Comprehensive Precalculus Study Guide: Algebra, Functions, Geometry, Sequences, and Applications

Algebraic Manipulation and Scientific Notation

Scientific Notation and Total Distance

Scientific notation is a way of expressing very large or very small numbers in the form $a \times 10^n$, where $1 \leq a < 10$ and $n$ is an integer.

• Key Point: To find the total distance traveled, sum all segments and express the result in scientific notation.

• Example: If Sally travels 2.1 km, then 3.0 km, then 20.4 km, the total is $2.1 + 3.0 + 20.4 = 25.5$ km, which is $2.55 \times 10^1$ km in scientific notation.

Ratios and Proportions

Ratios compare quantities and can be used to relate different groups in a population.

• Key Point: If the ratio of men to women is 2:3 and the ratio of adults to children is 5:4, you can set up equations to find other ratios.

• Example: If $M:W = 2:3$ and $(M+W):C = 5:4$, then $M = 2x$, $W = 3x$, $C = 4y$, and $M+W = 5y$. Solve for $x$ and $y$ to find $W:C$.

Index Laws and Simplification

Index laws (laws of exponents) are used to simplify expressions involving powers.

• Key Point: $rac{\sqrt{6x^7y^3}}{2x^2y^{1/2}}$ can be rewritten using index laws and radicals.

• Formula: $\sqrt{a} = a^{1/2}$, $a^m / a^n = a^{m-n}$

• Example: $\frac{\sqrt{6x^7y^3}}{2x^2y^{1/2}} = \frac{(6x^7y^3)^{1/2}}{2x^2y^{1/2}}$

Logarithmic Equations

Logarithms are the inverses of exponentials. Logarithmic equations can often be solved by applying log laws and exponentiation.

• Key Point: $\log_{10} x^2 - \log_{10} x - 12 = 0$ can be solved by combining logs and exponentiating both sides.

• Formula: $\log_a b - \log_a c = \log_a \left(\frac{b}{c}\right)$

Inequalities and Factorization

Solving Linear Inequalities

Linear inequalities can be solved similarly to equations, but the direction of the inequality reverses when multiplying or dividing by a negative number.

• Key Point: To solve $-2x + 3 > -1$, isolate $x$ and express the solution in interval notation.

• Example: $-2x + 3 > -1 \implies -2x > -4 \implies x < 2$ (note the reversal of the inequality).

Factorization and Simplification

Factorization involves expressing an algebraic expression as a product of its factors.

• Key Point: To simplify $\frac{x^2 - 3x - 18}{9 - x^2}$, factor both numerator and denominator and cancel common factors.

• Example: $x^2 - 3x - 18 = (x - 6)(x + 3)$, $9 - x^2 = (3 - x)(3 + x)$

Geometry: Area, Volume, and Trigonometry

Volume of Prisms and Trapezoidal Cross-Sections

The volume of a prism is the area of its cross-section multiplied by its length.

• Key Point: For a trapezoidal cross-section, area $A = \frac{1}{2}(a + b)h$ where $a$ and $b$ are parallel sides, $h$ is height.

• Formula: $\text{Volume} = \text{Area}_{\text{cross-section}} \times \text{length}$

Circle Sectors and Radians

The area of a sector of a circle is proportional to the angle at the center.

• Key Point: $\text{Area of sector} = \frac{\theta}{360^\circ} \pi r^2$ (if $\theta$ in degrees)

• Example: For a sector with $r = 5$ cm and $\theta = 1.6$ radians, convert to degrees if necessary.

Trigonometry in Triangles

Trigonometric ratios relate the angles and sides of right triangles.

• Key Point: Use $\sin$, $\cos$, and $\tan$ to find unknown sides or angles.

• Formula: $\sin \theta = \frac{\text{opposite}}{\text{hypotenuse}}$, $\cos \theta = \frac{\text{adjacent}}{\text{hypotenuse}}$, $\tan \theta = \frac{\text{opposite}}{\text{adjacent}}$

Sequences and Series

Arithmetic Sequences

An arithmetic sequence has a constant difference between consecutive terms.

