 7\sqrt{4 \cdot 2} + \sqrt{7} = 7(2\sqrt{2}) + \sqrt{7} = 14\sqrt{2} + \sqrt{7}
  • This expression cannot be simplified further because the radicals are not like terms.

• Simplifying Expressions with Radicals:

  • 4\sqrt[3]{2} + \sqrt[3]{2} = (4+1)\sqrt[3]{2} = 5\sqrt[3]{2}

• Simplifying Variable Expressions with Radicals:

  • \sqrt[3]{27y^3p^6} = \sqrt[3]{3^3y^3(p^2)^3} = 3y\sqrt[3]{(p^2)^3} = 3yp^2

Solving Radical Equations

• Solving Radical Equations:

  • \sqrt{10x + 20} = 10
    • Square both sides: 10x + 20 = 100
    • Subtract 20 from both sides: 10x = 80
    • Divide by 10: x = 8

• Solving Radical Equations:

  • -7\sqrt[3]{10x} - 63 = -14
    • Add 63 to both sides: -7\sqrt[3]{10x} = 49
    • Divide by -7: \sqrt[3]{10x} = -7
    • Cube both sides: 10x = -343
    • Divide by 10: x = -34.3

• Solving Radical Equations with Extraneous Solutions:

  • 6x + 1 = x - 9
    • Square both sides: (6x + 1)^2 = (x - 9)^2
    • 36x^2 + 12x + 1 = x^2 - 18x + 81
    • 35x^2 + 30x - 80 = 0
    • 7x^2 + 6x - 16 = 0
    • (x-2)(7x+8)=0
    • x = 2, -8/7

• Solving Equations with Rational Exponents:

  • 3x^{3/4} - 14 = 67
    • Add 14 to both sides: 3x^{3/4} = 81
    • Divide by 3: x^{3/4} = 27
    • Raise both sides to the power of 4/3: x = 27^{4/3} = (27^{1/3})^4 = 3^4 = 81

• Solving for x in Terms of y:

  • y = f(x) = \frac{5}{9}x + 2
    • Solve for x: y - 2 = \frac{5}{9}x
    • Multiply by 9/5: x = \frac{9}{5}(y - 2) = \frac{9y - 18}{5}
    • Find input(s) when output is -7: x = \frac{9(-7) - 18}{5} = \frac{-63 - 18}{5} = \frac{-81}{5}

Inverse Functions

• Finding the Inverse of a Linear Function:

  • f(x) = -3x + 3
    • Replace f(x) with y: y = -3x + 3
    • Swap x and y: x = -3y + 3
    • Solve for y: x - 3 = -3y
    • y = \frac{x - 3}{-3} = \frac{x}{-3} + 1 = -\frac{1}{3}x + 1
    • g(x) = -\frac{1}{3}x + 1

• Finding the Inverse of a Nonlinear Function:

  • f(x) = (x + 4)^3
    • Replace f(x) with y: y = (x + 4)^3
    • Swap x and y: x = (y + 4)^3
    • Take the cube root: \sqrt[3]{x} = y + 4
    • Solve for y: y = \sqrt[3]{x} - 4
    • g(x) = \sqrt[3]{x} - 4

Exponential Functions

• Rewriting Exponential Functions:

  • y = a(8)^{t/11} = a(8^{1/11})^t
    • 8^{1/11} \approx 1.208
    • y = a(1.208)^t = a(1 + 0.208)^t
    • Growth rate: 20.8%

• Compound Interest Formula:

  • A = P(1 + \frac{r}{n})^{nt}
    • Where:
      • A = balance after t years
      • P = principal
      • r = annual interest rate
      • n = number of times interest is compounded per year
      • t = number of years
  • Given:
    • P = $7500
    • r = 8.1% = 0.081
    • n = 12
    • t = 6
  • A = 7500(1 + \frac{0.081}{12})^{12 \cdot 6} \approx $12,173.61

Exponential Expressions and Logarithms

• Simplifying Exponential Expressions:

  • \frac{54e^9}{6e^5} = \frac{54}{6} \cdot \frac{e^9}{e^5} = 9e^{9-5} = 9e^4

• Modeling Exponential Growth

  • A = 3400e^{0.07t}
    • Where:
      • A (in dollars) is the balance of the account after t years
  • Comparing Principal
    • Your account's initial principal: 3400
    • Sister's Account's Principal: Based on graph, is >".$3400
  • Balance after 10 years
    • Your account: A = 3400e^{0.07(10)} \approx $6871.67
    • Sister's Account: Based on the graph, the balance after 10 years < $6871.67
  • Conclusion
    • Your sister's account has a greater principal
    • Your account has a greater balance after 10 years.

