Comprehensive Precalculus Final Exam Study Guide

Zero-Product Property

Definition and Application

The Zero-Product Property states that if the product of two or more factors is zero, then at least one of the factors must be zero.

• Key Point: If $ab = 0$, then $a = 0$ or $b = 0$.

• Application: Used to solve polynomial equations by factoring.

• Example: Solve $x^2 - 5x = 0$ by factoring: $x(x-5) = 0$ so $x = 0$ or $x = 5$.

Order of Operations

PEMDAS/BODMAS

Order of operations ensures consistent results when evaluating expressions.

• Parentheses/Brackets

• Exponents/Orders

• Multiplication and Division (left to right)

• Addition and Subtraction (left to right)

• Example: $3 + 6 \times (5 + 4) \div 3^2 = 3 + 6 \times 9 \div 9 = 3 + 6 \times 1 = 3 + 6 = 9$

Equations

Types and Solving Methods

• Identity: True for all values (e.g., $x + 0 = x$)

• Conditional: True for some values (e.g., $x + 2 = 5$)

• Inconsistent: No solution (e.g., $x + 1 = x + 2$)

• Solving Linear Equations: Isolate the variable using inverse operations.

• Solving Quadratic Equations: Factor, complete the square, or use the quadratic formula.

• Quadratic Formula: $x = \frac{ -b \pm \sqrt{b^2 - 4ac} }{2a}$

• Completing the Square: Transform $ax^2 + bx + c = 0$ into $(x + d)^2 = e$ form.

• Solving Higher Degree Polynomials: Factor and apply the zero-product property.

• Solving Rational Expressions: Find a common denominator, clear denominators, and solve.

• Solving Radical Equations: Isolate the radical, square both sides, and check for extraneous solutions.

• Extraneous Solutions: Solutions that arise from the solving process but do not satisfy the original equation.

Inequalities

Properties and Solution Sets

• Properties: Addition, subtraction, multiplication/division (reverse sign when multiplying/dividing by a negative).

• Solution Sets: Expressed in interval notation or on a number line.

• Absolute Value Inequalities: $|x| < a$ means $-a < x < a$; $|x| > a$ means $x < -a$ or $x > a$.

The Coordinate Plane

Points, Distance, and Midpoints

• Coordinates: $(x, y)$ pairs on the xy-plane.

• Distance Formula: $d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$

• Midpoint Formula: $M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)$

Graphs of Equations

Plotting and Intercepts

• Graphs: Visual representations of equations on the coordinate plane.

• Independent Variable: Usually $x$; Dependent Variable: Usually $y$.

• Plotting: Substitute values for $x$ to find corresponding $y$ values.

• x-Intercept: Where $y = 0$.

• y-Intercept: Where $x = 0$.

Symmetry

Types and Function Properties

• Symmetry across a Line: e.g., y-axis symmetry: $f(-x) = f(x)$ (even function).

• Symmetry about a Point: e.g., origin symmetry: $f(-x) = -f(x)$ (odd function).

• Even Functions: Symmetric about the y-axis.

• Odd Functions: Symmetric about the origin.

Circles

Equation and Properties

• Standard Form: $(x - h)^2 + (y - k)^2 = r^2$ where $(h, k)$ is the center and $r$ is the radius.

• Completing the Square: Used to rewrite general equations into standard form.

Lines

Forms and Properties

• Slope-Intercept Form: $y = mx + b$

• Point-Slope Form: $y - y_1 = m(x - x_1)$

• Slope: $m = \frac{y_2 - y_1}{x_2 - x_1}$

• Parallel Lines: Same slope, different y-intercepts.

• Perpendicular Lines: Slopes are negative reciprocals: $m_1 \cdot m_2 = -1$

Functions

Definition and Representation

• Domain: Set of all possible input values ($x$).

• Range: Set of all possible output values ($y$).

• Ways to Express: Equations, tables, graphs, mappings.

• Function Notation: $f(x)$

• Vertical Line Test: A graph represents a function if no vertical line intersects it more than once.

Increasing vs Decreasing Functions

Behavior on Intervals

• Increasing: $f(x_1) < f(x_2)$ for $x_1 < x_2$

• Decreasing: $f(x_1) > f(x_2)$ for $x_1 < x_2$

Relative Maxima and Minima

Turning Points

• Relative Maximum: Highest point in a local region.

• Relative Minimum: Lowest point in a local region.

Average Rate of Change

Definition and Interpretation

• Formula: $\frac{f(b) - f(a)}{b - a}$

• Secant Line: Line through $(a, f(a))$ and $(b, f(b))$

• Physical Interpretation: Average speed, growth rate, etc.

Function Transformations

Types of Transformations

• Vertical Shifts: $f(x) + k$

• Horizontal Shifts: $f(x - h)$

• Reflections: Across x-axis: $-f(x)$; across y-axis: $f(-x)$

• Vertical Stretch/Compression: $a f(x)$

• Horizontal Stretch/Compression: $f(bx)$

Compositions of Functions

Definition and Notation

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

• Order matters: $f(g(x)) \neq g(f(x))$ in general.

Inverse Functions

Finding and Interpreting Inverses

• One-to-One: Each output corresponds to exactly one input.

• Finding Inverse: Swap $x$ and $y$, solve for $y$.

• Domain and Range: Swap for the inverse function.

• Graphical Interpretation: Reflection across $y = x$.

• Restricted Domains: Sometimes needed to ensure invertibility.

Quadratic Functions

Properties and Applications

• Standard Form: $f(x) = ax^2 + bx + c$

• Vertex Form: $f(x) = a(x - h)^2 + k$

• Vertex: $\left( -\frac{b}{2a}, f\left(-\frac{b}{2a}\right) \right)$

• Roots: Solutions to $f(x) = 0$

• Applications: Maximum/minimum problems, projectile motion, area optimization.

