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

UNIT 3: How Do We Transform Functions?

Graph Transformations from Graphs and Symbols

Understanding how to transform functions is essential in College Algebra. Transformations allow us to modify parent functions to create new graphs and equations.

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

• Transformation Types: Includes translations (shifts), reflections, stretches, and compressions.

• Graphing Transformed Functions: Use the graph of a parent function to graph a transformed function, e.g., given the graph of $f(x)$, graph $f(x - 2) + 3$.

• Equation of Transformed Parent Function: Use the equation to determine features of the graph.

Example: If $f(x) = x^2$, then $g(x) = (x - 2)^2 + 3$ is the graph of $f(x)$ shifted right by 2 units and up by 3 units.

Steps for Graphing and Describing Transformations

• Identify a parent function given a graph or equation.

• List the transformations applied to a parent function.

• Graph a parent function and its transformation step by step.

• Write an equation for a transformation given a parent function and characteristics.

• Describe the steps of the transformation using mathematical language.

Example: Given $f(x)$, graph $f(x - 2) + 3$ and label all intermediate steps.

UNIT 4: What Can We Learn from an Equation?

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.

• Algebraically solve for the domain of a function given its equation.

• Write domains in interval notation.

Example: The domain of $f(x) = \frac{1}{x-3}$ is all real numbers except $x = 3$; in interval notation: $(-\infty, 3) \cup (3, \infty)$.

Determining x- and y-intercepts from Equations

Intercepts are points where the graph crosses the axes.

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

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

• Apply to linear, quadratic, higher-order polynomial, rational, radical, logarithmic, and exponential functions.

Example: For $f(x) = x^2 - 4$, x-intercepts are found by solving $x^2 - 4 = 0$; $x = 2$ and $x = -2$.

End Behavior of a Polynomial

The end behavior describes how the function behaves as $x$ approaches infinity or negative infinity.

• Identify the degree and leading coefficient of a polynomial.

• Describe end behavior using arrow notation.

Example: For $f(x) = 2x^3$, 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 graphed using slope and intercepts.

• Identify slope ($m$) and y-intercept ($b$) in $y = mx + b$.

• Use slope-intercept form to graph the function.

• Convert between slope-intercept and standard forms.

• Write equations given points and slope.

Example: For $y = 2x + 1$, slope is 2, y-intercept is 1.

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

Quadratic functions can be written and graphed in multiple forms.

• Standard form: $y = ax^2 + bx + c$

• Vertex form: $y = a(x-h)^2 + k$

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

• Identify axis of symmetry: $x = -\frac{b}{2a}$

• Find vertex, intercepts, and sketch the graph.

Example: For $y = (x-1)^2 + 2$, vertex is at $(1, 2)$.

Convert Between Forms of Quadratic Equations

• Convert from vertex form to standard form by expanding.

• Convert from standard form to vertex form by completing the square.

Example: $y = x^2 + 4x + 1$ can be rewritten in vertex form as $y = (x+2)^2 - 3$.

Relate Equations to Graphs of Exponential Functions

Exponential functions model growth and decay and can be transformed and graphed.

• Identify parent exponential function: $f(x) = a \cdot b^x$

• Apply transformations: shifts, stretches, reflections.

• Identify horizontal asymptote.

Example: For $f(x) = 2^{x} + 3$, the horizontal asymptote is $y = 3$.

Relate Exponential and Logarithmic Forms of Equations

Logarithmic functions are the inverses of exponential functions.

• Parent logarithmic function: $f(x) = \log_b(x)$

• Graph using transformation rules.

• Identify vertical asymptote.

Example: For $f(x) = \log_2(x-1)$, the vertical asymptote is $x = 1$.

Relate Equations to Graphs of Polynomials in Factored or Standard Form

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

• Find zeros and their multiplicities from the factored form.

• Identify cross/touch behavior at zeros.

• Evaluate test points between zeros.

• Sketch the graph in factored form.

• Apply the Remainder and Factor Theorems.

• Use synthetic division to test factors and find zeros.

• Apply the Rational Roots Theorem and factoring techniques.

Method	Description
Rational Roots Theorem	Find possible rational zeros using factors of constant and leading coefficient.
Synthetic Division	Test possible zeros and factor polynomials efficiently.
Factoring	Express polynomial as a product of factors to find zeros.

Example: For $f(x) = (x-2)^2(x+1)$, zeros are $x = 2$ (multiplicity 2, touch) and $x = -1$ (cross).

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