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 manipulate the graphs of parent functions to create new functions and analyze their properties.

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

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

• Graphing Transformations:

  • Use the graph of a parent function to graph a transformed function (e.g., given the graph of f(x), graph f(x - 2) + 3).

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

• Equation of a Transformed Parent Function: Use the equation to determine features of a graph.

Example: If f(x) = x^2, then g(x) = (x - 2)^2 + 3 is the graph of f(x) shifted right by 2 units and up by 3 units.

Skills for Transforming Functions

• Identify a parent function given an equation.

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

• Identify and perform transformations in the correct order.

• Explain why the order in which transformations are performed matters.

• Match graphs to their equations and vice versa for transformed functions.

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

• Graph a function from start to finish using transformations, including:

  • Identifying the parent function.

  • Performing transformations one step at a time, showing and labeling all intermediate steps on the graph.

  • Describing steps of the transformation using mathematical language.

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 rational function given its equation.

• Write domains in interval notation.

Example: For f(x) = 1/(x-3), the domain is all real numbers except x = 3, or $(-\infty, 3) \cup (3, \infty)$.

Determining x- and y-Intercepts from an Equation

Intercepts are points where the graph crosses the axes.

• x-intercepts: Set y = 0 and solve for x.

• y-intercepts: Set x = 0 and solve for y.

• 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 $x^2 - 4 = 0$.

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 of a polynomial.

• Describe end behavior using informal (arrow notation) language.

Example: For f(x) = 2x^3 - x, as $x \to \infty$, $f(x) \to \infty$; as $x \to -\infty$, $f(x) \to -\infty$.

UNIT 5: How are Different Representations of Functions Connected?

Relate Linear Equations to Graphs

Linear equations can be represented in various forms and are closely connected to their graphs.

• Identify the slope and y-intercept of a linear equation in slope-intercept form: $y = mx + b$.

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

• Convert between slope-intercept and standard forms: $Ax + By = C$.

• Write an equation of a line given a point and the slope.

• Interpret slope and y-intercept in context (e.g., revenue, profit).

Example: For y = 2x + 3, 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 several forms, each revealing different properties.

• Standard form: $y = ax^2 + bx + c$

• Vertex form: $y = a(x - h)^2 + k$

• Factored form: $y = a(x - r_1)(x - r_2)$

• Identify the axis of symmetry: $x = h$ in vertex form, $x = -\frac{b}{2a}$ in standard form.

• Find the vertex, intercepts, and direction of opening.

• Convert between forms by completing the square or factoring.

Example: For y = (x - 1)^2 + 2, vertex is (1, 2), axis of symmetry is x = 1.

Convert Between Forms of Quadratic Equations

• Convert from vertex form to standard form by expanding.

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

• Convert from standard form to factored form by factoring.

Relate Equations to Graphs of Exponential Functions

Exponential functions have the form $y = ab^x$ and model rapid growth or decay.

• Identify a parent exponential function and graph it using transformation rules.

• Identify the horizontal asymptote for an exponential function once graphed.

Example: For y = 2^x, the horizontal asymptote is $y = 0$.

Relate Exponential and Logarithmic Forms of Equations

Logarithmic functions are the inverses of exponential functions.

• Identify a parent logarithmic function and use its equation to create a table of values.

• Graph a logarithmic function using transformation rules.

• Identify the vertical asymptote for a logarithmic function once graphed.

Example: For y = \log_2(x), the vertical asymptote is $x = 0$.

Relate Equations to Graphs of Polynomials in Factored or Standard Form

Polynomials can be analyzed by their zeros, multiplicities, and end behavior.

• Find the zeros of a polynomial and their multiplicities given the equation in factored form.

• Identify cross/touch behavior at each zero.

• Evaluate test points between zeros to find other function values.

• Sketch the graph of a polynomial in factored form.

• Apply the Remainder and Factor Theorems:

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

  • Identify if a factor is a factor of a polynomial using synthetic division.

• Find zeros using:

  • The Rational Roots Theorem

  • Long or Synthetic Division

  • Factoring (Zero Product Property)

Example: For f(x) = (x - 2)^2(x + 1), zeros are x = 2 (multiplicity 2, touch), x = -1 (cross).

Additional info: This study guide is based on a syllabus or test review sheet for College Algebra, focusing on function transformations, graphing, and algebraic analysis of equations.