Polynomial Functions

Degree, Roots, and Behavior

• Leading Term: Term with the highest degree.

• End Behavior: Determined by leading term's degree and sign.

• Roots: $c$ is a root if $f(c) = 0$

• Multiplicity: Even multiplicity: graph bounces; odd: crosses the axis.

• Number of Roots: Up to $n$ for degree $n$.

• Turning Points: Up to $n-1$ for degree $n$.

Polynomial Division

Long and Synthetic Division

• Long Division: Divide polynomials similarly to numbers.

• Synthetic Division: Shortcut for dividing by linear factors.

Rational Functions

Asymptotes and Discontinuities

• Vertical Asymptotes: Where denominator is zero (and numerator is not zero).

• Holes: Common factors in numerator and denominator.

• Horizontal Asymptotes: Determined by degrees of numerator and denominator.

• Oblique Asymptotes: When degree of numerator is one more than denominator.

Exponential Functions

Definition and Properties

• General Form: $f(x) = a^x$, $a > 0$

• Base: The constant $a$

• Domain: $(-\infty, \infty)$

• Range: $(0, \infty)$

• Euler's Number: $e \approx 2.718$

• Transformations: Shifts, stretches, and reflections apply as with other functions.

Properties of Exponents

Rules

• $a^m \cdot a^n = a^{m+n}$

• $\frac{a^m}{a^n} = a^{m-n}$

• $(a^m)^n = a^{mn}$

• $(ab)^n = a^n b^n$

• $a^0 = 1$ (if $a \neq 0$)

• $a^{-n} = \frac{1}{a^n}$

Applications of Exponential Functions

Compound Interest, Half-Life, and Population Growth

• Compound Interest (Annually): $P(t) = P_0(1 + r)^t$

• Compound Interest (Continuously): $P(t) = P_0 e^{rt}$

• Half-Life: $P(t) = P_0 e^{kt}$, where $P(t^*) = \frac{1}{2}P_0$ at half-life $t^*$

• Population Growth/Decay: $P(t) = P_0 e^{kt}$

Logarithms

Definition and Properties

• Definition: $y = \log_a x$ means $a^y = x$

• Properties:

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

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

  • $\log_a (x^r) = r \log_a x$

• Common Logarithm: $\log_{10} x$

• Natural Logarithm: $\ln x = \log_e x$

• Domain: $(0, \infty)$

• Range: $(-\infty, \infty)$

• Graphs: Logarithmic functions increase slowly, pass through $(1, 0)$.

Solving Exponential and Logarithmic Equations

Methods

• Isolate the exponential or logarithmic part.

• Apply logarithms to both sides if necessary.

• Check for extraneous solutions, especially with logarithms.

Inverse of Exponential and Logarithmic Functions

Relationship

• Inverse of $f(x) = e^x$ is $f^{-1}(x) = \ln x$

Angles

Measurement and Conversion

• Standard Position: Vertex at origin, initial side on positive x-axis.

• Initial vs Terminal Side: Where angle starts and ends.

• Coterminal Angles: Differ by multiples of $360^\circ$ or $2\pi$ radians.

• Degrees and Radians: $180^\circ = \pi$ radians

• Conversion: Degrees to radians: $\theta_{\text{rad}} = \theta_{\text{deg}} \cdot \frac{\pi}{180}$

Arc Length and Area of a Sector

Formulas

• Arc Length: $S = r\theta$ (with $\theta$ in radians)

• Area of Sector: $A = \frac{1}{2} r^2 \theta$ (with $\theta$ in radians)

Unit Circle

Key Angles and Coordinates

• Know angles that are multiples of $30^\circ$ and $45^\circ$ in degrees and radians.

• Know exact and approximate coordinates for these angles.

• Use the unit circle to evaluate trigonometric functions.

Trigonometric Functions

Definitions and Graphs

• Six Basic Functions: $\sin$, $\cos$, $\tan$, $\csc$, $\sec$, $\cot$

• SOHCAHTOA: Mnemonic for right triangle definitions.

• Domain and Range: $y = \sin t$ and $y = \cos t$ have domain $(-\infty, \infty)$, range $[-1, 1]$

• Graphs: Know basic shapes and transformations (amplitude, period, phase shift).

Inverse Trigonometric Functions

Properties and Evaluation

• Restricted Domains: Needed to make trig functions invertible.

• Evaluation: Use unit circle and reference angles.

• Compositions: $\sin(\arcsin x) = x$ for $x$ in $[-1, 1]$

Simplifying Trig Expressions and Verifying Identities

Standard Trig Identities

• Pythagorean Identities: $\sin^2 x + \cos^2 x = 1$

• Reciprocal Identities: $\csc x = \frac{1}{\sin x}$, $\sec x = \frac{1}{\cos x}$, $\cot x = \frac{1}{\tan x}$

• Quotient Identities: $\tan x = \frac{\sin x}{\cos x}$, $\cot x = \frac{\cos x}{\sin x}$

• Even/Odd Identities: $\sin(-x) = -\sin x$, $\cos(-x) = \cos x$

• Sum and Difference Formulas: Provided as needed.

• Double Angle: $\sin 2x = 2\sin x \cos x$, $\cos 2x = \cos^2 x - \sin^2 x$

• Half Angle: $\sin^2 x = \frac{1 - \cos 2x}{2}$, $\cos^2 x = \frac{1 + \cos 2x}{2}$

Solving Right Triangles and Applications

Physical Applications

• Use trigonometric ratios to find missing sides or angles in right triangles.

• Apply to real-world problems (e.g., height, distance, angle of elevation).