College Algebra Study Guide: Transformations, Domains, and Graphs of Functions

UNIT 3: How Do We Transform Functions?

Graph Transformations from Graphs and Symbols

Understanding how functions are transformed is essential in College Algebra. Transformations include shifts, stretches, compressions, and reflections, which alter the appearance and position of a graph.

• Parent Function: The simplest form of a function, such as $f(x) = x$, $f(x) = x^2$, or $f(x) = |x|$.

• Transformation Types:

  • Vertical and Horizontal Shifts: Moving the graph up/down or left/right.

  • Reflections: Flipping the graph over the x-axis or y-axis.

  • Stretches and Compressions: Changing the steepness or width of the graph.

• Identifying Transformations: Given a graph or equation, determine the parent function and describe the transformations applied.

• Example: Given $f(x) = 2(x - 3)^2 + 1$, identify the parent function $f(x) = x^2$, a horizontal shift right by 3, vertical stretch by 2, and vertical shift up by 1.

Using Equations to Determine Transformations

Equations can be used to analyze and graph transformations step by step, labeling all changes and describing them mathematically.

• Order of Transformations: Apply transformations in the correct sequence for accurate graphing.

• Matching Graphs to Equations: Use knowledge of transformations to connect equations to their graphs.

• Describing Steps: Use mathematical language to explain each transformation.

• Example: For $g(x) = -f(x + 2) + 3$, reflect $f(x)$ over the x-axis, shift left by 2, and up by 3.

Determining Domain from an Equation

The domain of a function is the set of all possible input values (x-values) for which the function is defined.

• Algebraic Methods:

  • Solve for the domain of rational functions (denominator ≠ 0).

  • Solve for the domain of radical functions (expression under even root ≥ 0).

  • Solve for the domain of logarithmic functions (argument > 0).

  • Identify the domain of polynomial functions (all real numbers).

• Interval Notation: Write domains using interval notation, e.g., $(-cfty, 2) 5ccup (2, 5cfty)$.

• Example: For $f(x) = \frac{1}{x-3}$, domain is $x \neq 3$ or $(-\infty, 3) \cup (3, \infty)$.

Determining x- and y-Intercepts from an Equation

Intercepts are points where the graph crosses the axes. They are useful for sketching and analyzing functions.

• y-Intercept: Set $x = 0$ and solve for $y$.

• x-Intercepts: Set $y = 0$ and solve for $x$.

• Types of Functions: Find intercepts for linear, quadratic, polynomial, rational, radical, logarithmic, and exponential functions.

• Example: For $f(x) = x^2 - 4$, x-intercepts are $x = 2$ and $x = -2$; y-intercept is $y = -4$.

End Behavior of a Polynomial

End behavior describes how a polynomial function behaves as $x$ approaches $+5cfty$ or $-5cfty$.

• Degree and Leading Coefficient: These determine the direction of the graph's ends.

• Factored and Standard Form: Analyze end behavior from both forms.

• Up/Down Language: Use terms like "rises left, falls right" to describe end behavior.

• Example: For $f(x) = -x^3$, as $x \to \infty$, $f(x) \to -\infty$; as $x \to -\infty$, $f(x) \to \infty$.

UNIT 5: How are Different Representations of Functions Connected?

Relate Linear Equations to Graphs

Linear equations can be represented in various forms and are directly connected to their graphs.

• Slope-Intercept Form: $y = mx + b$, where $m$ is slope and $b$ is y-intercept.

• Standard Form: $Ax + By = C$.

• Graphing: Use slope and intercepts to sketch the line.

• Applications: Write and analyze cost, revenue, and profit functions.

• Example: For $y = 2x + 3$, slope is 2, y-intercept is 3.

Relate Equations to Graphs of Quadratics in Standard, Vertex, and Factored Form

Quadratic functions can be written in multiple forms, each revealing different properties of the graph.

• Standard Form: $y = ax^2 + bx + c$

• Vertex Form: $y = a(x-h)^2 + k$

• Factored Form: $y = a(x-r_1)(x-r_2)$

• Vertex: The point $(h, k)$ in vertex form.

