Comprehensive Study Guide for Rational Functions, Sequences, Probability, and Trigonometry

Defining and Identifying Undefined Rational Expressions

A rational expression is defined as a fraction where the numerator and denominator are polynomials. Such expressions are considered undefined when the denominator is equal to zero, as division by zero is mathematically impossible. For example, the expression $\frac{x^2}{(x-1)(x+2)}$ becomes undefined when $(x-1)(x+2) = 0$. By applying the zero-product property, we find that at $x = 1$ and $x = -2$, the denominator vanishes, rendering the entire expression undefined. Similarly, for the expression $\frac{x+2}{x^2+4x+4}$, the denominator factors into $(x+2)^2$. Setting this to zero reveals that the value making the expression undefined is $x = -2$. In a functional context, such as $f(x) = \frac{4}{x+5}$, the domain is restricted, and the function is undefined at $x = -5$.

Simplifying and Performing Operations on Rational Expressions

Simplifying rational expressions involves factoring both the numerator and the denominator and canceling common factors. For instance, simplifying $\frac{x-1}{x^2-6x+5}$ requires factoring the quadratic denominator into $(x-1)(x-5)$. The $(x-1)$ terms cancel out, leaving the simplified result of $\frac{1}{x-5}$. When dividing rational expressions, the rule is to multiply the first expression by the reciprocal of the second. In the case of $\frac{a^2}{b} \div \frac{x}{b}$, the operation becomes $\frac{a^2}{b} \times \frac{b}{x}$, which simplifies to $\frac{a^2}{x}$. For multiplication, such as $\frac{12a^6}{5} \cdot \frac{10}{24a}$, the coefficients and variables are reduced to result in a simplified value, notably arriving at $a^5$. Complex fractions, such as $\frac{1+\frac{1}{y}}{1-\frac{1}{y}}$, can be simplified by multiplying the numerator and denominator by the common denominator $y$, yielding $\frac{y+1}{y-1}$.

Least Common Multiple (LCM) and Rational Addition

Finding the Least Common Multiple (LCM) of monomials is essential for adding or subtracting rational expressions with unlike denominators. To find the LCM, one must identify the highest power of every factor present in the expressions. For example, for the monomials $5x^3y^2$ and $20x^2y^6$, the LCM is $20x^3y^6$. When simplifying an addition problem like $\frac{7}{ab} - 5$, one must find a common denominator, which is $ab$. The expression is rewritten as $\frac{7 - 5ab}{ab}$. This ensures that rational terms can be combined into a single, simplified rational expression.

Characteristics of Rational Functions and Discontinuities

Rational functions often possess asymptotes and points of discontinuity based on the behavior of their denominators. A vertical asymptote occurs at values that make the denominator zero (provided factors do not cancel). For the function $f(x) = \frac{1}{x+1} + 5$, a vertical asymptote exists at $x = -1$. A horizontal asymptote is determined by comparing the degrees of the numerator and denominator. For $f(x) = \frac{x+3}{2x-2}$, since the degrees are equal, the horizontal asymptote is the ratio of the leading coefficients, $y = \frac{1}{2}$. Conversely, if the degree of the numerator is less than the denominator, such as in $f(x) = \frac{2x^2}{x^3-2x^2}$, the horizontal asymptote is $y = 0$. A point of discontinuity, or a "hole," occurs when a factor in the numerator and denominator cancels out. For $f(x) = \frac{x^2-4x}{x-2}$ (re-examining the transcript example $f(x) = \frac{x^2-4}{x-2}$), the factor $(x-2)$ cancels, leaving a hole at $x = 2$.

Types of Variation in Rational Relationships

Direct variation describes a relationship where two variables increase or decrease proportionally, represented by $y = kx$. If $y = 24$ when $x = 8$, the constant $k$ is $3$. Thus, when $x = 48$, $y = 3 \times 48 = 144$. Inverse variation occurs when one variable decreases as the other increases, modelled by $y = \frac{k}{x}$. For example, if $y = -12$ when $x = 2$, then $k = -24$, and when $x = 6$, $y = -4$. Joint variation involves three variables, such as $y = kxz$. If $y = 70$ when $x = 10$ and $z = 4$, the constant $k$ is $1.75$. Combined variation merges these concepts, often written as $y = \frac{kx}{z}$. A specific case provided shows if $y = 20$ when $x = 2$ and $z = 4$, solving for $y$ when $x = 10$ and $z = -5$ yields $y = -80$.

