UNIT 3: How Do We Transform Functions?

Graph Transformations from Graphs and Symbols

Understanding how functions change under various transformations is a key skill in College Algebra. Transformations include shifts, stretches, compressions, and reflections.

• Parent Function: The simplest form of a function, such as , , or .

• Transformation Identification: Recognize and describe transformations (vertical/horizontal shifts, stretches/compressions, reflections) applied to a parent function.

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

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

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

Using the Equation of a Transformed Parent Function

Equations can be used to determine features of transformed graphs, such as intercepts, vertex, and asymptotes.

• Identify a parent function given an equation.

• Identify and perform transformations in the correct order.

• Match graphs to equations using transformation knowledge.

• Write equations for a function given a parent function and characteristics of the transformed graph.

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

Example: For , is reflected over the x-axis, vertically stretched by 2, shifted left by 1, and up by 4.

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 rational functions (denominator ≠ 0).

• Algebraically solve for the domain of radical functions (even index: radicand ≥ 0).

• Algebraically solve for the domain of log functions (argument > 0).

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

• Write domains in interval notation.

Example: For , domain is or .

Determining x- and y-intercepts from an Equation

Intercepts are points where the graph crosses the axes.

• Algebraically solve for y-intercepts (set ) and x-intercepts (set ).

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

Example: For , x-intercepts are and .

End Behavior of a Polynomial

End behavior describes how a function behaves as approaches infinity or negative infinity.

• Identify degree and leading coefficient from standard or factored form.

• Use these to determine end behavior (up/down) 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 slope-intercept form () or standard form ().

• Identify slope and y-intercept from slope-intercept form.

• Use slope-intercept form to graph the function.

• Convert between slope-intercept and standard forms.

• Write equations of lines given a point and slope.

• Write equations for cost, revenue, and profit functions.

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

Relate Equations to Graphs of Quadratics

Quadratic functions can be written in standard (), vertex (), or factored form ().

• Identify vertex (max/min), axis of symmetry, and direction of opening.

• Algebraically solve for y-intercept and x-intercepts.

• Sketch graphs and solve application problems (e.g., area, max/min).

• Convert between forms by factoring or completing the square.

Example: has vertex at (1,2).

Relate Equations to Graphs of Exponential Functions

Exponential functions have the form .

• Identify parent exponential function and graph using transformation rules.

• Identify horizontal asymptote after graphing.

Example: has horizontal asymptote .

Relate Exponential and Logarithmic Forms of Equations

Logarithmic functions are the inverses of exponential functions.

• Identify parent logarithmic function and use it to create tables of values.

• Graph logarithmic functions using transformation rules.

• Identify vertical asymptote after graphing.

Example: has vertical asymptote .

Relate Equations to Graphs of Polynomials in Factored or Standard Form

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

• Find zeros and their multiplicities from factored form.

• Identify cross/touch behavior at each zero.

• Evaluate test points between zeros to determine function values.

• Sketch graphs in factored form.

• Apply Remainder and Factor Theorems to polynomial functions.

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

Example: For , zero at (multiplicity 2, touch), zero at (multiplicity 1, cross).

Table: Methods for Finding Zeros of Polynomials

Additional info: Factoring may include difference of squares, sum/difference of cubes, and factoring trinomials.