Algebra 2/Trig Semester Review Notes

Section 6.1: Roots and Radical Expressions

• Value of -0.027:

  • The cube root of -0.027 is -0.3, since $(-0.3)^3 = -0.027$.

• Simplified form of $\sqrt{4x^2y^4}$:

  • $\sqrt{4x^2y^4} = 2|x|y^2$ because the square root of $4$ is $2$, the square root of $x^2$ is $|x|$ (absolute value of $x$), and the square root of $y^4$ is $y^2$.

• Real solutions of the equation $x^4 = 81$:

  • The real solutions are $x = -3$ and $x = 3$, since $(-3)^4 = 81$ and $(3)^4 = 81$.

Section 6.2: Multiplying and Dividing Radical Expressions

• Simplest form of $-4\sqrt{49x^5}$ / $\sqrt{7x^2}$:

  • $-4\sqrt{49x^5} / \sqrt{7x^2} = -4 * 7x^2\sqrt{x} / (x\sqrt{7}) = -28x^2\sqrt{x} / (x\sqrt{7}) = -4x\sqrt{49x^4}$ simplified form can be $-7x\sqrt{7x}$

• Simplest form of $\sqrt{80x^7y^6}$:

  • $\sqrt{80x^7y^6} = \sqrt{16 * 5 * x^6 * x * y^6} = 4x^3y^3\sqrt{5x}$.

• Simplest form of $\sqrt{25xy^2} * \sqrt{15x^4}$:

  • $\sqrt{25xy^2} * \sqrt{15x^4} = 5y\sqrt{x} * x^2\sqrt{15} = 5x^2y\sqrt{15x}$.

• Simplest form of $\sqrt{75x^5} / \sqrt{12xy^2}$:

  • $\sqrt{75x^5} / \sqrt{12xy^2} = (5x^2\sqrt{3x}) / (2y\sqrt{3x}) = (5x^2) / (2y)$.

• Simplest form of $\sqrt{4xy^2} / \sqrt{2xy^2}$:

  • $\sqrt{4xy^2} / \sqrt{2xy^2} = \sqrt{2}$.

Section 6.3: Binomial Radical Expressions

• Simplest form of $2\sqrt{72} - 3\sqrt{32}$:

  • $2\sqrt{72} - 3\sqrt{32} = 2\sqrt{36 * 2} - 3\sqrt{16 * 2} = 2 * 6\sqrt{2} - 3 * 4\sqrt{2} = 12\sqrt{2} - 12\sqrt{2} = 0$.

• Simplest form of $(2 - \sqrt{7})(1 + 2\sqrt{7})$:

  • $(2 - \sqrt{7})(1 + 2\sqrt{7}) = 2 + 4\sqrt{7} - \sqrt{7} - 2 * 7 = 2 + 3\sqrt{7} - 14 = -12 + 3\sqrt{7}$.

• Simplest form of $(\sqrt{2} + \sqrt{7})(\sqrt{2} - \sqrt{7})$:

  • $(\sqrt{2} + \sqrt{7})(\sqrt{2} - \sqrt{7}) = 2 - 7 = -5$.

• Simplest form of $7 / (2 + \sqrt{5})$:

  • $7 / (2 + \sqrt{5}) = 7 * (2 - \sqrt{5}) / ((2 + \sqrt{5})(2 - \sqrt{5})) = 7(2 - \sqrt{5}) / (4 - 5) = 7(2 - \sqrt{5}) / (-1) = -14 + 7\sqrt{5}$.

• Simplest form of $8\sqrt{5} - 4\sqrt{5} - 2\sqrt{5}$:

  • $8\sqrt{5} - 4\sqrt{5} - 2\sqrt{5} = (8 - 4 - 2)\sqrt{5} = 2\sqrt{5}$.

Section 6.4: Rational Exponents

• Simplest form of $125^{2/3}$:

  • $125^{2/3} = (5^3)^{2/3} = 5^2 = 25$.

• Simplest form of $x^{1/3} * y^{2/3}$:

  • There missing a options to answer in the text. Need more information to answer.

• Simplest form of $x^{1/3} / x^{3/4}$:

  • $x^{1/3} / x^{3/4} = x^{(1/3 - 3/4)} = x^{(4/12 - 9/12)} = x^{-5/12} = 1 / x^{5/12}$.

• Simplest form of $(x^2y^4)^{1/2}$:

  • $(x^2y^4)^{1/2} = xy^2$ .

