College Algebra

Chapter P: Prerequisites

• Real Numbers and Properties

  • Sets of numbers: Natural numbers, Integers, Rational numbers, and Irrational numbers form the set of Real Numbers $\mathbb{R}$.

  • Algebraic properties: Commutative, Associative, and Distributive properties govern real number operations.

  • Distance on the real line is defined using absolute value: $d(A, B) = |b - a|$.

• Exponents and Radicals

  • Laws of exponents: $a^m \cdot a^n = a^{m+n}$, $\frac{a^m}{a^n} = a^{m-n}$, and $(a^m)^n = a^{m n}$.

  • Fractional exponents: $a^{m/n} = \sqrt[n]{a^m} = (\sqrt[n]{a})^m$.

  • Rationalizing denominators involves removing radicals from the denominator using conjugate expressions.

• Algebraic Expressions and Factoring

  • Polynomial expansion utilizes standard formulas such as $(a + b)^2 = a^2 + 2 a b + b^2$ and $(a + b)(a - b) = a^2 - b^2$.

  • Factoring techniques include finding the greatest common factor (GCF), factoring by grouping, trinomial factoring, and using sum/difference of cubes formulas: $a^3 \pm b^3 = (a \pm b)(a^2 \mp a b + b^2)$.

• Rational Expressions

  • Domain restrictions must be identified by finding values that cause division by zero.

  • Operations require finding the least common denominator (LCD) to add or subtract fractions.

Chapter 1: Equations and Inequalities

• Linear Equations and Mathematical Models

  • A linear equation in one variable has the form $a x + b = 0$ where $a \neq 0$.

  • Solving formula-based word problems involves modeling physical, financial, or geometric situations.

• Quadratic Equations

  • Factoring method: Set expression to $0$ and apply the zero-product property.

  • Completing the square: Transform $x^2 + b x$ into a perfect square trinomial by adding $(\frac{b}{2})^2$.

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

  • The discriminant $D = b^2 - 4 a c$ determines the nature of roots (two real, one repeated, or two complex conjugate roots).

• Complex Numbers

  • Expressed in standard form $a + b i$, where $i = \sqrt{-1}$ and $i^2 = -1$.

  • Conjugates $a + b i$ and $a - b i$ multiply to yield a real number $a^2 + b^2$.

• Other Types of Equations

  • Radical equations: Isolate radical terms and check for extraneous solutions.

  • Equations quadratic in form: Use substitution $u = g(x)$ to reduce degree.

• Inequalities

  • Linear inequalities are solved similarly to equations, flipping inequality symbols when multiplying or dividing by negative values.

  • Non-linear inequalities (polynomial/rational) require finding key critical numbers and constructing sign charts.

  • Absolute value inequalities: $|x| < c$ translates to $-c < x < c$, and $|x| > c$ translates to $x < -c$ or $x > c$.

Chapter 2: Coordinates and Graphs

• The Coordinate Plane and Graphs

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

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

  • Equation of a circle centered at $(h, k)$ with radius $r$: $(x - h)^2 + (y - k)^2 = r^2$.

• Lines

  • Slope formula: $m = \frac{y<em>2 - y</em>1}{x<em>2 - x</em>1}$.

  • Point-slope form: $y - y<em>1 = m (x - x</em>1)$.

  • Slope-intercept form: $y = m x + b$.

  • Parallel lines have equal slopes $m<em>1 = m</em>2$; perpendicular lines have negative reciprocal slopes $m<em>1 \cdot m</em>2 = -1$.

• Functions and Transformations

  • Definition: A rule assigning each element $x$ in domain $D$ to exactly one element $f(x)$ in range $R$.

  • Vertical Line Test determines if a graph represents a function.

  • Transformations:

  • Vertical shifts: $y = f(x) + c$.

  • Horizontal shifts: $y = f(x - c)$.

  • Reflections: $y = -f(x)$ (across x-axis), $y = f(-x)$ (across y-axis).

  • Stretching/Compressing: $y = c f(x)$ or $y = f(c x)$.

• Combining Functions

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

  • One-to-one functions pass the Horizontal Line Test and possess inverses $f^{-1}(x)$ satisfying $f(f^{-1}(x)) = x$.

Chapter 3: Polynomial and Rational Functions

• Quadratic Functions

  • Standard/Vertex form: $f(x) = a (x - h)^2 + k$ with vertex at $(h, k)$.

  • Maximum or minimum value occurs at $x = -\frac{b}{2 a}$.

• Polynomial Functions of Higher Degree

  • End behavior determined by the leading coefficient and degree $n$ (Leading Coefficient Test).

  • Real zeros: Rational Zeros Theorem tests possible rational roots $\pm \frac{p}{q}$.

  • Synthetic Division simplifies division of polynomials by linear factors $x - c$.

  • Factor Theorem: $x - c$ is a factor of $P(x)$ if and only if $P(c) = 0$.

• Rational Functions

  • Domain excludes values causing denominator $Q(x) = 0$.

  • Vertical Asymptotes occur where denominator is zero (after simplifying).

  • Horizontal Asymptotes determined by comparing degrees of numerator and denominator.

