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 functions and analyze their graphs.

• Parent Function: The simplest form of a function in a family (e.g., for quadratics).

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

• Graphing Transformed Functions: Use the graph of a parent function to graph a transformed function, such as or .

• Equation of Transformed Parent Function: Use the equation to determine features of the graph, such as shifts and stretches.

Example: Given , the graph of is the graph of shifted 2 units to the right.

Steps for Graphing and Describing Transformations

• Identify the parent function from a graph or equation.

• List all transformations applied to the parent function.

• Graph the parent function and then apply each transformation step-by-step.

• Label all intermediate steps and features on the graph.

• Describe the transformation using mathematical language.

Example: For , reflect over the x-axis, stretch vertically by 2, and shift left by 3 units.

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.

• For rational functions, exclude values that make the denominator zero.

Example: For , the domain is all real numbers except ; in interval notation: .

Determining x- and y-intercepts from Equations

Intercepts are points where the graph crosses the axes.

• x-intercept: Set and solve for .

• y-intercept: Set and solve for .

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

Example: For , x-intercepts are found by solving (), y-intercept is .

End Behavior of a Polynomial

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

• Identify the degree and leading coefficient of the polynomial.

• Use these to determine if the graph rises or falls at the ends.

• Describe using arrow notation (e.g., as , ).

Example: For , as , .

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 their slope and intercepts.

• Identify slope () and y-intercept () in .

• Use slope-intercept form to graph the line.

• Convert between slope-intercept and standard forms.

• Write equations given points and slope.

Example: For , 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 features of the graph.

• Standard form:

• Vertex form:

• Factored form:

• Identify axis of symmetry:

• Find vertex, intercepts, and direction of opening.

• Convert between forms by completing the square or factoring.

Example: For , vertex is at , .

Relate Equations to Graphs of Exponential Functions

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

• Identify parent exponential function, e.g., .

• Apply transformation rules to graph exponential functions.

• Identify horizontal asymptote.

Example: For , the horizontal asymptote is .

Relate Exponential and Logarithmic Forms of Equations

Logarithmic functions are the inverses of exponential functions and can be graphed using transformation rules.

• Identify parent logarithmic function, e.g., .

• Graph logarithmic functions using transformations.

• Identify vertical asymptote (usually for ).

Example: For , vertical asymptote is .

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.

• Find zeros and their multiplicities from the factored form.

• Identify cross/touch behavior at each zero.

• Evaluate test points between zeros to determine function values.

• Sketch the graph in factored form.

• Apply the Remainder and Factor Theorems using synthetic division.

• Find zeros using Rational Roots Theorem, synthetic division, and factoring.

Example: For , zero at has multiplicity 2 (touches), zero at has multiplicity 1 (crosses).

Table: Polynomial Zeros and Behavior

Additional info: The study notes above expand on the syllabus outline, providing definitions, examples, and formulas for key College Algebra concepts relevant to function transformations, equations, and graphing.