• Simplest form of $(-32x^{10}y^{35})^{1/5}$:

  • $(-32x^{10}y^{35})^{1/5} = -2x^2y^7$.

Section 6.5: Solving Square Root and Other Radical Equations

• Solve for x: $\sqrt{2x - 4} - 3 = 1$

  • $\sqrt{2x - 4} = 4$
  • $2x - 4 = 16$
  • $2x = 20$
  • $x = 10$

• Solve for x: $4(x - 2)^{3/2} = 100$

  • $(x - 2)^{3/2} = 25$
  • $x - 2 = 25^{2/3}$
  • $x = 25^{2/3} + 2$

• Solve for x: $12x - 6 = 3 - x$

  • There is no radical in this equation.
  • $13x = 9$
  • $x = 9/13$

• Solve for x: $2(x+3)^{3/5} = 54$

  • $(x+3)^{3/5} = 27$
  • $x+3 = 27^{5/3}$
  • $x+3 = (3^3)^{5/3}$
  • $x+3 = 3^5$
  • $x+3 = 243$
  • $x = 240$

Section 6.7: Inverse Relations and Functions

• Inverse of the relation: ${( -2, 3), (-1, 1), (0, -1), (2, -2) }$

  • To find the inverse, swap the x and y coordinates of each ordered pair.
  • Inverse: ${(3, -2), (1, -1), (-1, 0), (-2, 2) }$

• Inverse of the function: $y = 5(x - 3)$

  • Swap x and y: $x = 5(y - 3)$
  • Solve for y: $x/5 = y - 3$
  • $y = (x/5) + 3$

• Function with domain $x \ge 5$ that is the inverse of $y = \sqrt{x + 5}$:

  • Swap x and y: $x = \sqrt{y + 5}$
  • Solve for y: $x^2 = y + 5$
  • $y = x^2 - 5$
  • Since the original function has a range of $y \ge 0$, the inverse has a domain of $x \ge 0$. But there stated domain is $x \ge 5$. Need more info to solve.
  • I Think it's a mistake and the answer is $y = x^2 - 5$

• Domain and range of the inverse of the function: $y = \sqrt{x - 5}$

  • The original function has a domain of $x \ge 5$ and a range of $y \ge 0$.
  • Therefore, the inverse function has a domain of $x \ge 0$ and a range of $y \ge 5$.

Section 6.8: Graphing Radical Functions

• Graph of $y = \sqrt{x + 4}$:

  • The graph is a square root function shifted 4 units to the left.

• Graph of $y = \sqrt{x - 3} - 2$:

  • The graph is a square root function shifted 3 units to the right and 2 units down.

• Graph of $y = 1 - \sqrt{x + 3}$:

  • The graph is a square root function shifted 3 units to the left, reflected over the x-axis, and shifted up 1 unit.

• Description of $y = \sqrt{9x - 3}$ to make it easy to graph using transformations of its parent function:

  • $y = \sqrt{9x - 3} = \sqrt{9(x - 1/3)} = 3\sqrt{x - 1/3}$. This is the graph of $y = 3\sqrt{x}$, shifted right 1/3 unit.

Chapter 7: Semester Review

Section 7.1: Exploring Exponential Models

• Function representing exponential decay with a y-intercept of 2:

  • Exponential decay has a base between 0 and 1. A y-intercept of 2 means the function is of the form $y = 2(b)^x$ where 0 < b < 1.
  • $y= 2(\frac{1}{2})^x$

• Determine whether each function represents exponential growth or decay, and determine the y-intercept for each function.

  • a. $y = 3(7)^x$: Exponential growth, y-intercept is 3.
  • b. $y = 4(2.5)^x$: Exponential growth, y-intercept is 4.
  • c. $y = 5(0.75)^x$: Exponential decay, y-intercept is 5.
  • d. $y = 0.5(0.2)^x$: Exponential decay, y-intercept is 0.5.

• Deposit of $3000 in a savings account with a 4% annual interest rate after 10 years:

  • $A = P(1 + r)^t$ where A is the amount after t years, P is the principal, r is the interest rate, and t is the time in years.
  • A = 3000(1 + 0.04)^{10} = 3000(1.04)^{10} \approx $4440.73

• Population of Bainsville is 2000. The population is supposed to grow by 10% each year for the next 5 years. How many people will live in Bainsville in 5 years?

