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MAT091 :: Lecture Note :: Week 14
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Overview

Review for Final Exam

A variable is used to represent an unknown quantity in a mathematical expression (or equation). Variables are typically named using a single letter. Values are assigned to variables using substitution (or replacement).

The terms of a mathematical expression are separated by addition or subtraction operators.

The factors of a term are two (or more) numbers or variables that are multiplied together. A number that is a factor along with a variable is called a coefficient.

A polynomial is a mathematical expression that is "constructed from one or more variables and constants, using only the operations of addition, subtraction, multiplication, and constant positive whole number exponents."

The degree of a polynomial is the "largest exponent on any of the variables contained in the expression."

A mathematical function "maps" a set of inputs and to a set of corresponding outputs. Functions are used to "model" real-world situations (applications, scenarios, stories, behaviors, etc.).

Functions can be respresented using tables, graphs, a set of ordered-pairs, or a algebraic expression (symbolic rule).

Functions are typically named using a single letter (usually lower-case, but sometimes upper-case).

   function notation:   y = f(x)
   ... f is the function name
   ... y is the output
   ... x is the input

   * every input produces one and only one output

   * the domain of a function is a set of valid input values
     [i.e. it's the input values for which a function is define]

   * the range of a fuction is a set of its output values

   * inputs are indepedent variables
   * outputs are dependent variables
     [outputs depend on inputs]

   * inputs are plotted (graphed) along horizonal-axis 
     [horizontal-axis is the x-axis]
   * outputs are plotted (graphed) on vertical-axis
     [vertical-axis is the y-axis]

   ordered-pair:  (x, y) ... input is x, output is y

   ordered-pair:  (x, f(x)) ... input is x, output is f(x)

   (3, 5) implies input 3 produces the known output 5
   (3, f(3)) implies input 3 produces the unknown output value f(3)

   f(x) is the output from function f when the input is x
   5 = f(3) implies function f outputs 5 when the input is 3  (3, 5)
   g(4) = 8 implies function g outputs 8 when the input is 4  (4, 8)

   f(x) = some_math_expression, then the definition of f(x) is known
   e.g. f(x) = 5(x) - 3  
        ...the math expression 5(x) - 3 is the definition for function f
        ...function f outputs 3 less than 5 times its input
        ...f(2) = 5(2) - 3 = 10 - 3 = 7      i.e. f(2) = 7    (2, 7)
        ...f(-1) = 5(-1) - 3 = -5 - 3 = -8   i.e. f(-1) = -8  (-1, -8)

Linear functions are a special form of function.

Linear functions have a constant numeric rate of change that is defined by their slopes.

Linear functions are 1st-degree polynomials.

Linear functions, when graphed, are straight lines.

   linear equation standard form:  Ax + By + C = 0 ...or... Ax + By = C

   slope y-intercept form:  y = mx + b  ...or...  f(x) = mx + b
                            y = b + mx  ...or...  f(x) = b + mx

                slope:  m              [δoutput/δinput; rise/run]
   vertical-intercept:  (0, b)         [y-axis intercept]
 horizontal-intercept:  (-b/m, 0)      [x-axis intercept]

    The slope is the rate of change between two points.

slope formula :  m = (y2 - y1) / (x2 - x1)

   m > 0  implies an increasing function
   m < 0  implies a decreasing function
   m = 0  implies a constant function

Increasing function: Outputs increase when the inputs increase.

Decreasing function: Outputs decrease when the inputs increase.

Constant function: Outputs never change when the inputs change.

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