Solving Rational Equations and Inequalities

Rational equations are solved by multiplying the entire equation by the least common denominator to eliminate the fractions. For the equation $\frac{x-1}{x+1} = \frac{6}{5}$, cross-multiplication results in $5(x-1) = 6(x+1)$, leading to $5x - 5 = 6x + 6$, and eventually $x = -11$. For inequalities like $\frac{9}{m-5} < 3$, one must consider both the solution set and the excluded values (where the denominator is zero). In this instance, the excluded value is $m = 5$. The solution to the inequality is partitioned into ranges, such as $m < 5$ or $m > 8$. A specific true/false note confirms that in the inequality $\frac{3}{x-4} > 1$, the excluded value is indeed $x = 4$.

Arithmetic Sequences and Series

An arithmetic sequence is a sequence of numbers where each term is derived by adding a fixed constant called the common difference ($d$). Given the sequence $11, 15, 19, \dots$, the common difference is $d = 4$. The next four terms are calculated as $23, 27, 31, 35$. The $n$-th term formula is defined as $a_n = a_1 + (n-1)d$. For a sequence where $a_1 = 5$ and $d = 4$, the 10th term ($a_{10}$) is calculated as $5 + (10-1)4 = 41$. If the 10th term is $15$ and the first term is $-3$, the difference is found by solve $-3 + 9d = 15$, yielding $d = 2$. Arithmetic means are terms inserted between two given values; between $6$ and $30$, two means would be $14$ and $22$. The sum of an arithmetic series is given by $S_n = \frac{n(a_1 + a_n)}{2}$. For the series $2 + 4 + 6 + \dots + 100$, there are $50$ terms, and the sum is $2550$. Sigma notation ($\sum$) is used for sums; for example, $\sum_{k=5}^{12} (3k + 7)$ has $12 - 5 + 1 = 8$ terms.

Geometric Sequences and Series

A geometric sequence is formed by multiplying each term by a constant ratio ($r$). In the sequence $12, 36, 108, 324, \dots$, the ratio is $r = 3$. The $n$-th term is found using $a_n = a_1 r^{n-1}$. For a sequence with $a_1 = 5$ and $r = 2$, the 6th term is $5(2)^5 = 160$. The $n$-th term formula for the sequence $5, 10, 20, 40, \dots$ is $a_n = 5(2)^{n-1}$. Geometric means involve filling gaps using the ratio; between $1$ and $27$, the geometric means are $3$ and $9$. The sum of a finite geometric series is $S_n = \frac{a_1(1-r^n)}{1-r}$. If $a_1 = 7$, $r = 3$, and $n = 4$, the sum is $280$. If a series is described by $\sum_{n=1}^{7} 4(-3)^{n-1}$, the calculated sum is $-2186$.

Infinite Geometric Series and Convergence

An infinite geometric series converges and has a defined sum only if the absolute value of the common ratio is less than one ($|r| < 1$). The sum of an infinite series is given by the formula $S = \frac{a_1}{1-r}$. For a series where $a_1 = 25$ and $r = \frac{1}{2}$, the sum is $\frac{25}{1 - 0.5} = 50$. If a series follows $\sum_{n=1}^{\infty} 10(\frac{1}{5})^{n-1}$, the sum is $\frac{10}{1 - 0.2} = 12.5$ (or $\frac{25}{2}$). If $|r| ≥ 1$, the series is divergent and no sum exists. Repeating decimals can be expressed as infinite geometric series. For instance, the decimal $0.\overline{11}$ is equivalent to the fraction $\frac{1}{9}$.

Binomial Theorem and Mathematical Induction

The Binomial Theorem provides a method for expanding expressions of the form $(a + b)^n$. Instead of Pascal's Triangle, combinations are used to find coefficients. The first term in the expansion of $(x + 1)^{10}$ is $x^{10}$. To find the third term of $(x + y)^3$, we use the formula for terms, resulting in $3xy^2$. Beyond expansions, the transcript notes the Principle of Mathematical Induction as a crucial technique for proving mathematical sentences related to the set of natural numbers ($n$). This principle involves proving a base case for $n=1$ and then showing that if it holds for some $k$, it must hold for $k+1$. Conversely, a counterexample can disprove a statement, such as checking if $n^2 + n + 11$ is prime for specific values like $n=5$.

