Algebra 2 Notes

Inverse Variation

• Equation: $y = \frac{a}{x}$, where $a$ is the constant of variation.

• Example: If $y = 3$ when $x = -8$, then $3 = \frac{a}{-8}$, so $a = -24$. The equation is $y = \frac{-24}{x}$.

• To find $y$ when $x = 4$, substitute into the equation: $y = \frac{-24}{4} = -6$.

Joint Variation

• Equation: $z = axy$, where a is the constant of variation.

• Example: If $z = -72$ when $x = 9$ and $y = -4$, then $-72 = a(9)(-4)$, so $-72 = -36a$ and $a = 2$. The equation is $z = 2xy$.

• To find $z$ when $x = 6$ and $y = 4$, substitute into the equation: $z = 2(6)(4) = 48$.

• Process: General equation, plug in values to find $a$, substitute $a$ back into the equation, then solve for the unknown variable.

Combined Variation

• d varies directly with m and inversely with the cube of p: $d = \frac{am}{p^3}$.

• r varies jointly with y and the square root of x: $r = a y \sqrt{x}$.

Graphing Rational Functions

• General form: $y = \frac{a}{x-h} + k$, where $(h, k)$ is the shift from the parent function $y = \frac{a}{x}$.

• The vertical asymptote is at $x = h$, and the horizontal asymptote is at $y = k$.

• Example: $y = \frac{-2}{x-1} + 3$. Vertical asymptote: $x = 1$. Horizontal asymptote: $y = 3$. The graph is shifted right 1 unit and up 3 units.

• Create a table of values using the parent function $y = \frac{-2}{x}$ with the new origin at $(1, 3)$ to plot points on the graph.

• Domain: All real numbers except $x = 1$. In interval notation: $(-\infty, 1) \cup (1, \infty)$.

• Range: All real numbers except $y = 3$. In interval notation: $(-\infty, 3) \cup (3, \infty)$.

Simplifying Rational Expressions

• Factoring is a key skill.

Difference of Two Squares

• $x^2 - 16 = (x+4)(x-4)$

Trinomial Factoring

• $x^2 - 7x + 12 = (x-4)(x-3)$

Example

• $\frac{x^2 - 16}{x^2 - 7x + 12} = \frac{(x+4)(x-4)}{(x-4)(x-3)} = \frac{x+4}{x-3}$

• Excluded values are values that make the denominator zero before simplification.

Factoring out Greatest Common Factor

• $5x + 20 = 5(x+4)$

Trial and Error Factoring for Trinomials with Leading Coefficient

• $3x^2 + 11x - 4 = (3x-1)(x+4)$

Dividing Rational Expressions

• Keep it, change it, flip it.

• $\frac{A}{B} \div \frac{C}{D} = \frac{A}{B} \cdot \frac{D}{C}$

• Factor and cancel common factors in numerator and denominator.

Subtracting Rational Expressions

• Find the common denominator and combine the numerators.

• $\frac{3}{x-5} - \frac{2}{x+4} = \frac{3(x+4)}{(x-5)(x+4)} - \frac{2(x-5)}{(x-5)(x+4)} = \frac{3x+12-2x+10}{(x-5)(x+4)} = \frac{x+18}{(x-5)(x+4)}$

Solving Rational Equations

Proportions

• Cross multiply: If $\frac{A}{B} = \frac{C}{D}$, then $AD = BC$.

• Check for extraneous solutions by making sure the solution does not make the denominator equal to zero.

Clearing Denominators

• Multiply all terms by the common denominator to eliminate fractions.

• Check for extraneous solutions.

• Example: Solve $\frac{3}{6} - \frac{x}{3} = \frac{5}{18}$. Multiply by 18: $9 - 6x = 5$.

• Solve for $x$: $-6x = -4$, so $x = \frac{2}{3}$.

Clearing Denominators with Quadratics

• Factor the denominators to find the common denominator.

• Multiply all terms by the common denominator.

• Solve the resulting equation (may be quadratic).

• Check for extraneous solutions.

Distance and Midpoint Formulas

• Distance formula: $d = \sqrt{(x<em>2 - x</em>1)^2 + (y<em>2 - y</em>1)^2}$

• Midpoint formula: $(\frac{x<em>1+x</em>2}{2}, \frac{y<em>1+y</em>2}{2})$

Probability

• Probability = $\frac{\text{Number of successful outcomes}}{\text{Total possible outcomes}}$

Fundamental Counting Principle

• If there are $n<em>1$ ways to do one thing, $n</em>2$ ways to do another, and so on, then there are $n<em>1 \cdot n</em>2 \cdot …$ ways to do all of them.