• Rewriting Exponential Equations in Logarithmic Form:

  • 36^{3/2} = 216
  • \log_{36} 216 = \frac{3}{2}

Logarithmic Expressions

• Expanding Logarithmic Expressions:

  • \log3 \frac{x^6}{5y} = \log3 x^6 - \log_3 (5y)
  • = 6\log3 x - (\log3 5 + \log_3 y)
  • = 6\log3 x - \log3 5 - \log_3 y

• Condensing Logarithmic Expressions:

  • \log5 6 + 3\log5 3 - \log5 9 = \log5 6 + \log5 3^3 - \log5 9
  • = \log5 6 + \log5 27 - \log_5 9
  • = \log5 (6 \cdot 27) - \log5 9
  • = \log5 162 - \log5 9
  • = \log5 \frac{162}{9} = \log5 18

Solving Exponential and Logarithmic Equations

• Solving Exponential Equations:

  • 7^x = 75
  • Take the logarithm of both sides (base 7): x = \log_7 75
  • Using the change of base formula: x = \frac{\log 75}{\log 7} \approx 2.219

• Solving Logarithmic Equations:

  • \log8 2x + \log8 (x + 4) = 2
  • Combine using the product rule: \log_8 [2x(x + 4)] = 2
  • Convert to exponential form: 2x(x + 4) = 8^2
  • 2x^2 + 8x = 64
  • 2x^2 + 8x - 64 = 0
  • x^2 + 4x - 32 = 0
  • (x + 8)(x - 4) = 0
  • x = -8, 4
  • Since logarithm can not be defined for negative numbers: x = 4

• Newton's Law of Cooling:

  • T = (T0 - TR)e^{-rt} + T_R
    • T_0 = 375°F
    • T_R = 66°F
    • T = 138°F at t = 15 minutes
  • Find r:
    • 138 = (375 - 66)e^{-15r} + 66
    • 72 = 309e^{-15r}
    • e^{-15r} = \frac{72}{309}
    • -15r = \ln(\frac{72}{309})
    • r = -\frac{1}{15} \ln(\frac{72}{309}) \approx 0.105
  • Find T after 25 minutes:
    • T = (375 - 66)e^{-0.105(25)} + 66
    • T \approx 93°F

• Writing Exponential Functions:

  • y = ab^x
  • Given points (1, 3) and (3, 75):
    • 3 = ab^1
    • 75 = ab^3
  • Divide the second equation by the first:
    • \frac{75}{3} = \frac{ab^3}{ab}
    • 25 = b^2
    • b = 5
  • Substitute back into the first equation:
    • 3 = a(5)
    • a = \frac{3}{5}=0.6
  • y = 0.6(5^x)

• Inverse Variation:

  • y = \frac{k}{x}
  • y = 6 when x = 3
  • 6 = \frac{k}{3}
  • k = 18
  • y = \frac{18}{x}
  • When x = 8
  • y = \frac{18}{8} = \frac{9}{4}

Rational Functions

• Graphing Rational Functions:

  • g(x) = -\frac{7}{x + 1} - 6
    • Domain: all real numbers except -1
    • Range: all real numbers except -6

• Graphing Rational Functions:

  • f(x) = \frac{3x - 1}{x - 2}
    • Domain: all real numbers except 2
    • Range: all real numbers except 3

• Simplifying Rational Expressions:

  • \frac{x^2 - 5x - 14}{x^2 + 4x + 4} = \frac{(x - 7)(x + 2)}{(x + 2)(x + 2)} = \frac{x - 7}{x + 2}

Operations with Rational Expressions

• Multiplying Rational Expressions:

  • \frac{3x^9y^8}{5x^5y^4} \cdot \frac{5x^4y^2}{12x^3y^4} = \frac{3 \cdot 5 x^{9+4} y^{8+2}}{5 \cdot 12 x^{5+3} y^{4+4}}
  • = \frac{15x^{13}y^{10}}{60x^8y^8} = \frac{x^{13-8}y^{10-8}}{4} = \frac{x^5y^2}{4}

• Multiplying Rational Expressions:

  • \frac{3x^2 - 24x}{x^2 - 6x - 16} \cdot \frac{x^2 - 8x - 20}{5x^2}
  • = \frac{3x(x - 8)}{(x - 8)(x + 2)} \cdot \frac{(x - 10)(x + 2)}{5x^2}
  • = \frac{3x(x - 8)(x - 10)(x + 2)}{5x^2(x - 8)(x + 2)}
  • = \frac{3(x - 10)}{5x}

• Dividing Rational Expressions:

