alg 2 review

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 $\sqrt{4} = 2$, $\sqrt{x^2} = |x|$ (absolute value of x), and $\sqrt{y^4} = y^2$.

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

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

Section 6.2: Multiplying and Dividing Radical Expressions

• Simplest form of $-4\sqrt{9x^{\frac{13}{7x^2}}}$:

  • $-4\sqrt{\frac{9x^{13}}{7x^2}} = -4\sqrt{\frac{9}{7}x^{11}} = -4 \cdot \frac{3}{\sqrt{7}} x^{\frac{11}{2}}$

• Simplest form of $\sqrt{80x^7y^6}$:

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

• Simplest form of $\sqrt[3]{25xy^2} \cdot \sqrt[3]{15x^4}$:

  • $\sqrt[3]{25xy^2 \cdot 15x^2} = \sqrt[3]{75x^3y^2} = x\sqrt[3]{75y^2} = 5x\sqrt[3]{3y^2}$.

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

  • $\frac{\sqrt{75x^5}}{\sqrt{12xy^2}} = \sqrt{\frac{75x^5}{12xy^2}} = \sqrt{\frac{25x^4}{4y^2}} = \frac{5x^2}{2|y|}$.

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

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

Section 6.3: Binomial Radical Expressions

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

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

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

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

• Simplest form of $\frac{7}{2+\sqrt{5}}$

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

• Simplest form of $\sqrt[3]{5} - 4\sqrt[3]{40} - 2\sqrt[3]{135}$:

  • $\sqrt[3]{5} - 4\sqrt[3]{40} - 2\sqrt[3]{135} = \sqrt[3]{5} - 4\sqrt[3]{8\cdot 5} - 2\sqrt[3]{27 \cdot 5} = \sqrt[3]{5} - 4(2\sqrt[3]{5}) - 2(3\sqrt[3]{5}) = \sqrt[3]{5} - 8\sqrt[3]{5} - 6\sqrt[3]{5} = -13\sqrt[3]{5}$.

Section 6.4: Rational Exponents

• Simplest form of $12^3 \cdot 45^3 \cdot 50^3$:

  • $12^\frac{2}{3} \cdot 45^\frac{2}{3} \cdot 50^\frac{2}{3} = (12 \cdot 45 \cdot 50)^\frac{2}{3} = (27000)^\frac{2}{3} = (30^3)^\frac{2}{3} = 30^2 = 900$.

• Simplest form of $x^\frac{1}{3} y^\frac{2}{3}$:

  • $x^\frac{1}{3} y^\frac{2}{3} = (xy^2)^\frac{1}{3} = \sqrt[3]{xy^2}$.

• Simplest form of $x^\frac{1}{3} \cdot x^\frac{2}{4}$:

  • $x^\frac{1}{3} \cdot x^\frac{2}{4} = x^\frac{1}{3} \cdot x^\frac{1}{2} = x^{\frac{1}{3}+\frac{1}{2}} = x^{\frac{2+3}{6}} = x^{\frac{5}{6}} = \sqrt[6]{x^5}$.

• Simplest form of $\frac{x^\frac{1}{3}}{x^2y^4}$

  • $\frac{x^\frac{1}{3}}{x^2y^4} = x^\frac{1}{3} x^{-2} y^{-4} = x^{\frac{1}{3} - 2} y^{-4} = x^{\frac{1-6}{3}} y^{-4} = x^{-\frac{5}{3}} y^{-4} = \frac{1}{x^\frac{5}{3}y^4}$.

• Simplest form of $(-32x^{10}y^{15})^\frac{1}{5}$:

  • $(-32x^{10}y^{15})^\frac{1}{5} = (-2^5 x^{10}y^{15})^\frac{1}{5} = -2x^{\frac{10}{5}}y^{\frac{15}{5}} = -2x^2y^3$.