Fundamental Counting Principle and Permutations

The fundamental counting principle states that if there are $n$ ways to make one choice and $m$ ways to make another, there are $n \times m$ total outcomes. For tossing a coin twice, there are $2 \times 2 = 4$ possible outcomes. If a clothing set consists of $4$ pairs of pants, $3$ shirts, and $5$ pairs of shoes, there are $4 \times 3 \times 5 = 60$ possible outfits. Permutations are used when order matters. Linear permutations of $n$ items are $n!$. For 5 players standing in a line, the probability that a specific person is in the middle and another is to their right is $\frac{1}{20}$. For word permutations with repeated letters, such as "Horse" (الالحصان), the probability is based on the total permutations of those letters. Circular permutations for $n$ objects result in $(n-1)!$ ways to arrange them; for 5 people sitting around a round table, there are $4! = 24$ ways. However, if a fixed reference point exists (like a seat by a window), it reverts to linear permutations ($n!$).

Combinations and Probability Theory

Combinations differ from permutations because the order of selection does not matter. The formula for combinations is $_nC_r = \frac{n!}{r!(n-r)!}$. For example, choosing 3 students out of 10 results in $_{10}C_3 = 120$ ways. Therefore, the probability of selecting three specific people is $\frac{1}{120}$. In general probability, the likelihood of an event is the ratio of the number of favorable outcomes to the total outcomes in the sample space. Choosing 2 students from 20 results in a probability of $\frac{1}{190}$ for a specific pair. Geometric probability involves areas or lengths; for a square with area $9\,\text{cm}^2$ containing a circle with area $3\,\text{cm}^2$, the probability of a point falling in the circle is $\frac{3}{9} = \frac{1}{3}$. Conditional probability, denoted $P(A|B)$, is the probability of event A occurring given that B has already occurred. If two dice are thrown and the sum is 9, the probability that one die shows a 5 is $\frac{1}{2}$.

Dependent, Independent, and Mutually Exclusive Events

Independent events are those where the outcome of one does not affect the other, such as rolling a die multiple times. The probability of rolling a specific number on the tenth roll remains $\frac{1}{6}$ (or $\frac{1}{2}$ for odd) regardless of previous rolls. Dependent events occur when choices are made without replacement. Mutually exclusive events cannot happen at the same time; the probability of either happening is the sum of their individual probabilities. For non-mutually exclusive events, you must subtract the overlap: $P(A \cup B) = P(A) + P(B) - P(A \cap B)$. If the probability of rain is $75\%$, the complement (probability of no rain) is $25\%$. Finally, a compound event is explicitly defined as a combination of two or more simple events.

Trigonometric Ratios and Right Triangle Trigonometry

In right-angled triangles, trigonometric functions are defined by the ratios of the sides. For an angle $\theta$, $\sin(\theta)$ is opposite/hypotenuse, $\cos(\theta)$ is adjacent/hypotenuse, and $\tan(\theta)$ is opposite/adjacent. Reciprocal ratios include \csc(\theta) = \frac{1}{\sin(\theta)}, $\sec(\theta) = \frac{1}{\cos(\theta)}$, and $\cot(\theta) = \frac{1}{\tan(\theta)}$. Given $\cos(\theta) = \frac{4}{5}$ in the first quadrant, we can calculate $\sin(\theta) = \frac{3}{5}$ using the Pythagorean identity $\sin^2(\theta) + \cos^2(\theta) = 1$. To find missing side lengths, one might use equations like $\cos(60^∘) = \frac{4}{x}$ or $\tan(60^∘) = \frac{x}{4}$, depending on the given side and angle. For area, the formula for a triangle is $A = \frac{1}{2}ab \sin(C)$. Applying this to a triangle with sides $12$ and $15$ and a $15^∘$ angle results in an area of approximately $23.3\,\text{cm}^2$.