Ordering Items

• The number of ways to order n items is $n!$

Combinations

• The number of ways to choose r items from n items when the order does not matter.

• $nCr = \frac{n!}{(n-r)!r!}$

Permutations

• The number of ways to choose r items from n items when the order matters.

• $nPr = \frac{n!}{(n-r)!}$

Binomial Expansion Theorem

• $(a+b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k$

• \binom{n}{k} is the binomial coefficient, also written as $nCk$.

Finding a Specific Term

• To find the coefficient of $x^4$ in $(2x-3)^7$, use the term $n choose k, a^{n-k}, b^k$. Find the correct combination to get the power needed.
  *$\binom{7}{3} (2x)^4 (-3)^3 = 35 \cdot 16x^4 \cdot (-27), = -15120x^4$ The coefficient is $-15120$.

• Pascal's triangle can also be used to determine coefficients.

Binomial Probability

• Probability of exactly k successes in n trials.

• $\binom{n}{k} p^k (1-p)^{n-k}$, where p is the probability of success on a single trial.

• Example: If you score a free throw 80% of the time, the probability of scoring exactly seven out of 10 free throws is $\binom{10}{7} (0.8)^7 (0.2)^3$ approximately 20%.

Statistics

Measures of Central Tendency

• Mean: Average.

• Median: Middle value when data is ordered. If there are two middle values, take their average.

• Mode: Value that occurs most often. There can be multiple modes.

Measures of Dispersion

• Range: Max - Min.

• Standard deviation: How spread out the data is from the mean. $\sqrt{\frac{\sum(x<em>i - \bar{x})}{N-1}}$ where the numerator is the sum of the deviations from the mean ( $x</em>i - \bar{x}$ ).

• Outlier: Value that is far from the other data values.

Effects of Transformations on Statistics

• Adding a constant to each data point adds that constant to the mean, median, and mode, but does not change the standard deviation or range.

• Multiplying each data point by a constant multiplies the mean, median, mode, range, and standard deviation by that constant.

Normal Distribution

• Empirical Rule: 68% of data within 1 standard deviation of the mean, 95% within 2 standard deviations, 99.7% within 3 standard deviations.

Z Score

• $z = \frac{x - \mu}{\sigma}$, where $x$ is the data point, $\mu$ is the mean, and $\sigma$ is the standard deviation.

Margin of Error

• $\pm \frac{1}{\sqrt{n}}$, where n is the sample size.

Sequences and Series

• Sequence: List of numbers.

• Series: Sum of numbers in a sequence.

• Summation notation: $\sum<em>{i=1}^{n} a</em>i$, where $a_i$ is a formula for the ith term.

Arithmetic Sequences

• Explicit formula: $a<em>n = a</em>1 + d(n-1)$, where $a_1$ is the first term and d is the common difference.

Geometric Sequences

• Explicit formula: $a<em>n = a</em>1 (r)^{n-1}$, where $a_1$ is the first term and r is the common ratio.

Finite Geometric Series

Formula: $S<em>n = a</em>1 \frac{(1 - r^n)}{(1 - r)}$

Infinite Geometric Series

Formula: $S =  \frac{a_1}{(1 - r)}$. 

Note: The ratio has to be between 1 and -1 for the formula to be valid.

The Unit Circle Basics

• Cosecant: Flip Sine

• Secant: Flip Cosine

• Cotangent: Flip Tangent

Right Triangle Trigonometry

SOH CAH TOA
Sine = Opposite / Hypotenuse
Cosine = Adjacent / Hypotenuse
Tangent = Opposite / Adjacent

The Pythagorean Theorem

$a^2 + b^2 = c^2$

Solving Triangles

• All angles have to add up to 180 Degrees.

• Be cautious when using inverse trigonometric functions, as there may be more solutions than the calculator provides.

Angle Conversion

• from Degrees to Radians, multiply by $\frac{\pi}{180}$

• from Radians to Degrees, multiply by $\frac{180}{\pi}$

Arc Length (s) and Sector Area Formulas

• Arclength in radians: $s = r \theta$

• Area of sector:$A= \frac{1}{2} r^2 \theta$
  (Note: angle is in radians)

Inverse Trigonometric Functions and their Ranges:

Sine

Range: [-pi/2, pi/2]

Cosine

Range: [0, pi]

Tangent

Range: (-pi/2, pi/2)

NOTE

To find the proper angle you always have to consider which quadrant the unit circle angle resides in. See the various trigonometric functions and their signs in the document.

Laws

Law of Sines (non-right angles)

*Note: The formula below will not work with variables that are side side side. It will not work and you will not be able to resolve the equation.
$\frac{sinA}{a} = \frac{sinB}{b} = \frac{sinC}{c}$