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: $4(x-2)^\frac{3}{2} = 100$

  • $(x-2)^\frac{3}{2} = 25$

  • $x-2 = 25^\frac{2}{3}$

  • $x = 25^\frac{2}{3} + 2$

• Solve for x: $\sqrt[3]{12x-6} = 3 - x$

  • $12x-6 = (3-x)^3$

  • $12x-6 = 27 - 27x + 9x^2 - x^3$

  • $x^3 - 9x^2 + 39x - 33 = 0$

• Solve for x: $2(x+3)^\frac{4}{3} = 54$

  • $(x+3)^\frac{4}{3} = 27$

  • $x+3 = 27^\frac{3}{4}$

  • $x = 27^\frac{3}{4} - 3$

• Solve for x: $5x - 3 = \sqrt{2x+3}$

  • $(5x-3)^2 = 2x+3$

  • $25x^2 - 30x + 9 = 2x+3$

  • $25x^2 - 32x + 6 = 0$

Section 6.7: Inverse Relations and Functions

• What is the inverse of the relation?

  • To find the inverse of a relation, swap the x and y coordinates. If the original relation is {(x, y)}, the inverse is {(y, x)}.

• What is the inverse of the function? $y = 5(x-3)$

  • $y = 5(x-3)$

• $x = 5(y-3)$

  • $\frac{x}{5}= y -3$

  • $y = \frac{x}{5} + 3$

• What function with domain $x \ge 5$ is the inverse of $y=\sqrt{x+5}$?

  • Original: $y = \sqrt{x+5}$

  • Inverse: $x = \sqrt{y+5}$

  • $x^2 = y + 5$

  • $y = x^2 - 5$

  • Since the original function has a range of $y \ge 0$, the inverse function has a domain of $x \ge 0$. However, since the problem specifies that the domain is $x \ge 5$, there might be an error in the problem statement because the domain should be $x\ge0$.

• What is the domain and range of the inverse of the function $y = \sqrt{x} - 5$?

  • The original function $y = \sqrt{x} - 5$ has a domain of $x \ge 0$ and a range of $y \ge -5$.

  • The inverse function will have a domain of $x \ge -5$ and a range of $y \ge 0$.

Section 6.8: Graphing Radical Functions

• $y = \sqrt{x+4}$:

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

• $y = \sqrt{x-3} - 2$:

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

• $y = 1 - \sqrt{x+3}$:

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

• $y = \sqrt{9x - 3}$:

• To make the function easier to graph using transformations, factor out the 9: $y = \sqrt{9(x - \frac{1}{3})} = 3\sqrt{x-\frac{1}{3}}$

  • This represents the graph of $y = 3\sqrt{x}$, shifted right $\frac{1}{3}$ units.

Chapter 7: Semester Review

Section 7.1: Exploring Exponential Models

• Exponential Decay with 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. Therefore, $y = 2(\frac{1}{2})^x$ represents exponential decay with a y-intercept of 2.

• Exponential Growth or Decay:

  • a. $y = 3(7)^x$: Exponential growth, y-intercept = 3

  • b. $y = 4(2.5)^x$: Exponential growth, y-intercept = 4

  • c. $y = 5(0.75)^x$: Exponential decay, y-intercept = 5

  • d. $y = 0.5(0.2)^x$: Exponential decay, y-intercept = 0.5

• Compound Interest:

  • The formula for compound interest is $A = P(1 + r)^t$, where:

    • A = amount after t years

    • P = principal (initial deposit)

    • r = annual interest rate

    • t = number of years

  • Given P = $3000, r = 4% = 0.04, t = 10 years:

  • A = 3000(1 + 0.04)^{10} = 3000(1.04)^{10} \approx $4440.73

• Population Growth:

  • The population of Bainsville is growing by 10% each year. The formula for population growth is $P(t) = P_0(1+r)^t$

  • $P_0 = 2000$ and $r=0.10$, $t = 5$ years.

  • $P(5)=2000(1+0.10)^5 = 2000(1.1)^5 \approx 3221$

• Depreciation:

  • The formula for depreciation is $V(t) = V_0(1-r)^t$

  • Given, V_0 = $48,000, $r = 15$, and $t=5$ years after purchase.

  • V(5) = 48000(1 - 0.15)^5 = 48000(0.85)^5 = 48000(0.4437) \approx $21,300

Section 7.2: Properties of Exponential Functions

• Sarah's Account:

  • Sarah deposits \frac{3}{4} \cdot $1200 = $900 into a bank account.