Angle Measurements and Arc Lengths

Angles can be measured in degrees or radians, where $180^∘ = \pi\,\text{radians}$. To convert $75^∘$ to radians, multiply by $\frac{\pi}{180}$, resulting in $\frac{5\pi}{12}$. Conversely, $\frac{3\pi}{2}$ radians equals $270^∘$. A full rotation of the earth constitutes $2\pi\,\text{radians}$ or $360^∘$. Angles that share the same terminal side are called coterminal angles. For example, $420^∘$ is coterminal with $60^∘$ ($420 - 360 = 60$). Arc length ($s$) on a circle with radius $r$ and central angle $\theta$ (in radians) is given by $s = r\theta$. For a circle with a radius of $5$ and an angle of $\frac{2\pi}{3}$, the arc length is $\frac{10\text{π}}{3}$. The transcript also highlights calculations for a circle with radius $27$ and an angle of $\frac{\pi}{3}$, resulting in an arc length of $9\text{π}$.

Trigonometric Functions on the Unit Circle

The unit circle is a circle centered at the origin with a radius of $1$. On this circle, the coordinates of any point are given by $(\cos(\theta), \sin(\theta))$. If a point on the unit circle is $(-\frac{9}{41}, \frac{40}{41})$, then $\sin(\theta) = \frac{40}{41}$. Signs of trig functions vary by quadrant: in the first, all are positive; in the second, only sine and cosecant are positive; in the third, tangent and cotangent are positive; and in the fourth, cosine and secant are positive. For example, $\cos(120^∘) = -\frac{1}{2}$ because it is in the second quadrant. A reference angle ($\theta'$) is the acute angle formed by the terminal side and the x-axis. For $150^∘$, the reference angle is $30^∘$. Quadrantal angles, such as $540^∘$, fall directly on an axis.

Graphing and Inverse Trigonometric Functions

The amplitude of a trigonometric function is half the distance between its maximum and minimum values, while the period is the horizontal distance over which the function repeats. For the function $y = 4 \text{ sin}(5\theta)$, the amplitude is $4$ and the period is $\frac{360^∘}{5} = 72^∘$. Inverse trigonometric functions, such as $y = \text{ sin}^{-1}(x)$, are used to find angle measures from known ratios. For example, $\sin^{-1}(\frac{\sqrt{3}}{2})$ equals $60^∘$, and $\sin^{-1}(-\frac{\sqrt{2}}{2})$ equals $-45^∘$. A specific complex calculation provided is $\text{sin}(\text{cos}^{-1}(\frac{72}{108}))$ or similar compositions like $\text{tan}(\text{tan}^{-1}(\frac{1}{2})) = \frac{1}{2}$.

Laws of Sines and Cosines

The Law of Sines states that $\frac{a}{\text{sin}(A)} = \frac{b}{\text{sin}(B)} = \frac{c}{\text{sin}(C)}$. It is used when we know two angles and any side (AAS or ASA) or two sides and an opposite angle (SSA). The SSA case can lead to zero, one, or two solutions. For instance, given $a = 19$, $b = 12$, and $A = 139^∘$, there is only one solution because $a > b$. The Law of Cosines, $a^2 = b^2 + c^2 - 2bc \text{ cos}(A)$, is applied when we know three sides (SSS) or two sides and the included angle (SAS). It is used, for example, to find a missing side like $a = 13, b = 24, c = 24$ using the corresponding angle.

Questions & Discussion

Q: How many outcomes are possible when rolling a standard number cube 4 times? Using the fundamental counting principle, because each cube has 6 sides, the result is $6 \times 6 \times 6 \times 6 = 6^4 = 1296$.

Q: What is the number of ways 4 people can sit in a circular arrangement if one seat is fixed near a door? Since the door acts as a fixed reference point, the arrangement is treated as a linear permutation, making the number of ways $4! = 24$.

Q: Is a geometric series convergent if its ratio is 1? No, the transcript notes that for a series to be convergent, the absolute value of the ratio must be strictly less than one ($|r| < 1$). A series like $1 + 1 + 1 + \dots$ is not convergent.

Q: What is the probability of picking a blue section on a spinner with 360 degrees if the blue section is 30 degrees? The probability is the ratio of the sector's angle to the full circle: $\frac{30^∘}{360^∘} = \frac{1}{12}$.

Q: How do you find the probability of not landing on a certain color (e.g., purple)? This is found by using the complement rule: $1 - P(\text{purple})$. If the probability of purple is $0.25$, the probability of not purple is $0.75$.