
Relation:
Set of ordered pairs to represent informations
Function:A relation in which no two of the ordered pairs have the same first element
*Every relations is NOT a function!!*
Example of Function Graph- Temperature vs. Time
Domain:The set of all first elements, referred to as the independent variable
Range:The set of all second elements,dependent
Example) Let A and B be two sets. A function
f from A to B is a relation between A and B such that for each a A there
is one and only one associated b B. The set A is called the domain of the
function, B is called its range. Often a function is denoted as y = f(x) or simply f(x),
indicating the relation { (x, f(x)) }.
N:A natural number is any of the numbers 0, 1, 2, 3... that can be used to measure the size of finite sets, N means: {0,1,2,3,...}
Z:The integers, or whole numbers, consist of the natural numbers (0, 1, 2, ...) and the negative whole numbers (-1, -2, -3, ...).
Q:A rational number is a real number that can be expressed as the ratio between two integers (integers are a subset of the rational numbers), usually written as a fraction a / b, where the denominator (here b) is not equal to zero. A real number that is not rational, is an irrational number.
R:the real numbers are, intuitively, numbers that are in one-to-one correspondence with the points on a line-- the number line.
A linear function is one whose graph is a straight line (hence the term "linear").
A linear function is one that can be written in the form f(x) = mx + b Function form *Example: f(x) = 3x - 1 m = 3, b = -1
y = mx + b Equation form *Example:y = 3x - 1 where m and b are fixed numbers (the names m and b are traditional).
Example Here is a partial table of values
of the linear function f(x) = 3x - 1. Fill in the missing values.
Plotting a few of these points gives the following graph.
x
-4
-3
-2
-1
0
1
2
3
4
Y
-13
-10
-7
-4
-1
2
5
8
11
The graph of a quadratic function is a curve called a parabola.
Parabolas may open upward or downward and vary in "width"
or "steepness", but they all have the same basic "U"
shape. The picture below shows three graphs, and they are all parabolas.
All parabolas are symmetric with respect to a line called the axis of symmetry. A parabola intersects its axis of symmetry at a point called the vertex of the parabola.
You know that two points determine a line. This means that if you are given any two points in the plane, then there is one and only one line that contains both points. A similar statement can be made about points and quadratic functions.
Many quadratic functions can be graphed easily by hand using
the techniques of stretching/shrinking and shifting (translation) the parabola
y = x2 .
Example 1.
Sketch the graph of y = x2/2. Starting with the graph of y = x2, we shrink by a factor of one half. This means that for each point on the graph of y = x2, we draw a new point that is one half of the way from the x-axis to that point.
=>When we have a special case of hybrid functions is when each rule is a linear function
Consider the function y = 2x + 3 on the interval (-3, 1) and
the function y = 5 (a horizontal line) on the interval (1, 5). Let's graph
those two functions on the same graph. Note that they span the interval from
(-3, 5). Since the graphs do not include the endpoints, the point where each
graph starts and then stops are open circles Graph of the piecewise function
y = 2x + 3 on the interval (-3, 1) and y = 5 on the interval (1, 5)
The graph depicted above is called piecewise because it consists of two or more pieces. Notice that the slope of the function is not constant throughout the graph. In the first piece, the slope is 2 or 2/1, while in the second piece, the slope is 0. However, at the point where they adjoin, when we substitute 1 in for x, we get y = 5 for both functions, so they share the point (1, 5).
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Graph
Your Own Function!
Examples to
Work On
Ex: a.)
f(x)=((1/2)^x-3+2
b.) g(x)=(1/2)^x
| Domain f (x) | (-∞,∞) |
| Range f(x) | [2,∞) | Asymptotes f(x) | y=2 | X Intercept f(x) | N/A | transformation from g(x)to f(x) | up 2, right 3 |
| a.) h:y →3x^2+5 and b.) k:t→1/2(x-4) | |
| 1.) h(2)=12 | 2.) h=1.5x^2+1/2 |
| cell 1 | D: [0,2) |
| R: [-1,∞) |
Problem: If it costs $3 to send parcels weighing between at least 1kg but less than 3kg, $7 for parcels weighing at least 3kg but less than 5kg, $13 for parcels weighing at least 5kg but less than 8kg and so on, we have the following graph: