Sequences Series Geometric Common ratio

TOPIC #1: POLYNOMIAL OPERATIONS

• Factoring Trinomials
  • Example: $x^2 + 7x + 10 = (x+5)(x+2)$
• Grouping
  • Rearrange first, necessary.
  • Example:
    • $4x^3 - x^2 + 16x - 4$
    • Rearrange: $4x^3 + 16x - x^2 - 4$
    • $4x(x^2 + 4) - 1(x^2 + 4)$
    • $(x^2 + 4)(4x - 1)$
• AC Method
  • Example:
    • $2x^2 + x - 3$
    • $2x^2 + 3x - 2x - 3$
    • $x(2x + 3) - 1(2x + 3)$
    • $(2x + 3)(x - 1)$
• Difference of Two Perfect Squares
  • Example:
    • $4x^2 - 49$
    • $(2x + 7)(2x - 7)$
• Long Division
  • Put zeros in for missing terms.
  • Example:
    • Divide $2x^4 + 4x^2 - 1$ by $x+1$
    • Quotient: $2x^3 - 2x^2 + 6x - 6$
    • Remainder: 5
• Remainder Theorem
  • The function's value will tell you the remainder.
  • If remainder = 0, then you have found a root and a factor.
  • Example:
    • Is $x-5$ a factor of $2x^3 - 4x^2 - 7x - 10$?
    • Answer: $2(5)^3 - 4(5)^2 - 7(5) - 10 = 105$. No, not a factor.
  • Graphically: If $x+1$ is a factor of $f(x)$, then the remainder (y-value) is 0.
• Synthetic Division
  • Example:
    • Divide $2x^4 + 0x^3 + 4x^2 + 0x - 1$ by $x+1$
    • Quotient: $2x^3 - 2x^2 + 6x - 6$
    • Remainder: 5
• Solving Fractional Equations
  • Get a common denominator.
  • Set numerators equal.
  • Solve and reject solutions that make the denominator undefined.
  • Extraneous solutions are solutions that get rejected!

TOPIC #2: RATIONAL EXPRESSIONS / ALGEBRAIC FRACTIONS

• Example:
  • $\frac{x}{x-5} + \frac{3}{x+2} = \frac{7x}{x^2 - 3x - 10}$
  • $\frac{x(x+2)}{(x-5)(x+2)} + \frac{3(x-5)}{(x-5)(x+2)} = \frac{7x}{(x-5)(x+2)}$
  • Undefined when $x^2 + 2x - 8 = 0$, which means $(x+4)(x-2) = 0$, so $x = -4$ and $x = 2$
  • Based on the denominators, $x \neq 5$ and $x \neq -2$
  • $x(x+2) + 3(x-5) = 7x$
  • $x^2 + 2x + 3x - 15 = 7x$
  • $x^2 - 2x - 15 = 0$
  • $(x-5)(x+3) = 0$
  • $x = 5$ or $x = -3$
  • $x = 5$ is extraneous, so reject it.

