Graphing Basic Functions with Translations and Refections

Notes on graphing functions using transformations. We cover rigid translations and reflections as well as domain and range.

Notes:
   1. Relations, Graphs, and Functions
   2. Graphing the Basic Functions
   3. Using Transformations to Graph Functions

YouTube Video Tutorials:

Simplifying Rational Expressions

Given a rational expression, the quotient of two polynomials, we will factor the numerator and denominator if we can and then cancel factors that are exactly the same.

Rational Expressions and Equations Playlist on YouTube

When evaluating rational expressions, plug in the appropriate values either before simplifying or after, the result will be the same.  Although, it is more efficient to simplify first then evaluate.

We can see that when evaluating, the result will be the same whether or not we simplify first.  It turns out that not all numbers can be used when we evaluate.

The point is that not all real numbers will be defined in the above rational expression.  In fact, there are two restrictions to the domain, -2 and 3/5.  These values, when plugged in, will result in zero in the denominator.  Another way to say this is that the domain consists of all real numbers except for −2 and 3/5.

Tip: To find the restrictions, set each factor in the denominator equal to zero and solve. The factors in the numerator do not contribute to the list of restrictions.

Simplify and state the restrictions to the domain.

Even if the factor cancels it still contributes to the list of restrictions.  Basically, it is important to remember the domain of the original expression when simplifying. Also, we must use caution when simplifying, please do not try to take obviously incorrect shortcuts like this:

Since subtraction is not commutative, we must be alert to opposite binomial factors.  For example, 5 − 3 = 2 and 3 − 5 = −2. In general,

Simplify and state the restrictions to the domain.

At this point, we evaluate using function notation.

Video Examples on YouTube:

Introduction to Functions

You may have noticed that we have been evaluating or "plugging-in" lots of numbers to find the corresponding y-values.  We have been doing this to find points on the graph (x, y).  This process can be streamlined using function notation.

This new notation above is read, "y is equal to f of x".  A function is a rule that uniquely assigns one output to every one input. So, at this point we can think of our non-vertical lines as functions.

INTRO TO FUNCTIONS PLAYLIST

The domain of a function is the set of inputs, usually the x-values.  The range of a function is the set of outputs, usually the y-values.

Evaluate the given function.

Find x.

Do not let the function notation discourage you, it takes some time to get used to.  The main thing to remember is that y = f(x); sometimes x is given and sometimes y or f(x) is given.

Graph the given linear function.

Word Problem. Bill has a popular software company which sells copies of its program for $149.  If the initial start up cost for the company was $10,000 and it costs $12 to produce each copy:
   a.  Find a cost function C(x) that models this business.
   b.  Find a revenue function R(x) that models this business.
   c.  Find the profit function P(x) using your functions above.
   d.  Find the profit when 1000 programs are produced and sold.
   e.  Find the number of programs that must be sold to break even.

Solution:

a. Cost Function - Include all fixed and variable costs of production.

b. Revenue Function - Include all proceeds from sales.

c. Profit Function - Revenue less cost of production.

d. Here x = 1,000 programs produced and sold.

e. Break Even - Occurs when profit is equal to zero.

Video Examples on YouTube:  Intro to Functions PlayList