  • \frac{3x}{ -4x - 24} \div \frac{x^2 + 3x}{x^2 + 9x + 18} = \frac{3x}{-4(x+6)} \div \frac{x(x+3)}{(x+6)(x+3)}
  • = \frac{3x}{-4(x+6)} \cdot \frac{(x+6)(x+3)}{x(x+3)}
  • = \frac{3x(x+6)(x+3)}{-4x(x+6)(x+3)} = -\frac{3}{4}

Rational Equations

• Solving Rational Equations:

  • \frac{x + 2}{2x} = \frac{6 + x}{2x + 9}
  • Cross multiply: (x+2)(2x+9) = 2x(6+x)
  • 2x^2 + 9x + 4x + 18 = 12x + 2x^2
  • 2x^2 + 13x + 18 = 12x + 2x^2
  • x = -18

• Solving Rational Equations:

  • - \frac{8}{x+7} = \frac{6}{x} - \frac{11}{1}
  • Multiply by x(x+7) to eliminate the fractions:
    • -8x = 6(x+7) - 11x(x+7)
    • -8x = 6x + 42 - 11x^2 - 77x
    • 11x^2 + 63x - 42 = 0
    • (11x-6)(x+7)=0
    • x= 6/11, -7

• Solving Rational Equations:

  • \frac{20}{x-5} = \frac{32x^2}{x^2 - 25} - \frac{16x}{x+5}
  • \frac{20}{x-5} = \frac{32x^2}{(x-5)(x+5)} - \frac{16x}{x+5}
  • Multiply both sides by (x-5)(x+5)
    • 20(x+5) = 32x^2 - 16x(x-5)
    • 20x + 100 = 32x^2 - 16x^2 + 80x
    • 16x^2 + 60x - 100 = 0
    • 4x^2 + 15x - 25 = 0
    • (4x + 20)(x - 5/4)
    • x=-5, 5/4

Trigonometry

• Trigonometric Functions:

  • \cot \theta = \frac{8}{3}

  • In a right triangle: \cot \theta = \frac{\text{adjacent}}{\text{opposite}}

  • Let adjacent side = 8, opposite side = 3

  • Hypotenuse = \sqrt{8^2 + 3^2} = \sqrt{64 + 9} = \sqrt{73}

  • \tan \theta = \frac{3}{8}

  • \sin \theta = \frac{3}{\sqrt{73}} = \frac{3\sqrt{73}}{73}

  • \csc \theta = \frac{\sqrt{73}}{3}

  • \cos \theta = \frac{8}{\sqrt{73}} = \frac{8\sqrt{73}}{73}

  • \sec \theta = \frac{\sqrt{73}}{8}

• Solving Right Triangles

  • \angle D = 26°

  • \angle F = 90°

  • e = 7

  • \angle E = 180° - (90° + 26°) = 64°

  • \frac{d}{\sin D} = \frac{e}{\sin E} = \frac{f}{\sin F}

  • \frac{d}{\sin 26°} = \frac{7}{\sin 64°}

  • d = \frac{7 \sin 26°}{\sin 64°} \approx 3.41

  • \frac{f}{\sin 90°} = \frac{7}{\sin 64°}

  • f = \frac{7}{\sin 64°} \approx 7.79

• Radian Conversion:

  • 325° \cdot \frac{\pi}{180°} = \frac{325\pi}{180} = \frac{65\pi}{36} \text{ radians}

• Unit Circle:

  • \theta = -540° = -540° + 360° + 360° = 180°
  • sin(-540°) = 0
  • csc(-540°) = undefined
  • cos(-540°) = -1
  • sec(-540°) = -1
  • tan(-540°) = 0
  • cot(-540°) = undefined

• Reference Angles:

  • \cos \frac{5\pi}{3}
  • Reference angle: \frac{5\pi}{3}
  • Quadrant: IV
  • \cos \frac{\pi}{3} = \frac{1}{2} = 0.5

Trigonometric Functions and Graphs

• Period of Cosine:

  • g(x) = \cos 8x
  • Period = \frac{2\pi}{B} = \frac{2\pi}{8} = \frac{\pi}{4}
  • Horizontal shrink by a factor of \frac{1}{8}

• Transformations of Sine Functions:

  • g(x) = 2\sin 3x + 1
  • Amplitude = 2
  • Period = \frac{2\pi}{3}
  • Vertical shift up by 1

• Trigonometric Functions:

  • \cos \theta = -\frac{5}{13}
  • \frac{\pi}{2} < \theta < \pi
    • \theta is in Quadrant II
  • \sin \theta = \frac{12}{13}
  • \csc \theta = \frac{13}{12}
  • \tan \theta = -\frac{12}{5}
  • \cot \theta = -\frac{5}{12}
  • \sec \theta = -\frac{13}{5}$$