  • $A = P(1 + r)^t$ $A = 2000(1 + 0.10)^{5} = 2000(1.10)^{5} \approx 3221$

• Manufacturer bought a new rolling press for $48,000. It has depreciated in value at an annual rate of 15%. What is its value 5 years after purchase? Round to the nearest hundred dollars.

  • $A = P(1 - r)^t$ $A = 48000(1 - 0.15)^{5} = 48000(0.85)^{5} \approx 21,300$

Section 7.2: Properties of Exponential Functions

• Sarah received a paycheck for $1200. She deposited 1/4 of the money into a bank account. The account has an interest rate of 6% compounded continuously. This is the first and last deposit that Sarah makes into this account. How much money will be in the account in 15 years?

  • Principal Deposit: 1200 * (1/4) = $300
  • Formula for the continuous compounded interest is: $A = Pe^{rt}$ , where P is the principal, r is the annual interest rate, and t is the time in years.
  • A = 300 * e^{0.06 * 15} = $300 * e^{0.9} \approx $736.64

• Bram invested $10,000 in an account that earns simple 5% interest annually.

  • a. How much interest does the account earn in the first 10 years? Round to the nearest dollar.
    • Simple Interest Formula: $I = Prt$ , where P is the principal, r is the annual interest rate, and t is the time in years.
    • I= 10000 * 0.05 * 10 = $5000
  • b. How much more would the account earn in interest in the first 10 years if the interest compounded continuously? Round to the nearest dollar.
    • $A= Pe^{rt}$, where P is the principal, r is the annual interest rate, and t is the time in years.
    • $A = 10000 * e^{0.05 * 10}$
    • $A = 10000 * e^{0.5}$
    • A \approx $16,487.21
    • Interest Earned with Compound Continuously: Interest = A - P = $16487.21 - $10000 = $6487.21
    • Additional Interest Earned: $6487.21 - $5000 = $1487.21

Section 7.3: Logarithmic Functions as Inverses

• Write each equation in logarithmic form.

  • $100 = 10^2$ is equivalent to $log_{10}100 = 2$
  • $9^3 = 729$ is equivalent to $log_{9}729 = 3$
  • $64^{\frac{4}{3}}$ is equivalent to $log_{64}4 = \frac{2}{3}$
  • $\frac{1}{9} = 3^{-2}$ is equivalent to $log_{3}(\frac{1}{9} = -2)$

• Evaluate each logarithm.

  • $log 1000$ is equivalent to $log_{10}1000 = 3$
  • $log_{4}256 = 4$ is because $4^4 = 256$
  • $log_{27}9 = \frac{2}{3}$ is because $(27)^{\frac{2}{3}} = 9$
  • $log_{\frac{1}{2}}6 = -2$ is because $(\frac{1}{2})^{-2} = 4$

Section 7.4: Properties of Logarithms

• Write each expression as a single logarithm.

  • $log 8 + log 3 = log(8*3) = log 24$
  • $4(log<em>{2}x + log</em>{2}3) = 4log<em>{2}(x*3) = log</em>{2}(3x)^4 = log_{2}(81x^4)$
  • $log 4 + log 2 - log5 = log((4*2)/5) = log(\frac{8}{5})$

• Expand each logarithm.

  • $log<em>{b}3m^2p^2 = log</em>{b}3 + 2log<em>{b}m + 2log</em>{b}p$
  • $log(\frac{(xy)^4}{2}) = 4logx + 4logy - log2$
  • $log(4mn)^3 = 3log4 + 3logm + 3logn$

• Which expression is the correct expansion of $log_{4}(3x)^2$ ?

  • $2(log<em>{4}3 + log</em>{4}x)$

• Which statement correctly expresses $4log_{3}x + 7logy$ as a single logarithm?

  • $log_{3}(x^4y^7)$

Section 7.5: Exponential and Logarithmic Equations

• If $9^x = 243$, what is the value of x?

  • $9^x = 243$ can be written as $(3^2)^x = 3^5$
  • $3^{2x} = 3^5$
  • $2x = 5$
  • $x = \frac{5}{2} = 2.5$

• If $2^{3x+2} = 64$, what is the value of x?

  • $2^{3x+2} = 64$
  • $2^{3x+2} = 2^6$
  • $3x+2 = 6$
  • $3x = 4$
  • $x = \frac{4}{3}$

• If $log(3x+25) = 2$, what is the value of x?

  • Assuming the base is 10, $log_{10}(3x+25) = 2$
  • $3x+25 = 10^2$
  • $3x+25 = 100$
  • $3x = 75$
  • $x = 25$

• Which best approximates the solution of $16^{2x} = 124$?