TOPIC #3: EXPONENTS, RADICALS, + IMAGINARY NUMBERS

• Exponent Rules
  • Add exponents: $x^3 \cdot x^4 = x^7$
  • Subtract exponents: $\frac{x^5}{x^2} = x^3$
  • Multiply exponents: $(x^3)^2 = x^6$
  • Zero exponent: $x^0 = 1$
  • Fractional exponents: $x^{\frac{m}{n}} = \sqrt[n]{x^m} = (\sqrt[n]{x})^m$
    • Denominator is the root, numerator is the power.
• Simplifying using Exponents
  • Much easier to work with fractional exponents than radicals!
  • Example: Convert to fractions, subtract exponents.
• Simplifying Imaginary Numbers
  • $i = \sqrt{-1}$
  • $i^2 = -1$
  • Set calculator to a+bi mode!
  • Example: $(2 - 3ki)^2 = (2 - 3ki)(2 - 3ki)$
    • $= 4 - 6ki - 6ki + 9k^2i^2$
    • $= 4 - 12ki + 9k^2(-1)$
    • $= 4 - 12ki - 9k^2$
    • This is simplified completely.
• Radical Equations
  • Isolate the radical.
  • Square both sides.
  • Solve and check; usually have to reject solutions.
  • Check back to radicals.
  • Example:
    • $\sqrt{x+2} + 4 = x$
    • $\sqrt{x+2} = x - 4$
    • $(\sqrt{x+2})^2 = (x-4)^2$
    • $x+2 = x^2 - 8x + 16$
    • $x^2 - 9x + 14 = 0$
    • $(x-7)(x-2) = 0$
    • $x = 7$ or $x = 2$
    • Check:
      • $\sqrt{7+2} + 4 = 7 \Rightarrow 3 + 4 = 7$
      • $\sqrt{2+2} + 4 = 2 \Rightarrow 2 + 4 = 2$ (reject!)
• Quadratics with Imaginary Solutions
  • $x^2 + 4 = 0$
  • $x^2 = -4$
  • $x = \pm \sqrt{-4}$
  • $x = \pm 2i$
  • Example:
    • $x^2 - 2x + 3 = 0$
    • $x = \frac{2 \pm \sqrt{4 - 4(1)(3)}}{2(1)}$
    • $= \frac{2 \pm \sqrt{-8}}{2}$

TOPIC #4: SOLVING EQUATIONS + INEQUALITIES

• Solving by Factoring
  • Finding roots means x-intercepts, solutions, or zeros.
  • Example:
    • $x^4 - 4x^3 - 9x^2 + 36x = 0$
    • $x(x^3 - 4x^2 - 9x + 36) = 0$
    • $x[x^2(x-4) - 9(x-4)] = 0$
    • $x(x^2 - 9)(x-4) = 0$
    • $x = 0$, $x = \pm 3$, $x = 4$
• Circle-Line System
  • Algebraically:
    • Solve the line for x or y.
    • Substitute into the circle equation.
    • Solve for points of intersection.
    • Example:
      • $(x+1)^2 + (y-2)^2 = 25$
      • $y - x = -2 \Rightarrow y = x - 2$
      • $(x+1)^2 + (x-2-2)^2 = 25$
      • $(x+1)^2 + (x-4)^2 = 25$
      • $x^2 + 2x + 1 + x^2 - 8x + 16 = 25$
      • $2x^2 - 6x - 8 = 0$
      • $x^2 - 3x - 4 = 0$
      • $(x-4)(x+1) = 0$
      • $x = 4 \Rightarrow y = 4 - 2 = 2$
      • $x = -1 \Rightarrow y = -1 - 2 = -3$
      • Solutions: $(4, 2)$ and $(-1, -3)$
      • Check by plugging back into the original equations.
  • Graphically:
    • 0 solutions (imaginary solutions)
    • 1 solution
    • 2 solutions

TOPIC #5: SYSTEMS OF EQUATIONS

• Systems of 3 Equations
  • Pick 2 equations and eliminate one variable.
  • Pick 2 different equations and eliminate the SAME variable.
  • Now solve your 2 equations + 2 unknowns system for two solutions.
  • Find the third solution by plugging back into one of the original equations.
  • Check using MATRIX.
  • Do NOT rely on your calc!
  • These are often asked in the short answer sections.
  • Example Equations:
    • $-6x + 5y + 2z = -11$
    • $-2x + y + 4z = -9$
    • $4x - 5y + 5z = -4$
  • Steps:
    • Equation 1 + Equation 3:
      • $-6x + 5y + 2z = -11$
      • $4x - 5y + 5z = -4$
      • $-2x + 7z = -15$
    • Equation 2:
      • $5(-2x + y + 4z = -9)$
      • $-10x + 5y + 20z = -45$
    • $4x - 5y + 5z = -4$
    • $-6x + 25z = -49$
  • Solve for z:
    • $-3(-2x + 7z = -15)$
    • $-6x + 25z = -49$
    • $6x - 21z = 45$
    • $-6x + 25z = -49$
    • $4z = -4 \Rightarrow z = -1$
  • Solve for x:
    • $-2x + 7(-1) = -15$
    • $-2x - 7 = -15$
    • $x = 4$
  • Solve for y:
    • $-2(4) + y + 4(-1) = -9$
    • $-8 + y - 4 = -9$
    • $y - 12 = -9 \Rightarrow y = 3$
  • Solution: $(4, 3, -1)$
• Calculate Intersection
  • Where does $f(x) = g(x)$?
  • Also useful if you have two weird equations set equal to each other.
  • 2nd TRACE #5: INTERSECT
    • First Curve: move cursor to where you think the intersection is, Enter.
    • Second Curve: move cursor to where you think the intersection is, Enter.
    • Guess: Enter.

