Precalculus Final Exam Review: Key Concepts and Applications

Review of Precalculus Concepts

Equations, Inequalities, and Applications

This section covers solving linear and quadratic equations, inequalities, and their applications in various contexts.

• Linear Equations: An equation of the form $ax + b = 0$ can be solved for $x$ by isolating the variable.

• Quadratic Equations: Equations of the form $ax^2 + bx + c = 0$ can be solved using factoring, completing the square, or the quadratic formula: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

• Inequalities: Solve inequalities by similar methods as equations, but remember to reverse the inequality sign when multiplying or dividing by a negative number.

• Applications: Problems may involve finding the slope-intercept form of a line, solving for unknowns in word problems, or interpreting solutions in context.

Example: Find the slope-intercept form of the line passing through $(-1,7)$ and $(4,27)$. Use the formula $m = \frac{y_2 - y_1}{x_2 - x_1}$, then $y = mx + b$.

The Rectangular Coordinate System, Lines, and Circles

This topic explores graphing equations, finding equations of lines, and understanding the geometry of circles.

• Coordinate System: The plane is defined by the $x$ and $y$ axes. Points are given as $(x, y)$.

• Lines: The slope-intercept form is $y = mx + b$. The point-slope form is $y - y_1 = m(x - x_1)$.

• Circles: The standard equation is $(x - h)^2 + (y - k)^2 = r^2$, where $(h, k)$ is the center and $r$ is the radius.

Example: Find the equation of a circle with center $(2, -3)$ and radius $5$:$(x - 2)^2 + (y + 3)^2 = 25$

Functions and Their Properties

Functions describe relationships between variables. Key properties include domain, range, composition, and inverses.

• Definition: A function $f$ assigns each input $x$ exactly one output $f(x)$.

• Domain and Range: The domain is the set of all possible inputs; the range is the set of all possible outputs.

• Composition: $(f \circ g)(x) = f(g(x))$

• Inverse Functions: If $f(x)$ is invertible, $f^{-1}(y)$ gives the input $x$ for output $y$.

Example: If $f(x) = 2x + 3$, then $f^{-1}(y) = \frac{y - 3}{2}$

Polynomial and Rational Functions

Polynomials are expressions of the form $a_nx^n + ... + a_1x + a_0$. Rational functions are ratios of polynomials.

• Polynomial Functions: The degree is the highest power of $x$. Roots are found by factoring or using the quadratic formula.

• Rational Functions: Functions of the form $\frac{P(x)}{Q(x)}$, where $P$ and $Q$ are polynomials.

• Domain: Exclude values that make the denominator zero.

Example: $f(x) = \frac{x^2 - 1}{x - 1}$ is undefined at $x = 1$.

Exponential and Logarithmic Functions

Exponential functions have the form $f(x) = a^x$. Logarithmic functions are the inverses: $f(x) = \log_a x$.

• Exponential Growth/Decay: $y = a b^x$ models growth if $b > 1$, decay if $0 < b < 1$.

• Logarithms: $\log_a x$ is the exponent to which $a$ must be raised to get $x$.

• Properties:

  • $\log_a (xy) = \log_a x + \log_a y$

  • $\log_a \left(\frac{x}{y}\right) = \log_a x - \log_a y$

  • $\log_a (x^r) = r \log_a x$

Example: Solve $2^x = 8$: $x = \log_2 8 = 3$

Trigonometric Functions and Applications

Trigonometry studies the relationships between angles and sides in triangles, and the properties of periodic functions.

• Basic Functions: $\sin x$, $\cos x$, $\tan x$

• Unit Circle: Defines trigonometric functions for all real numbers.

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

• Applications: Used to solve problems involving angles, lengths, and periodic phenomena.

Example: If $\theta$ is an angle in a right triangle with opposite side $3$and adjacent side $4$, then $\tan \theta = \frac{3}{4}$.

Graphs of Functions and Transformations

Understanding how functions are graphed and how transformations affect their appearance is essential in precalculus.

• Vertical and Horizontal Shifts: $f(x) + k$ shifts up, $f(x - h)$ shifts right.

• Reflections: $-f(x)$ reflects over the $x$-axis.

• Stretching and Compressing: $af(x)$ stretches vertically if $|a| > 1$, compresses if $0 < |a| < 1$.

Example: The graph of $y = \sin(x)$ shifted up by $2$units is$y = \sin(x) + 2$.

Systems of Equations and Matrices

Systems of equations can be solved using substitution, elimination, or matrix methods.

• Linear Systems: Two or more equations with two or more variables.

• Matrix Representation: Systems can be written as $AX = B$, where $A$ is a matrix of coefficients.

• Solution Methods: Use row reduction, inverse matrices, or Cramer's rule.

Example: Solve $\begin{cases} x + y = 3 \\ 2x - y = 0 \end{cases}$ by substitution or elimination.

Sequences, Series, and Probability

Sequences are ordered lists of numbers; series are sums of sequences. Probability measures the likelihood of events.

• Arithmetic Sequence: $a_n = a_1 + (n-1)d$

• Geometric Sequence: $a_n = a_1 r^{n-1}$

• Sum of Series: Arithmetic: $S_n = \frac{n}{2}(a_1 + a_n)$; Geometric: $S_n = a_1 \frac{1 - r^n}{1 - r}$

• Probability: $P(E) = \frac{\text{number of favorable outcomes}}{\text{total number of outcomes}}$

Example: Find the sum of the first $5$terms of the sequence $2, 4, 6, 8, 10$: $S_5 = \frac{5}{2}(2 + 10) = 30$

Tables and Data Interpretation

Many problems involve interpreting tabular data, identifying patterns, and constructing functions from data points.

x	f(x)
-2	2
1	3
3	4

Example: Use the table to find a function $f(x)$ that fits the data. Try linear or quadratic models as appropriate.

Graphical Analysis and Applications

Graphical problems require interpreting the meaning of slopes, intercepts, and areas under curves in applied contexts.

• Average Rate of Change: $\frac{f(b) - f(a)}{b - a}$

• Interpreting Graphs: Identify intervals of increase/decrease, maxima/minima, and points of intersection.

• Applications: Use graphs to model real-world phenomena such as population growth, motion, and economics.

Example: The slope of a tangent line to a curve at a point represents the instantaneous rate of change.

Trigonometric Applications and Geometry

Trigonometry is used to solve problems involving triangles, circles, and periodic motion.

• Law of Sines: $\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}$

• Law of Cosines: $c^2 = a^2 + b^2 - 2ab \cos C$

• Applications: Find unknown sides or angles in triangles, model periodic phenomena.

Example: Find the length of a side in a triangle given two sides and the included angle using the Law of Cosines.

Summary Table: Key Precalculus Formulas

Topic	Formula	Example
Linear Equation	$y = mx + b$	Find $m$ from two points
Quadratic Formula	$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$	Solve $x^2 + 2x + 1 = 0$
Exponential Growth	$y = ab^x$	$y = 2 \cdot 3^x$
Logarithm	$y = \log_a x$	$\log_2 8 = 3$
Trigonometric Ratio	$\sin \theta = \frac{\text{opposite}}{\text{hypotenuse}}$	$\sin 30^\circ = 0.5$
Average Rate of Change	$\frac{f(b) - f(a)}{b - a}$	Find change over interval

Additional info: These notes synthesize the main topics and problem types found in the provided Precalculus final exam review questions, covering all major areas relevant to a college-level precalculus course.