  • The formula for continuous compounding is $A = Pe^{rt}$

  • P = $900, r = 0.06, t = 15$</p></li><li><p>$A = 900e^{(0.06)(15)} = 900e^{0.9} \approx $2214.12$</p></li></ul></li><li><p>Bram's Investment:</p><ul><li><p>Simple interest formula:$I = PRT

  • Given: P = $10,000, R = 5% = 0.05, T = 10 years

  • a. Interest earned: I = 10000 \cdot 0.05 \cdot 10 = $5000

  • Continuous compounding Formula: $A=Pe^{rt}$

  • b. A = 10000e^{(0.05)(10)} = 10000e^{0.5} \approx $16,487.21

  • Interest Earned= A-P=16487.21-10000 = $6487.21

  • Difference: 6487.21 - 5000 = $1487.21

Section 7.3: Logarithmic Functions as Inverses

• Logarithmic Form:

  • $100 = 10^2$ becomes $\log_{10} 100 = 2$

  • $9^3 = 729$ becomes $\log_9 729 = 3$

  • $64^{\frac{2}{3}} = 16$ becomes $\log_{64} 16 = \frac{2}{3}$

  • $(1/3)^{-3}=27$ becomes $\log_{\frac{1}{3}} 27 = -3$

• Evaluate Logarithms:

  • $\log 1000 = \log_{10} 1000 = 3$ (since $10^3 = 1000$)

  • $\log_4 256 = 4$ (since $4^4 = 256$)

  • $\log_{27} 9 = \frac{2}{3}$ (since $27^{\frac{2}{3}} = (3^3)^{\frac{2}{3}} = 3^2 = 9$)

  • $\log_{125} 5 = \frac{1}{3}$ (since $125^{\frac{1}{3}} = (5^3)^{\frac{1}{3}} = 5$)

Section 7.4: Properties of Logarithms

• Single Logarithm:

  • $\log 8 + \log 3 = \log (8 \cdot 3) = \log 24$

  • $4(\log<em>2 x + \log</em>2 3) = 4\log<em>2 (3x) = \log</em>2 (3x)^4 = \log_2 (81x^4)$

  • $\log 4 + \log 2 - \log 5 = \log(4 \cdot 2) - \log 5 = \log 8 - \log 5 = \log \frac{8}{5}$

• Expand Logarithm:

  • $\log<em>b 3m^2p^3 = \log</em>b 3 + \log<em>b m^2 + \log</em>b p^3 = \log<em>b 3 + 2\log</em>b m + 3\log_b p$

  • $\log \frac{(xy)^4}{z^2} = \log (xy)^4 - \log z^2 = 4\log(xy) - 2\log z = 4(\log x + \log y) - 2\log z = 4\log x + 4\log y - 2\log z$

  • $\log(4mn)^3 = 3 \log(4mn) = 3(\log 4 + \log m + \log n) = 3\log 4 + 3 \log m + 3\log n$

• Correct expansion of $\log_4(3x)^2$:

  • $\log<em>4(3x)^2 = 2 \log</em>4 (3x) = 2(\log<em>4 3 + \log</em>4 x) = 2\log<em>4 3 + 2\log</em>4 x$

• Expressing $4 \log<em>3 x + 7 \log</em>3 y$ as a single logarithm:

  • $4 \log<em>3 x + 7 \log</em>3 y = \log<em>3 x^4 + \log</em>3 y^7 = \log_3 (x^4y^7)$

Section 7.5: Exponential and Logarithmic Equations

• If $9^x = 243$:

  • Since $9 = 3^2$ and $243 = 3^5$, we have $(3^2)^x = 3^5$, which simplifies to $3^{2x} = 3^5$. Thus, $2x = 5$, and $x = \frac{5}{2} = 2.5$

• If $2^{3x+2} = 64$:

  • Since $64 = 2^6$, we have $2^{3x+2} = 2^6$. Thus, $3x+2 = 6$, so $3x = 4$, and $x = \frac{4}{3}$

• If $\log(3x+25) = 2$:

  • Assuming base 10, we have $10^2 = 3x+25$, so $100 = 3x+25$. Then $75 = 3x$, and $x = 25$

• Approximating the solution of $16^{2x} = 124$:

  • Taking the logarithm of both sides: $\log(16^{2x}) = \log(124)$

  • $2x \log(16) = \log(124)$

  • $2x = \frac{\log(124)}{\log(16)}$

  • $x = \frac{\log(124)}{2\log(16)} \approx \frac{2.093}{2(1.204)} \approx \frac{2.093}{2.408} \approx 0.869$

