UNIT 3: How Do We Transform Functions?

Graph Transformations from Graphs and Symbols

Understanding how functions are transformed is a fundamental skill in College Algebra. Transformations include shifts, reflections, stretches, and compressions applied to parent functions.

• Parent Function: The simplest form of a function in a family (e.g., f(x) = x^2 for quadratics).

• Identifying Transformations: Recognize and describe transformations such as translations (shifts), reflections, stretches, and compressions from a graph or equation.

• Graphing Transformed Functions: Use the graph of a parent function to graph a transformed function, such as f(x) = x^2 - 3.

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

Example: If the parent function is f(x) = x^2 and the transformation is f(x) = (x - 2)^2 + 3, the graph shifts right by 2 units and up by 3 units.

Using the Equation of a Transformed Parent Function

Given the equation of a transformed function, you should be able to determine its features and graph it step by step.

• Identify the parent function from the equation.

• Identify and perform the transformations in the correct order.

• Match graphs to equations using knowledge of transformations.

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

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

Example: For g(x) = -2f(x + 1) - 4, the transformations are: shift left 1, reflect over the x-axis, vertical stretch by 2, and shift down 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, radical, and polynomial functions.

• Write domains in interval notation.

Example: For f(x) = 1/(x-2), the domain is all real numbers except x = 2:

Determining x- and y-Intercepts from an Equation

Intercepts are points where the graph crosses the axes.

• Algebraically solve for y-intercepts (set x = 0).

• Algebraically solve for x-intercepts (set y = 0 and solve for x).

• 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 (i.e., x = 2 and x = -2).

End Behavior of a Polynomial

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

• Identify the degree and leading coefficient from the equation.

• Use these to determine if the ends of the graph rise or fall.

Example: For f(x) = -2x^3 + x, as x 5crightarrow 5cinfty, f(x) 5crightarrow -5cinfty; as x 5crightarrow -5cinfty, f(x) 5crightarrow 5cinfty.

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 from slope-intercept form:

• Use the slope-intercept form to graph the function.

• Convert between slope-intercept and standard forms.

• Write equations of lines given points and slopes.

• Write and interpret cost, revenue, and profit functions.

Example: For y = 2x - 3, the slope is 2 and the y-intercept is -3.

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

Quadratic functions can be written and analyzed in different forms, each revealing different properties.

• Identify the vertex from vertex form:

• Identify axis of symmetry:

• Find y-intercept by setting

• Determine if the parabola opens up or down (sign of a).

• Write equations from graphs and vice versa.

• Solve application problems involving quadratics.

Example: For y = -3(x+1)^2 + 2, the vertex is (-1, 2), opens downward.

Convert Between Forms of Quadratic Equations

• Convert from vertex to standard form by expanding.

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

Example: Expand y = (x-2)^2 + 1 to get y = x^2 - 4x + 5.

Relate Equations to Graphs of Exponential Functions

Exponential functions model rapid growth or decay and have unique features such as asymptotes.

• Identify parent exponential functions and graph them using transformation rules.

• Identify horizontal asymptotes.

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

Relate Exponential and Logarithmic Forms of Equations

• Identify parent logarithmic functions and use them to create tables of values.

• Graph logarithmic functions using transformation rules.

• Identify vertical asymptotes.

Example: For f(x) = 5clog_2(x-1), the vertical asymptote is x = 1.

Relate Equations to Graphs of Polynomials in Factored or Standard Form

Polynomials can be analyzed for their zeros, multiplicities, and overall graph shape.

• Find zeros and their multiplicities from factored form.

• Identify cross/touch behavior at each zero.

• Evaluate test points between zeros to determine function sign.

• Sketch the graph using zeros and end behavior.

• Apply the Remainder and Factor Theorems.

• Test if a value is a zero or factor using synthetic division.

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

Example: For f(x) = (x-1)^2(x+2), zeros are x = 1 (multiplicity 2, touches) and x = -2 (crosses).

Table: Summary of Forms and Features of Functions

Additional info: This table summarizes the main forms and features of function types discussed in the syllabus.