TOPIC #6: FUNCTIONS

• Degree: highest exponent
  • Even degree:
    • Both ends in the same direction.
    • Positive: As $x \rightarrow \infty$, $f(x) \rightarrow \infty$. As $x \rightarrow -\infty$, $f(x) \rightarrow \infty$. End behavior: ↑↑
    • Negative: As $x \rightarrow \infty$, $f(x) \rightarrow -\infty$. As $x \rightarrow -\infty$, $f(x) \rightarrow -\infty$. End behavior:↓↓
  • Odd degree:
    • Both ends in opposite directions.
    • Positive: As $x \rightarrow \infty$, $f(x) \rightarrow \infty$. As $x \rightarrow -\infty$, $f(x) \rightarrow -\infty$. End behavior: ↗
    • Negative: As $x \rightarrow \infty$, $f(x) \rightarrow -\infty$. As $x \rightarrow -\infty$, $f(x) \rightarrow \infty$. End behavior: ↘
• Where is a function increasing/decreasing?
  • Identify intervals of increase and decrease on the graph.
• Relative Max / Min
  • 2nd TRACE #3 MIN
  • 2nd TRACE #4 MAX
    • Left Bound, Right Bound, Guess, Enter
• Multiplicity: how many times a root appears
  • If even, the graph is tangent (bounce).
  • If odd, the graph crosses through.
  • Example: $f(x) = (x-1)^2 (x+2) (x-4)$
    • $x = 1$, multiplicity 2, bounce
    • $x = -2$, multiplicity 1, cross
    • $x = 4$, multiplicity 1, cross
• General Cubic: Degree 3, shape: ⅃ or ↄ
• General Quartic: Degree 4, shape: W or M

TOPIC #7: EVEN/ODD FUNCTIONS

• Not the same as even/odd degree!
• Even Function
  • Symmetric about y-axis.
  • Examples: cosine, random graph symmetric about y-axis.
  • Algebraically: Plug in -x, if $f(-x) = f(x)$, then even.
    • Example: $f(x) = x^4 + 3x^2 + 2$
      • $f(-x) = (-x)^4 + 3(-x)^2 + 2 = x^4 + 3x^2 + 2 = f(x)$, therefore even.
• Odd Function
  • Symmetric about origin.
  • Examples: sine, random graph symmetric about origin.
  • Algebraically: Plug in -x, if $f(-x) = -f(x)$, then odd.
    • Example: $f(x) = 3x^3 - 6x$
      • $f(-x) = 3(-x)^3 - 6(-x) = -3x^3 + 6x = -f(x)$, therefore odd.

TOPIC #8: AVERAGE RATE OF CHANGE

• Slope!
• $m = \frac{y<em>2 - y</em>1}{x<em>2 - x</em>1}$
• If the y-values are not given, find them and do the slope formula.

TOPIC #9: INVERSE FUNCTIONS

• $f^{-1}(x)$
• Switch x and y.
• Solve for y.
• Example: $f(x) = 2x - 3$
  • $x = 2y - 3$
  • $x + 3 = 2y$
  • $y = \frac{1}{2}x + \frac{3}{2}$
• Example: $f(x) = \frac{x-1}{x+3}$
  • $x = \frac{y-1}{y+3}$
  • $x(y+3) = y-1$
  • $xy + 3x = y - 1$
  • $xy - y = -3x - 1$
  • $y(x-1) = -3x - 1$
  • $y = \frac{-3x-1}{x-1} = \frac{3x+1}{1-x}$
  • $f^{-1}(x) = \frac{-3x-1}{x-1} = \frac{3x+1}{1-x}$