Section 7.6: Natural Logarithms

• Tallahassee Population:

  • The formula for population growth is $P(t) = P_0(1 + r)^t$

  • Given: $P_0 = 168979$, $r = 0.01$, and we want to find t such that $P(t) = 180000$

  • $180000 = 168979(1 + 0.01)^t$

  • $\frac{180000}{168979} = (1.01)^t$

  • Taking the natural logarithm of both sides:

  • $\ln(\frac{180000}{168979}) = \ln((1.01)^t)$

  • $\ln(\frac{180000}{168979}) = t \ln(1.01)$

  • $t = \frac{\ln(\frac{180000}{168979})}{\ln(1.01)} \approx \frac{\ln(1.065)}{0.00995} \approx \frac{0.063}{0.00995} \approx 6.33$

  • It will take approximately 6.33 years for the population to reach 180,000.

Chapter 13: Semester Review

• Coterminal Angles:

• a. 20°: Positive: $20 + 360 = 380°$, Negative: $20 - 360 = -340°$

  • b. 265°: Positive: $265 + 360 = 625°$, Negative: $265 - 360 = -95°$

  • c. 305°: Positive: $305 + 360 = 665°$, Negative: $305 - 360 = -55°$

• Coordinates on the Unit Circle:

  • a. -150°: Reference angle is 30° in the third quadrant. Coordinates: $(\frac{-\sqrt{3}}{2}, -\frac{1}{2})$

  • b. 30°: Coordinates: $(\frac{\sqrt{3}}{2}, \frac{1}{2})$

  • c. 120°: Reference Angle is 60° in the second quadrant. Coordinates: $(\frac{-1}{2}, \frac{\sqrt{3}}{2})$

  • d. -45°: Reference angle is 45° in the fourth quadrant. Coordinates: $(\frac{\sqrt{2}}{2}, \frac{-\sqrt{2}}{2})$

• Degrees to Radians:

  • a. $\frac{\pi}{6} = \frac{\pi}{6} \cdot \frac{180}{\pi} = 30°$

  • b. $\frac{3\pi}{5} = \frac{3\pi}{5} \cdot \frac{180}{\pi} = 108°$

  • c. $\frac{5\pi}{12} = \frac{5\pi}{12} \cdot \frac{180}{\pi} = 75°$

• Values of Cosine and Sine:

  • a. $\theta = \frac{\pi}{6}$: $cos(\frac{\pi}{6}) = \frac{\sqrt{3}}{2}$, $sin(\frac{\pi}{6}) = \frac{1}{2}$

  • b. $\theta = \frac{3\pi}{4}$: $cos(\frac{3\pi}{4}) = -\frac{\sqrt{2}}{2}$, $sin(\frac{3\pi}{4}) = \frac{\sqrt{2}}{2}$

  • c. $\theta = \frac{\pi}{3}$: $cos(\frac{\pi}{3}) = \frac{1}{2}$, $sin(\frac{\pi}{3}) = \frac{\sqrt{3}}{2}$

  • d. $\theta = -\frac{2\pi}{3}$: $cos(-\frac{2\pi}{3}) = -\frac{1}{2}$, $sin(-\frac{2\pi}{3}) = -\frac{\sqrt{3}}{2}$

  • e. $\theta = \frac{5\pi}{6}$: $cos(\frac{5\pi}{6}) = -\frac{\sqrt{3}}{2}$, $sin(\frac{5\pi}{6}) = \frac{1}{2}$

  • f. $\theta = \frac{7\pi}{4}$: $cos(\frac{7\pi}{4}) = \frac{\sqrt{2}}{2}$, $sin(\frac{7\pi}{4}) = -\frac{\sqrt{2}}{2}$

• Amplitude and Period of Sine Functions:

  • a. $y = \frac{1}{2}sin(3\theta)$: Amplitude: $\frac{1}{2}$, Period: $\frac{2\pi}{3}$

  • b. $y = sin(5\theta)$: Amplitude: 1, Period: $\frac{2\pi}{5}$

  • c. $y = 4sin(\frac{\pi}{\theta})$ -- looks like an error it should be y = 4sin(\pi\theta): Amplitude: 4, Period: $\frac{2\pi}{\pi} = 2$