• Axis of Symmetry: $x = h$ in vertex form, $x = -\frac{b}{2a}$ in standard form.

• Direction: Opens upward if $a > 0$, downward if $a < 0$.

• Intercepts: Solve for x- and y-intercepts algebraically.

• Sketching: Use key points and symmetry to draw the graph.

• Example: $y = (x-1)^2 + 2$ has vertex at $(1,2)$, opens upward.

Convert Between Forms of Quadratic Equations

Converting between standard, vertex, and factored forms helps in graphing and solving quadratic equations.

• FOIL Method: Expand factored form to standard form.

• Completing the Square: Convert standard form to vertex form.

• Example: $y = x^2 + 4x + 4$ can be written as $y = (x+2)^2$.

Relate Equations to Graphs of Exponential Functions

Exponential functions model rapid growth or decay and have distinctive graphs.

• Parent Exponential Function: $f(x) = a^x$

• Transformations: Apply shifts and stretches using transformation rules.

• Horizontal Asymptote: The line $y = k$ that the graph approaches but never crosses.

• Example: $f(x) = 2^{x} + 3$ has a horizontal asymptote at $y = 3$.

Relate Equations to Graphs of Logarithmic Functions

Logarithmic functions are the inverses of exponential functions and have unique graph features.

• Parent Logarithmic Function: $f(x) = \log_a(x)$

• Transformations: Use transformation rules to graph variations.

• Vertical Asymptote: The line $x = h$ that the graph approaches.

• Example: $f(x) = \log_2(x-1)$ has a vertical asymptote at $x = 1$.

Relate Equations to Graphs of Polynomials in Factored or Standard Form

Polynomials can be analyzed using their factored or standard forms to find zeros, multiplicities, and graph behavior.

• Zeros and Multiplicities: Solve $f(x) = 0$ to find x-intercepts and their multiplicities.

• Cross/Touch Behavior: Even multiplicity means the graph touches the axis; odd multiplicity means it crosses.

• Test Points: Use values between zeros to determine graph position.

• Sketching: Draw the graph using zeros, multiplicities, and end behavior.

• Example: $f(x) = (x-2)^2(x+1)$ has zeros at $x=2$ (multiplicity 2, touches) and $x=-1$ (multiplicity 1, crosses).

Remainder and Factor Theorems for Polynomials

These theorems help determine factors and zeros of polynomials using division methods.

• Remainder Theorem: The remainder of $f(x)$ divided by $(x-c)$ is $f(c)$.

• Factor Theorem: If $f(c) = 0$, then $(x-c)$ is a factor of $f(x)$.

• Synthetic Division: A shortcut for dividing polynomials by linear factors.

• Testing Zeros: Use synthetic division to test if a value is a zero or factor.

• Example: For $f(x) = x^3 - 4x^2 + 5x - 2$, test $x=1$ using synthetic division.

Finding Zeros of Polynomials

Zeros can be found using several methods, including factoring, synthetic division, and the Rational Roots Theorem.

• Rational Roots Theorem: Possible rational zeros are factors of the constant term over factors of the leading coefficient.

• Factoring: Use factoring techniques for trinomials, sums/differences of cubes, and squares.

• Long Division: Divide polynomials to find factors and zeros.

• Example: For $f(x) = x^3 - 2x^2 - x + 2$, possible rational zeros are $\pm1, \pm2$.

Form of Function	Key Features	Transformation/Analysis
Linear ($y = mx + b$)	Slope, y-intercept	Graph using slope and intercept
Quadratic (Standard, Vertex, Factored)	Vertex, axis of symmetry, direction, intercepts	Convert between forms, sketch graph
Exponential ($y = a^x$)	Horizontal asymptote, growth/decay	Apply transformations, graph
Logarithmic ($y = \log_a(x)$)	Vertical asymptote, domain	Apply transformations, graph
Polynomial (Factored/Standard)	Zeros, multiplicities, end behavior	Factor, use theorems, sketch graph

Additional info: This guide expands on the syllabus outline by providing definitions, examples, and formulas for each topic, ensuring a self-contained study resource for College Algebra students.