Graphing Techniques: Transformations
We will use our knowledge of the shapes of these commonly used algebraic graphs and our understanding of
transformations to sketch transformed functions.
Example #1: Given the graph
Properties of Functions
Continuity: A function is continuous over an interval of its domain if its hand-drawn graph over that interval can
be sketched without lifting the pencil from the paper.
Properties of Rational Functions
A rational number is a ratio of integers. A rational function is a ratio of polynomials.
A rational function can be written in the form
p ( x)
R( x) =
where p ( x ) and q( x) are polynomials and q ( x)
The Graph of a Rational Function
Calculus provides the tools required to graph a rational function accurately. In precalculus, we can gather quite a
bit of information about a rational graph to get an idea of the general shape and posi
Show all work and use proper mathematical notation to earn full credit.
Write your answers on the lines provided.
f ( x) 5 x2 2 x 4 , find
f ( x h) f ( x )
. Simplify your answer. _
f ( x) x2 and g ( x) 3 6
Collaborative Worksheet #1
Write your answers on the blanks provided.
Show all work to receive credit.
Collaborative Name _
f ( x + h) f ( x )
Let f ( x) = 3x 4 x 1 , find
f ( x + h) f ( x)
. Simplify your answer.
Let f ( x) = 4 , f
The Graph of a Function
Vertical Line Test
y is a function of x if a vertical line intersects the graph at only one point.
Example #1: Determine the domain, range whether the relation is a function, intercepts, and x value giv
Library of Functions; Piecewise-defined Functions
Lets review the graphs of commonly seen
functions. Knowing these graphs will lay the
foundation for further graphing techniques.
Sketch each of t
A relation is a correspondence between two sets.
The domain of a function is the largest set of real numbers for which the function is defined._
The corresponding set of rea numbers that we obtain f