TOPIC #10: PARABOLAS - FOCUS/DIRECTRIX

• A parabola is a set of points equidistant from a point (focus) and a line (directrix).
• $p$ = distance from vertex to focus or directrix
• Equations:
  • $y = \pm \frac{1}{4p} (x-h)^2 + k$
  • $x = \pm \frac{1}{4p} (y-k)^2 + h$
  • $(h, k)$ is the vertex.
  • Graph always opens toward the focus!
  • y = + opens up.
  • y = - opens down.
  • x = + opens right.
  • x = - opens left.
• Example:
  • Vertex $(1, 2)$, $p = 2$, opens down.
  • Equation: $y = -\frac{1}{4(2)} (x-1)^2 + 2 = -\frac{1}{8} (x-1)^2 + 2$
• Converting to Focus/Directrix Form
  • Solve for the variable that is NOT being squared!
  • Example:
    • $(x-1)^2 = -8(y-2)$
    • $(x-1)^2 = -8y + 16$
    • $(x-1)^2 - 16 = -8y$
    • $y = -\frac{1}{8} (x-1)^2 + 2$
    • Vertex: $(1, 2)$, $p = 2$

TOPIC #6A: EXPONENTIALS

• Exponential Growth
  • Asymptote on x-axis ($y = 0$).
  • x-int: NONE
  • y-int: $(0, 1)$
  • As $x \rightarrow \infty$, $f(x) \rightarrow \infty$
  • As $x \rightarrow -\infty$, $f(x) \rightarrow 0$
  • Examples: $y = 2^x$, $y = e^x$
  • Table of values is increasing.
• Exponential Decay
  • Asymptote on x-axis ($y = 0$).
  • x-int: NONE
  • y-int: $(0, 1)$
  • As $x \rightarrow \infty$, $f(x) \rightarrow 0$
  • As $x \rightarrow -\infty$, $f(x) \rightarrow \infty$
  • Examples: $y = (\frac{1}{2})^x$, $y = 2^{-x}$
  • Table of values is decreasing.
• Know how to graph weird exponentials by typing them into your calculator and graphing values from the table!
• Asymptote
  • Usually on x-axis unless graph is shifted up or down.
  • Example: $y = 2^x + 2$. Asymptote is now at $y = 2$.
    • As $x \rightarrow \infty$, $f(x) \rightarrow \infty$
    • As $x \rightarrow -\infty$, $f(x) \rightarrow 2$
• Basic Exponential Growth / Decay Formulas
  • $A = A_0 (1 \pm r)^t$
    • $A$ = amount after t time
    • $A_0$ = initial amount
    • $r$ = rate as a decimal
    • $t$ = time
      *Example:
      *Solve for r:
      $150=130(1+r)^5$
      $\frac{150}{130}=(1+r)^5$
      $\sqrt[5]{\frac{150}{130}}=1+r$
      $r = 0.029 = 2.9033$
  • Half Life
    • $A = A_0 (\frac{1}{2})^{\frac{t}{v}}$
      • $A$ = amount after t time
      • $A_0$ = initial amount
      • $v$ = half life in time
        *If v = .00357 then rate as a decimial is $ln(.5)/v = -194.11$
  • Compound Interest
    • $A = P(1 + \frac{r}{n})^{nt}$
      • $A$ = amount after t time
      • $P$ = initial amount (principal)
      • $r$ = rate as a decimal
      • $n$ = number of times it is compounded per year.
        • $n = 1$ annually
        • $n = 2$ semiannually
        • $n = 4$ quarterly
        • $n = 12$ monthly
        • $n = 52$ weekly
  • Continuous Compound Interest
    • $A = Pe^{rt}$
      • $A$ = amount after t time
      • $P$ = initial amount (principal)
      • $r$ = rate as a decimal
      • $t$ = time
        *Example:
        *Solve for Ao:
        $150 = A<em>0(1.03)^5$$A</em>0=129.4$
• Exponential Time Adjustments
  *$40(1.65)^t$
  *Given $40(1.65)^t$ where t is in years. Convert to months $m = 12t$
  *$40(1.65)^{\frac{m}{12}} = 40(1.0426)^{\frac{12t}{12}}$
• Down Payment: how much money you pay first before taking out a loan.
  *House Costs: $200,000. Down payment is $50,000.
• Loan amount would be $150,0000 *Car Costs $18,000. Loan amount is $15,000. Down payment is $3,000.
  • Total Cost = Down Payment + Loan/Borrowed Amount
  • Down Payment = Total Cost - Loan/Borrowed Amount

