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}$$