Graphing Functions

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Graphing Functions
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Overview

• Functions
• The Cartesian Coordinate System
• Equations of Lines
• Graphs of Functions
• Algebra of functions
• Linear Functions
• Quadratic Functions
• Mathematical Models

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Functions

• A function is a rule that assigns to each element
  in a set A (the domain) one and only one element
  in a set B (the range).

A
B
f(x)
x
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If y f(x), then y is the dependent variable
and x is the independent variable.
The domain of the function f(x) is the set of all
values of x for which f(x) is a real number.
The range of the function f(x) is the set of
values assumed by y f(x) as x takes on all
values in its domain.
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Cartesian Coordinates

• A representation of points in a two-dimensional
  space (or plane).
• (x,y) coordinates an ordered pair of numbers
  representing a unique point in the plane.

y
The Origin
y-axis
x
0
x-axis
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y
y
(x0 ,y0)
(2 ,1)
y0
1
x
x
x0
0
0
2
1
-1
(1 ,-1)
y
Quadriant I (,)
Quadriant II (-,)
x
0
Quadriant IV (,-)
Quadriant III (-,-)
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Straight Lines
Let y mx b b intercept on y-axis (vertical
axis) m slope
y
y mx b
(x2,y2)
y2
(rise)
(x1,y1)
y1
Slope rise over run
b
(run)
x
0
x1
x2
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Equations of Lines
y f(x) mx b

• Straight lines are the graphical representation
  of linear functions.
• There are three ways to find the function of a
  straight line

Point-Slope Method If we are given the slope m
and a point on the line (x1,y1), then the
equation for the line is y y1 m (x x1)
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Slope-Intercept Method If we are given the slope
m and the intercept on the y-axis (vertical axis)
(0,b), then the equation for the line is y m x
b
Point-Point Method If we are given two points on
a straight line (x1,y1) and (x2,y2) then we can
find the slope m Then find the y-axis
intercept b by setting x x1 and y y1. b y1
mx1
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Graphs of Functions

• The set of all points (x,y) in the xy-plane such
  that x is in the domain of f and y f(x)

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x f(x) (x,y) 1 3 (1,3) 2 8 (2,8) 3 16 (3,16) 4
28 (4,28) 5 40 (5,40)
Vertical-Line Test A curve in the xy-plane is the
graph of a function yf(x) if and only if each
vertical line intersects it in at most one point
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Which of the following curves are not functions?
y
y
x
x
y
y
y
x
x
x
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Algebra of Functions
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Linear Functions
A linear equation may come in the form Ax By
C 0
It can be converted to the form y mx b b
intercept on y-axis (constant) m slope
(constant) Therefore defining y as a function of
x.
Note that b -C/B and m -A/B
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Parallel and Perpendicular Lines
Two distinct lines are parallel if and only if
their slopes are equal or their slopes are
undefined. Let the slope of line L1 be m1 and the
slope of line L2 be m2, if L1 is distinct from L2
and m1 m2, then L1 and L2 are parallel.
Let the slope of line L1 be m1 and the slope of
line L2 be m2, if L1 and L2 are two distinct
non-vertical lines then, L1 is perpendicular to
L2 if
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Simultaneous Equations
Sometimes you will be asked to solve for the
values of several variables that satisfy certain
conditions. Suppose we are asked to find x and y
such that
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Graphical Solution
y
y g(x)
b
(x,y)
d
x
0
y f(x)
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Finding the intersection
Step 1 Substitute equation (1) for the value of
y in equation (2)
Step 2 Solve for x.
Step 3 Substitute your solution for x in the
first equation to solve for y.
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Quadratic Functions
y
y
Axis of Symmetry
The Vertex coordinates
Vertex
x
x
The axis of symmetry
Axis of Symmetry
Vertex
a lt 0
a gt 0

• The graph of a quadratic function is called a
  parabola.

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• Polynomial Functions
• Rational Functions - the quotient of two
  polynomials
• Power Functions

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Mathematical Models
Formulate
Real-world Problem
Mathematical Model
Solve
Test
Solution of Mathematical Model
Solution of real-world problem
Interpret
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Constructing Mathematical Models

• Assign a letter to each variable mentioned in the
  problem.
• Find an expression for the quantity sought
• Use the conditions given in the problem to write
  the quantity sought as a function f of one
  variable.
• Note any restrictions to be placed on the domain
  of f from the physical considerations of the
  problem.