TOPIC #6B: LOGARITHMS

• Log Graphs
  • $y = \log x$
    • Y-int: NONE
    • As $x \rightarrow \infty$, $f(x) \rightarrow \infty$
    • As $x \rightarrow 0$, $f(x) \rightarrow -\infty$
    • Asymptote on y-axis ($x = 0$).
    • x-int: $(1, 0)$
  • A shift right or left will move the asymptote.
    • $y = \log(x-2)$
      • Asymptote is at $x = 2$
      • As $x \rightarrow \infty$, $f(x) \rightarrow \infty$
      • As $x \rightarrow 2$, $f(x) \rightarrow -\infty$
      • Since it was moved right 2 units.
• Logs + Exps are INVERSES!
  *Inverse of $y = 2^x$ is $y = log_2x$
  *Inverse of $y = e^x$ is $y = lnx$
• ExpLog Conversion $a^b = c$ converts to $log<em>ac = b$ *Solve for X: $log</em>8x = 2 \rightarrow 8^2 = x \rightarrow x = 64$
• Special Logs
  • Common log: base 10, $log_{10}x = log x$
  • Natural log: base e, $log_ex = \ln x$
• Solving for Exponents using Logs
  • Isolate the exponential first (divide any coefficients out, move constants over).
  • Use explog conversion.
    *Examples:
    *Basic: $2^{X+3} = 6 \rightarrow log<em>26 = x+3 \Rightarrow x = log</em>26 - 3 = -.415$
    *$100 = 80e^{2t} \rightarrow 1.25 = e^{2t} \Rightarrow ln(1.25) = 2t\Rightarrow t=.112$
    *Equations w/ Fractional Exp.
    *Isolate Variable
    *Raise to reciprocal power:
    Example: $2x^{\frac{3}{2}} + 4 = 36 \Rightarrow x^{\frac{3}{2}} = 16 \Rightarrow (x^{\frac{3}{2}})^{\frac{2}{3}} = 16^{\frac{2}{3}} \Rightarrow x = 64$
    *Solving Exponentials by Changing Bases
    *get common base.
    *Example: $4^{9x-5} = 32^{x+8} \Rightarrow (2^2)^{9x-5} = (2^5)^{x+8} \Rightarrow 18x-10 = 5x+40$
    $13x = 50 \Rightarrow x = \frac{50}{13}$

TOPIC #7: SEQUENCES + SERIES

• Arithmetic Sequence
  • Created by adding a certain value (common difference, d) to the previous term.
  • TERM FORMULA: $a<em>n = a</em>1 + (n-1)d$
    • (on ref. sheet) *Example: *$-7, -3, 1, 5, 9,… *d = +4 d = +4
      • Find 11th term: a_{11} = -7 + (11-1)(4) = -7 + 10(4) = 33
        *Explicit Formula - Written in terms of n:
        *-7, -3, 1, 5, 9,…
    • a_n = -7 + (n-1)(4) = 4n-11 * FINDING SUM:
    • Of first 11 terms:
      • \sum_{n=1}^{11} (4n-11) = 143
        Math --> summation
• Sequence a pattern of numbers in a certain order
• Series a sum of a sequence
• Geometric Sequence
  • Created by multiplying each term by a certain value (common ratio, r).
  • TERM FORMULA: an = a1 \cdot r^{n-1}
    • (on ref sheet) *Two Ways: Example *8, 2, .5, .125,…
      • Find 5th term: a5 = 8 (\frac{1}{4})^{5-1} = 8(\frac{1}{4})^4 = \frac{1}{32} *FINDING SUM: Find Sum of first 11 terms\sum{n=1}^{11}8(\frac{1}{4})^{n-1}=[10.66666412]$$
        Sum formula from ref sheet