• Key Point: $n$-th term: $a_n = a_1 + (n-1)d$

• Sum of first $n$ terms: $S_n = \frac{n}{2}(a_1 + a_n)$

• Example: For $a_1 = \frac{1}{2}$, $d = \frac{1}{3}$, $a_{15} = \frac{1}{2} + 14 \times \frac{1}{3}$

Geometric Sequences and Exponential Decay

In a geometric sequence, each term is a constant multiple of the previous term.

• Key Point: $n$-th term: $a_n = a_1 r^{n-1}$

• Exponential decay: $a_n = a_1 (\text{decay factor})^n$

• Example: If $a = 189.23 \times 10^6$ and reduces to $\frac{1}{6}$ every second, solve $a_n < 1$ for $n$.

Functions and Graphs

Polynomial Long Division and Factorization

Polynomial long division is used to divide polynomials, and factorization expresses a polynomial as a product of its factors.

• Key Point: To divide $x^2 + x^2 - 10x + 8$ by $(x-2)$, use long division.

• Factorization: $x^2 + x^2 - 10x + 8 = (x-2)(x-4)$ (example, actual factorization may differ).

Graphing Functions and Stationary Points

Stationary points occur where the derivative of a function is zero. The graph of a function can be sketched using intercepts, stationary points, and end behavior.

• Key Point: Find $f'(x)$, solve $f'(x) = 0$ for stationary points, and use these to sketch the graph.

Conic Sections: Ellipses and Parabolas

Ellipses

An ellipse is the set of all points such that the sum of the distances from two fixed points (foci) is constant.

• Standard form: $\frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1$

• Center: $(h, k)$

• Semi-axes: $a$ and $b$ are the lengths of the semi-major and semi-minor axes.

Parabolas

A parabola is the set of all points equidistant from a fixed point (focus) and a fixed line (directrix).

• Standard form: $y = a(x-h)^2 + k$ or $x = a(y-k)^2 + h$

• Completing the square: Used to rewrite quadratic equations in vertex form.

Vectors

Vector Operations and Unit Vectors

Vectors have both magnitude and direction. A unit vector has a magnitude of 1.

• Key Point: To find a unit vector in the direction of $\vec{b}$, divide $\vec{b}$ by its magnitude.

• Formula: $\text{Unit vector} = \frac{\vec{b}}{|\vec{b}|}$

Exponential and Logarithmic Applications

Population Growth and Exponential Models

Exponential growth can be modeled by $N = N_0 e^{kt}$, where $N_0$ is the initial amount, $k$ is the growth rate, and $t$ is time.

• Key Point: To find the time for a population to reach a certain value, solve for $t$ in the equation.

Compound Interest and Debt

Compound interest and debt accumulation can be modeled by $P_n = P_0 (1 + \frac{r}{100})^n$.

• Key Point: $P_0$ is the initial amount, $r$ is the interest rate per period, $n$ is the number of periods.

Integration and Calculus

Basic Integration

Integration is the reverse process of differentiation and is used to find areas under curves.

• Key Point: $\int x^n dx = \frac{x^{n+1}}{n+1} + C$ for $n \neq -1$

• Definite integrals: $\int_a^b f(x) dx$ gives the net area between $x=a$ and $x=b$.

Integration by Substitution

Integration by substitution is used when the integrand is a composite function.

• Key Point: Let $u = g(x)$, then $du = g'(x)dx$.

Applications of Integration

Integration can be used to find areas, volumes, and solve problems in physics and engineering.

Summary Table: Index Laws

Law	Example
$a^m \times a^n = a^{m+n}$	$x^2 \times x^3 = x^5$
$a^m / a^n = a^{m-n}$	$x^5 / x^2 = x^3$
$(a^m)^n = a^{mn}$	$(x^2)^3 = x^6$
$a^0 = 1$	$x^0 = 1$
$a^{-n} = 1/a^n$	$x^{-2} = 1/x^2$

Summary Table: Common Geometric Formulas

Shape	Area	Volume
Rectangle	$A = lw$	$V = lwh$
Triangle	$A = \frac{1}{2}bh$	-
Trapezium	$A = \frac{1}{2}(a+b)h$	-
Circle	$A = \pi r^2$	-
Prism	-	$V = \text{Area}_{\text{base}} \times \text{height}$

Additional info: Some calculus and advanced integration topics may exceed standard Precalculus, but are included for completeness as they appear in the provided materials.