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MAT091 :: Lecture Note :: Week 15
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(due Week #15)
(due Week #15)
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" realworld situations (applications, scenarios, stories, behaviors, etc.).
Functions can be respresented using tables, graphs, a set of orderedpairs, or a algebraic expression (symbolic rule).
Functions are typically named using a single letter (usually lowercase, but sometimes uppercase).
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 horizonalaxis [horizontalaxis is the xaxis] * outputs are plotted (graphed) on verticalaxis [verticalaxis is the yaxis] orderedpair: (x, y) ... input is x, output is y orderedpair: (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 1stdegree polynomials.
Linear functions, when graphed, are straight lines.
linear equation standard form: Ax + By + C = 0 ...or... Ax + By = C slope yintercept form: y = mx + b ...or... f(x) = mx + b y = b + mx ...or... f(x) = b + mx slope: m [δoutput/δinput; rise/run] verticalintercept: (0, b) [yaxis intercept] horizontalintercept: (b/m, 0) [xaxis intercept] The slope is the rate of change between two points. slope formula : m = (y_{2}  y_{1}) / (x_{2}  x_{1}) m > 0 implies an increasing function m < 0 implies a decreasing function m = 0 implies a constant functionIncreasing 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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