Chapter 4: Exponential and Logarithmic Functions

• Exponential Functions

  • Base $b > 0, b \neq 1$ formula: $f(x) = a b^x$. Natural exponential base $e \approx 2.71828$.

  • Compound interest continuous formula: $A(t) = P e^{r t}$.

• Logarithmic Functions

  • Inverse of exponential functions: $y = \log_b(x) \iff b^y = x$.

  • Natural logarithm: $\ln(x) = \log_e(x)$.

  • Properties of Logarithms:

  • Product Rule: $\log<em>b(x y) = \log</em>b(x) + \log_b(y)$.

  • Quotient Rule: $\log<em>b(\frac{x}{y}) = \log</em>b(x) - \log_b(y)$.

  • Power Rule: $\log<em>b(x^c) = c \log</em>b(x)$.

  • Change of Base Formula: $\log<em>b(x) = \frac{\log</em>a(x)}{\log_a(b)}$.

• Exponential and Logarithmic Equations

  • Solve by isolating exponential/logarithmic terms and taking logarithms or exponentiating both sides.

Chapter 5 & 6: Trigonometric Functions

• Angles and Radian Measure

  • Radian conversion: $\pi \text{ radians} = 180^\circ$.

  • Arc length formula: $s = r \theta$ (where $\theta$ is in radians).

• Right Triangle Trigonometry

  • $\sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}}$

  • $\cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}}$

  • $\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}$

• Unit Circle Approach

  • Circle equation: $x^2 + y^2 = 1$.

  • For a point $(x, y)$ corresponding to angle $t$: $\cos(t) = x$, $\sin(t) = y$, and $\tan(t) = \frac{y}{x}$.

  • Fundamental Pythagorean Identity: $\sin^2(t) + \cos^2(t) = 1$.

• Periodic Graphs

  • Sine and Cosine functions have period $2 \pi$ and amplitude $A$ for $y = A \sin(B (x - C))$.

Chapter 7: Analytic Trigonometry

• Trigonometric Identities

  • Reciprocal, Quotient, and Pythagorean identities.

  • Sum and Difference Formulas:

  • $\sin(\alpha \pm \beta) = \sin(\alpha) \cos(\beta) \pm \cos(\alpha) \sin(\beta)$

  • $\cos(\alpha \pm \beta) = \cos(\alpha) \cos(\beta) \mp \sin(\alpha) \sin(\beta)$

  • Double-Angle Formulas:

  • $\sin(2 \theta) = 2 \sin(\theta) \cos(\theta)$

  • $\cos(2 \theta) = \cos^2(\theta) - \sin^2(\theta)$

• Trigonometric Equations

  • Solved by isolating trigonometric terms, factoring, and using inverse trigonometric functions to find all solutions in specified intervals.

Chapter 8: Polar Coordinates and Vectors

• Polar Coordinates

  • Point representation: $(r, \theta)$.

  • Conversion to Rectangular: $x = r \cos(\theta)$, $y = r \sin(\theta)$.

  • Conversion to Polar: $r^2 = x^2 + y^2$, $\tan(\theta) = \frac{y}{x}$.

• Vectors

  • Represented in component form $\mathbf{v} = \langle a, b \rangle = a \mathbf{i} + b \mathbf{j}$.

  • Magnitude: $||\mathbf{v}|| = \sqrt{a^2 + b^2}$.

  • Dot Product: $\mathbf{u} \cdot \mathbf{v} = u<em>1 v</em>1 + u<em>2 v</em>2 = ||\mathbf{u}|| ||\mathbf{v}|| \cos(\theta)$.

Chapter 9: Systems of Equations and Inequalities

• Linear Systems

  • Solved using Substitution, Elimination, or Gaussian Elimination with matrices.

• Matrix Operations

  • Addition, scalar multiplication, and matrix multiplication.

  • Inverses of matrices $A^{-1}$ used to solve system $A \mathbf{x} = \mathbf{b}$.

Chapter 10: Analytic Geometry

• Conic Sections

  • Parabola: Standard form $(x - h)^2 = 4 p (y - k)$ with vertex $(h, k)$ and focus $(h, k + p)$.

  • Ellipse: Standard form $\frac{(x - h)^2}{a^2} + \frac{(y - k)^2}{b^2} = 1$ with foci at distance $c = \sqrt{a^2 - b^2}$ from center.

  • Hyperbola: Standard form $\frac{(x - h)^2}{a^2} - \frac{(y - k)^2}{b^2} = 1$ with foci at distance $c = \sqrt{a^2 + b^2}$ from center.

Chapter 11: Sequences and Series

• Sequences and Summation Notation

  • Sequence $a<em>n$, summation notation $\sum</em>{k=1}^n a_k$.

• Arithmetic Sequences

  • $n$-th term: an = a1 + (n - 1) d$.

  • Sum of n$terms:$Sn = \frac{n}{2} (a1 + a_n)$.</p></li></ul></li><li><p><strong>Geometric Sequences</strong></p><ul><li><p>$n$-th term:$an = a1 r^{n-1}$.</p></li><li><p>Sum of finite geometric series:$Sn = a1 \frac{1 - r^n}{1 - r}$.</p></li><li><p>Sum of infinite geometric series (when$|r| < 1$):$S = \frac{a_1}{1 - r}$$.