  • $16^{2x} = 124$
  • $log(16^{2x}) = log(124)$
  • $2x*log(16) = log(124)$
  • $x = \frac{log124}{2log16}$
  • $x \approx 1.150$

Section 7.6: Natural Logarithms

• In 2007, the population of Tallahassee, Florida was 168,979. Some researchers believe that the population of Tallahassee will increase at a rate of 1% each year for the 10 years following this. If the researchers are correct, how many years will it take for the population of Tallahassee to reach 180,000?

  • $P(t) = P_0e^{rt}$
  • $180000 = 168979e^{0.01t}$
  • $\frac{180000}{168979} = e^{0.01t}$
  • $ln(\frac{180000}{168979}) = 0.01t$
  • $t = \frac{ln(\frac{180000}{168979})}{0.01}$
  • $t \approx 6.34$

Algebra 2/Trig Chapter 13: Semester Review

Trigonometry Problems

Coterminal Angles:

1. Coterminal Angles: Angles that share the same terminal side.

   • a. $20°$
     • Positive coterminal angle: $20° + 360° = 380°$
     • Negative coterminal angle: $20° - 360° = -340°$
   • b. $265°$
     • Positive coterminal angle: $265° + 360° = 625°$
     • Negative coterminal angle: $265° - 360° = -95°$
   • c. $305°$
     • Positive coterminal angle: $305° + 360° = 665°$
     • Negative coterminal angle: $305° - 360° = -55°$

Coordinates on the Unit Circle:

2. Unit Circle: A circle with a radius of 1 centered at the origin.

   • a. $-150°$
     • Coordinates: $(\frac{-\sqrt{3}}{2}, \frac{-1}{2})$
   • b. $30°$
     • Coordinates: $(\frac{\sqrt{3}}{2}, \frac{1}{2})$
   • c. $120°$
     • Coordinates: $(\frac{-1}{2}, \frac{\sqrt{3}}{2})$

Radian to Degree Conversion:

3. Radian-Degree Conversion: To convert radians to degrees, multiply by $\frac{180}{π}$.

   • a. $\frac{π}{6}$
     • Degrees: $\frac{π}{6} * \frac{180}{π} = 30°$
   • b. $\frac{3π}{5}$
     • Degrees: $\frac{3π}{5} * \frac{180}{π} = 108°$
   • c. $\frac{5π}{12}$
     • Degrees: $\frac{5π}{12} * \frac{180}{π} = 75°$

Exact Values of Cosine and Sine:

4. Cosine and Sine Values: Determining exact values for given angles.

   • a. $\frac{π}{6}$
     • $cos(\frac{π}{6}) = \frac{\sqrt{3}}{2}, sin(\frac{π}{6}) = \frac{1}{2}$
   • b. $\frac{3π}{4}$
     • $cos(\frac{3π}{4}) = \frac{-\sqrt{2}}{2}, sin(\frac{3π}{4}) = \frac{\sqrt{2}}{2}$
   • c. $\frac{2π}{3}$
     • $cos(\frac{2π}{3}) = \frac{-1}{2}, sin(\frac{2π}{3}) = \frac{\sqrt{3}}{2}$
   • d. $-\frac{π}{3}$
     • $cos(-\frac{π}{3}) = \frac{1}{2}, sin(-\frac{π}{3}) = -\frac{\sqrt{3}}{2}$
   • e. $\frac{5π}{6}$
     • $cos(\frac{5π}{6}) = \frac{-\sqrt{3}}{2}, sin(\frac{5π}{6}) = \frac{1}{2}$
   • f. $\frac{7π}{4}$
     • $cos(\frac{7π}{4}) = \frac{\sqrt{2}}{2}, sin(\frac{7π}{4}) = \frac{-\sqrt{2}}{2}$

Amplitude and Period of Sine Functions:

5. Amplitude and Period: Key characteristics of sine functions.

   • a. $y = \frac{1}{2}sin(3θ)$
     • Amplitude: $\frac{1}{2}$, Period: $\frac{2π}{3}$
   • b. $y = sin(5θ)$
     • Amplitude: $1$, Period: $\frac{2π}{5}$
   • c. $y = 4sin(\frac{π}{3}θ)$
     • Amplitude: $4$, Period: $\frac{2π}{\frac{π}{3}} = 6$
   • d. $y = -\frac{3}{5}sin(θ)$
     • Amplitude: $\frac{3}{5}$, Period: $2π$
   • e. $y = -2sin(\frac{1}{3}θ)$
     • Amplitude: $2$, Period: $\frac{2π}{\frac{1}{3}} = 6π$
   • f. $y = πsin(2θ)$
     • Amplitude: $π$, Period: $\frac{2π}{2} = π$

