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 , , or .

• 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 , graph .

• Point Mapping: Given a point on a parent graph, identify its image on a transformed graph.

Example: If , then is the graph of shifted right by 2 units and up by 3 units.

Using Equations of Transformed Parent Functions

Equations can be used to determine features of a graph, such as intercepts, vertex, and symmetry.

• Identify a parent function given an equation.

• Identify the transformations applied to a parent function given its equation.

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

• Graph a function from start to finish using transformations, including labeling all intermediate steps.

• Describe steps of the transformation using mathematical language.

Example: Given , the transformation involves reflection over the x-axis, vertical stretch by 2, left shift by 1, and up shift by 5.

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

Determining x- and y-intercepts from Equation

Intercepts are points where the graph crosses the axes.

• Algebraically solve for the intercepts of linear, quadratic, higher-order polynomial, rational, radical, logarithmic, and exponential functions.

• Find x-intercepts by setting and solving for .

• Find y-intercepts by setting and solving for .

Example: For , x-intercepts are found by solving (), and the 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 a polynomial.

• Describe end behavior using arrow notation.

Example: For , as , ; 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 slope and intercepts.

• Identify slope and y-intercept from an equation.

• Use slope-intercept form () to graph.

• Convert between slope-intercept and standard forms.

• Write equations given points and slope.

Example: For , slope is 3, y-intercept is -2.

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

Quadratic functions can be written and graphed in multiple forms, each revealing different features.

• Standard form:

• Vertex form:

• Factored form:

• Identify axis of symmetry: in vertex form, in standard form.

• Find vertex, intercepts, and end behavior.

• Convert between forms by completing the square or factoring.

Example: For , vertex is at , .

Relate Equations to Graphs of Exponential Functions

Exponential functions model rapid growth or decay and have unique transformation properties.

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

• Graph exponential functions using transformation rules.

• 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 have their own parent forms and transformations.

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

• Graph logarithmic functions using transformation rules.

• Identify vertical asymptote.

Example: For , the vertical asymptote is .

Relate Equations to Graphs of Polynomials in Factored or Standard Form

Polynomials can be analyzed for their zeros, multiplicities, and graph behavior using their equations.

• Find zeros and their multiplicities from factored form.

• Identify cross/touch behavior at zeros.

• Evaluate test points between zeros to find function values.

• Sketch graphs in factored form.

• Apply Remainder and Factor Theorems using synthetic division.

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

Example: For , zeros are (multiplicity 2, touch) and (cross).

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