  • d. $y = -sin(\theta)$: Amplitude: 1, Period: $2\pi$

  • e. $y = -2sin(-\theta)$: Amplitude: 2, Period: $2\pi$

  • f. $y = \pi sin(2\theta)$: Amplitude: $\pi$, Period: $\frac{2\pi}{2} = \pi$

• Tangent Values:

  • a. $\tan(\frac{\pi}{6}) = \frac{\sqrt{3}}{3}$

  • b. $\tan(\frac{\pi}{3}) = \sqrt{3}$

  • c. $\tan(\frac{3\pi}{4}) = -1$

  • d. $\tan(\pi) = 0$

  • e. $\tan(\frac{5\pi}{6}) = -\frac{\sqrt{3}}{3}$

  • f. $\tan(-\frac{\pi}{4}) = -1$

• Period and Asymptotes of Tangent Functions: *General form of a tangent function $y = A tan(B\theta + C) + D$;

  • The period is $\frac{\pi}{|B|}$, where B is the coefficient of $\theta$
    *The Vertical asymptotes are at $B\theta + C = -\frac{\pi}{2}$ and $B\theta + C = \frac{\pi}{2}$

  • a. $y = 2\tan(\frac{\theta}{2})$:

    • Period: $\frac{\pi}{\frac{1}{2}} = 2\pi$

    • Asymptotes: $\frac{\theta}{2} = -\frac{\pi}{2}$ leading to $\theta = -\pi$
      $\frac{\theta}{2} = \frac{\pi}{2}$ leading to $\theta = \pi$

  • b. $y = -\tan(\frac{\theta}{2})$:

    • Period: $\frac{\pi}{\frac{1}{2}} = 2\pi$

    • Asymptotes: Same as (a): $\theta = -\pi$, $\theta = \pi$

  • c. $y = 4\tan(2\theta)$:

    • Period: $\frac{\pi}{2}$

    • Asymptotes: $2\theta = -\frac{\pi}{2}$ leading to $\theta = -\frac{\pi}{4}$

    • $2\theta = \frac{\pi}{2}$ leading to $\theta = \frac{\pi}{4}$

• Transformations of Trigonometric Functions:

  • a. $y = 3\cos x + 2$: Vertical stretch by a factor of 3, shifted up 2 units.

  • b. $y = -3\sin 2x - 7$: Reflected over the x-axis, vertically stretched by a factor of 3, horizontally compressed by a factor of 2, shifted down 7 units.

  • c. $y = 5\cos(\frac{1}{2}x) + 4$: Vertical stretch by a factor of 5, horizontally stretched by a factor of 2, shifted up 4 units.

• Equations from Transformations:

  • a. Sine function reflected over the x-axis, stretched vertically by a factor of 4, shifted up 5: $y = -4\sin x + 5$

  • b. Cosine function stretched vertically by a factor of 2, horizontally stretched by a factor of 3, shifted down 4: $y = 2\cos(\frac{1}{3}x) - 4$

• Cosine Functions from Graphs: *Write a cosine function for each graph: The cosine function starts at the top of the curve, or at the bottom if reflected.

  • a. From the graph: Vertical shift: +3 , Amplitude = 1 therefore: $y = cos(x) + 3$

  • b. Vertical shift: -1 , Amplitude = 4 therefore: $y = 4cos(x) - 1$

• Sine Functions from Graphs: *Write a sine function for each graph. The sine function starts in the middle of the curve.

  • a. Reflected about x-axis, vertical shift: -3 , Amplitude = 2 therefore: $y=-2sin(x) - 3$

  • b. Vertical shift: 1 , Amplitude = 3 Period goes from about -5 to 5 so the Period = 10, therefore, $\frac{2 \pi}{B} = 10$ solving for B get $B= \frac{ \pi}{5}$. $y=3sin( \frac{ \pi}{5}x) + 1$
    Graphing Sine Functions:

  • a. y=4sin(x) + 5 : The Vertical shift is +5, amplitude = 4, curve intersects the y-axis at +5

  • b. y=-2sin(2x)+2 : Reflected about the x-axis, then the graph is shifted up +2.