Graphs of Cosine Functions:

6. Sketching Cosine Functions: Over the interval from 0 to $2π$.

   • a. $y = cos(2θ)$
     • The graph of $y = cos(2θ)$ completes two full cycles in the interval from 0 to $2π$.
   • b. $y = -3cos(\frac{1}{2}θ)$
     • The graph of $y = -3cos(\frac{1}{2}θ)$ is vertically stretched by a factor of 3, reflected over the x-axis, and completes half a cycle in the interval from 0 to $2π$.

Exact Values of Tangent Functions:

7. Tangent Values: Using the unit circle.

   • a. $tan(\frac{π}{6}) = \frac{\sqrt{3}}{3}$
   • b. $tan(\frac{π}{3}) = \sqrt{3}$
   • c. $tan(\frac{3π}{4}) = -1$
   • d. $tan(\frac{5π}{6}) = -\frac{\sqrt{3}}{3}$
   • e. $tan(-\frac{π}{4}) = -1$

Period and Asymptotes of Tangent Functions:

8. Tangent Function Analysis: Period and asymptote determination.

   • a. $y = 2tan(\frac{θ}{2})$
     • Period: $\frac{π}{\frac{1}{2}} = 2π$
     • Asymptotes: at $θ = π$ and $θ = 3π$
   • b. $y = -tan(2θ)$
     • Period: $\frac{π}{2}$
     • Asymptotes: at $θ = \frac{π}{4}$ and $θ = \frac{3π}{4}$
   • c. $y = 4tan(2θ)$
     • Period: $\frac{π}{2}$
     • Asymptotes: at $θ = \frac{π}{4}$ and $θ = \frac{3π}{4}$

Transformations of Trigonometric Graphs:

9. Transformations: Describing graph changes.

   • a. $y = 3cos x + 2$
     • Vertical stretch by a factor of 3, vertical shift up 2.
   • b. $y = -3sin 2x - 7$
     • Vertical stretch by a factor of 3, reflection over the x-axis, horizontal compression by a factor of 2, vertical shift down 7.
   • c. $y = 5cos(\frac{1}{2}x) + 4$
     • Vertical stretch by a factor of 5, horizontal stretch by a factor of 2, vertical shift up 4.

Equations from Transformations:

10. Writing Equations: Based on described transformations.

    • a. Sine function reflected over the x-axis, stretched vertically by a factor of 4, shifted up 5:
      y = -4sin(x) + 5
    • b. Cosine function stretched vertically by a factor of 2, horizontally stretched by a factor of 3, and shifted down 4:
      y = 2cos(\frac{1}{3}x) - 4

Writing Cosine Functions from Graphs:

11. Cosine Equations: Matching graphs.

    • Analyze the amplitude, period, and any vertical shifts to write the appropriate cosine function for each graph.

Writing Sine Functions from Graphs:

12. Sine Equations: Matching graphs.

    • Analyze the amplitude, period, and any vertical shifts to write the appropriate sine function for each graph.

Graphing Sine Functions:

13. Graphing Procedures: Over the interval from 0 to $2π$.

    • a. $y = 4sin x + 5$
      • Graph a sine function with amplitude 4, shifted vertically up by 5 units.
    • b. $y = -2 sin 2x + 2$
      • Graph a sine function with amplitude 2, reflected over the x-axis, compressed horizontally by a factor of 2, and shifted vertically up by 2 units.
    • c. $y = - sin x - 4$
      • Graph a sine function reflected over the x-axis, and shifted vertically down by 4 units.

Writing Tangent Functions from Graphs:

14. Tangent Equations: Analyze the period and asymptotes to write the tangent function for each graph.

Period Lengths of Trigonometric Functions:

15. Period Lengths:

    • a. $y = -2 sin 2x + 2$
      • Period: $\frac{2π}{2} = π$
    • b. $y = 4cos(\frac{1}{3}x) - 7$
      • Period: $\frac{2π}{\frac{1}{3}} = 6π$
    • c. $y = tan πx$
      • Period: $\frac{π}{π} = 1$