UNIT 3: How Do We Transform Functions?

Graph Transformations from Graphs and Symbols

Understanding how to transform functions is a key skill 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.

• Identifying Transformations: Given a graph, determine which transformations have been applied to the parent function.

• Graphing Transformed Functions: Use the graph of a parent function to sketch a transformed function, e.g., given , graph .

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

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

Using Equations to Transform Functions

Transformations can also be performed algebraically using function equations.

• Identifying Parent Functions: Recognize the base function from its equation.

• Transformation Order: Apply transformations in the correct sequence (e.g., horizontal shifts before vertical shifts).

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

• Describing Transformations: Use mathematical language to describe each step of the transformation.

Example: For , the transformation involves a reflection over the x-axis, a vertical stretch by 2, a left shift by 1, and an up shift by 4.

Determining Domain from an Equation

Domain of Various Functions

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

• Rational Functions: Exclude values that make the denominator zero.

• Radical Functions: For even roots, exclude values that make the radicand negative.

• Log Functions: Exclude values that make the argument non-positive.

• Polynomial Functions: Domain is usually all real numbers unless otherwise restricted.

• Interval Notation: Write domains using interval notation, e.g., .

Example: For , the domain is .

Determining x- and y-Intercepts from an Equation

Intercepts of Various Functions

Intercepts are points where the graph crosses the axes.

• x-intercept: Set and solve for .

• y-intercept: Set and solve for .

• Types of Functions: Linear, quadratic, higher-order polynomial, rational, radical, logarithmic, exponential.

• Factoring: For quadratics, factor to find x-intercepts.

Example: For , x-intercepts are found by solving ; thus, and .

End Behavior of a Polynomial

Describing End Behavior

End behavior describes how the function behaves as approaches or .

• Degree and Leading Coefficient: These determine the end behavior.

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

• Arrow Notation: Use up/down arrows to describe behavior.

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 related to their graphs.

• Slope-Intercept Form:

• Standard Form:

• Identifying Slope and y-Intercept: From the equation, determine and .

• Graphing: Use slope and y-intercept to sketch the line.

• Converting Forms: Change between slope-intercept and standard forms.

• Applications: Write and solve linear cost, revenue, and profit functions.

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 expressed in multiple forms, each revealing different properties.

• Vertex Form:

• Standard Form:

• Factored Form:

• Vertex: The point in vertex form.

• Axis of Symmetry: in vertex form, in standard form.

• Direction: Upward if , downward if .

• Intercepts: Solve for x-intercepts by factoring or using the quadratic formula.

• Sketching: Use vertex, axis of symmetry, and intercepts to sketch the graph.

Example: For , vertex is , axis of symmetry is .

Convert Between Forms of Quadratic Equations

Quadratic equations can be converted between standard, vertex, and factored forms.

• FOIL Method: Expand factored form to standard form.

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

Example: can be written as .

Relate Equations to Graphs of Exponential Functions

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

• Parent Exponential Function:

• Transformations: Apply shifts, stretches, and reflections.

• Horizontal Asymptote: The line that the graph approaches but never crosses.

Example: For , the horizontal asymptote is .

Relate Equations to Graphs of Logarithmic Functions

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

• Parent Logarithmic Function:

• Transformations: Use transformation rules to graph variations.

• Vertical Asymptote: The line that the graph approaches but never crosses.

Example: For , the 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 determine zeros and graph behavior.

• Finding Zeros: Set the polynomial equal to zero and solve for .

• Multiplicity: The number of times a zero occurs; affects whether the graph crosses or touches the axis.

• Cross/Touch Behavior: Odd multiplicity crosses the axis; even multiplicity touches and turns.

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

• Sketching: Use zeros and multiplicities to sketch the graph.

• Remainder and Factor Theorems: Use synthetic division to test for zeros and factors.

• Finding Zeros Methods:

  • The Rational Roots Theorem

  • Long or Synthetic Division

  • Factoring (Zero Product Property)

Example: For , is a zero of multiplicity 2 (touches), is a zero of multiplicity 1 (crosses).

Additional info:

• Some context and terminology have been expanded for clarity and completeness.

• Examples and formulas have been added to illustrate key concepts.