  • c. y=−sinx−4. Reflected about the x axis, then the graph shifted down -4

• Tangent Functions: *a. y=tan(x): Has asymptotes that go through $\frac{\pi}{6}$, $\frac{-\pi}{6}$, Period equals$\pi$ *a. y=tan($\frac{1}{2}$x Has asymptotes that go through $\frac{\pi}{3}$, $\frac{-\pi}{3}$, Vertical stretch = -5 *Period lengths of functions:

  • a. $y = -2 sin 2x + 2$. The period is is \frac{2\pi}{

Here's a simplified guide, using examples from Sections 6 and 7, to help you easily solve these types of problems:

Section 6.5: Solving Radical Equations

• Isolate the Radical:
  Get the square root or cube root part alone on one side.
  Example: In \sqrt{2x-4} - 3 = 1$, first add 3 to both sides to get$\sqrt{2x-4} = 4$.</p></li><li><p>Raise to the Power:<br>If it's a square root, square both sides. If it's a cube root, cube both sides.<br>Example: Square both sides of$\sqrt{2x-4} = 4$to get$2x - 4 = 16$.</p></li><li><p>Solve for x:<br>Solve the equation you're left with.<br>Example: From$2x - 4 = 16$, add 4 to both sides to get$2x = 20$, then divide by 2 to find$x = 10$.</p></li><li><p>Check Your Answer:<br>Put your answer back into the original equation to make sure it works.<br>Example: Plug$x = 10$into$\sqrt{2x-4} - 3 = 1$to check if it's correct.</p></li></ul><p>Section 6.7: Finding Inverse Functions</p><ul><li><p>Swap x and y:<br>Rewrite the equation with$x$in place of$y$and$y$in place of$x$.<br>Example: Change$y = 5(x - 3)$<br>to$x = 5(y - 3)$.</p></li><li><p>Solve for y:<br>Get$y$by itself on one side.<br>Example: For$x = 5(y - 3)$, divide by 5 to get$\frac{x}{5} = y - 3$, then add 3 to both sides to get$y = \frac{x}{5} + 3$.</p></li></ul><p>Section 6.8: Graphing Radical Functions</p><ul><li><p>Know the Basic Shape:<br>Understand what a basic square root or cube root graph looks like.</p></li><li><p>Spot the Shifts:<br>See if anything is added or subtracted inside or outside the root.<br>Inside shifts the graph left or right; outside shifts it up or down.<br>Example: In$y = \sqrt{x + 4}$, the +4 shifts the graph 4 units to the left.</p></li><li><p>Reflections:<br>A negative sign in front of the root flips the graph upside down.</p></li></ul><p>Section 7.1: Exponential Growth and Decay</p><ul><li><p>Growth vs. Decay:<br>Look at the base of the exponent.<br>If the base is greater than 1, it's growth. If it's between 0 and 1, it's decay.<br>Example:$y = 3(7)^x$is growth because 7 is greater than 1.</p></li><li><p>Compound Interest:<br>Use the formula$A = P(1 + r)^t$.<br>A is the final amount, P is the starting amount, r is the interest rate, and t is the time in years.</p></li><li><p>Population Growth:<br>Apply the formula$P(t) = P_0(1+r)^t$</p></li><li><p>Depreciation:<br>Apply the formula$V(t) = V_0(1-r)^t$</p></li></ul><p>Section 7.3: Logarithms</p><ul><li><p>Convert Forms:<br>Switch between exponential and logarithmic forms.<br>Example: If$100 = 10^2$, then$\log_{10} 100 = 2$.</p></li><li><p>Evaluate:<br>Ask yourself, \"What power do I need to raise the base to, to get this number?\"<br>Example: To find$\{log 1000 , think, \"10 to what power is 1000?\" The answer is 3.

Section 7.4: Using Log Properties

• Expand:
  Break down complex logs into simpler parts using properties.
  Example: \log (xy)$becomes$\log x + \log y$.</p></li><li><p>Combine:<br>Use log properties to merge multiple logs into one.<br>Example:$\log 8 + \log 3$becomes$\log (8 \cdot 3) = \log 24$.</p></li></ul><p>Section 7.5: Solving Exponential and Log Equations</p><ul><li><p>Same Base:<br>Get both sides of an exponential equation to have the same base.<br>Example: If$9^x = 243$, rewrite as$(3^2)^x = 3^5$, so$2x = 5$.</p></li><li><p>Convert to Exponential:<br>Change a log equation into exponential form to solve.<br>Example: If$\log(3x + 25) = 2$, rewrite as$10^2 = 3x + 25$$ .