Embed Size (px)
DESCRIPTION
Cartesian Coordinate plane The Cartesian Coordinate plane is composed of the – x-axis (horizontal) and the – y-axis (vertical), – They meet at the origin (0,0) – Divide the plane into four quadrants. – Ordered pairs graphed on the plane can be represented by (x,y).
Citation preview
Warm Up
To help guide this chapter, a project (which will be explained after the warm up) will help guide Chapter 2
On the front page of your 3 page packet, fill out:1. Your name2. A name for your business3. Two (appropriate) products to sell4. The price for each product (the price must be more than $1 and less than $10
You have five minutes to do this. If you finish, read through the rest of the packet
CHAPTER 2.1, 2.2
RELATIONS AND LINEAR FUNCTIONS
Cartesian Coordinate plane The Cartesian Coordinate plane is composed
of the – x-axis (horizontal) and the – y-axis (vertical), – They meet at the origin (0,0) – Divide the plane into four quadrants.– Ordered pairs graphed on the plane
can be represented by (x,y).
Cartesian Coordinate Plane
VOCABULARY
RELATION – SET OF ORDERED PAIRS– example: {(4,5), (–2,1), (5,6), (0,2)}
DOMAIN – SET OF ALL X’S– D: {4, –2, 5, 0}
RANGE – SET OF ALL Y’S– R: {5, 1, 6, 2}
A relation can be shown by a mapping, a graph, equations, or a list (table).
Mapping-shows how each member of the domain is paired with each member of the range.
Function
A function is a special type of relation. – By definition, a function exists if and
only if every element of the domain is paired with exactly one element from the range.
– That is, for every x-coordinate there is exactly one y-coordinate. All functions are relations, but not all relations are functions.
One-to-one Mapping
– example: {(4,5), (–2,1), (5,6), (0,2)}
Function
Example– B: {(4,5), (–2,1), (4,6), (0,2)} – Not a function because the domain 4
is paired with two different ranges 5 & 6
Vertical Line Test
The vertical line test can be applied to the graph of a relation.
If every vertical line drawn on the graph of a relation passes through no more than one point of the graph, then the relation is a function.
Graphing by domain and range
Y=2x+1– Make a table to find ordered pairs that satisfy the
equation– Find the domain and range– Graph the ordered pairs– Determine if the relation is a function
More Vocab. Function Notation A function is commonly denoted by f. In
function notation, the symbol f (x), is read "f of x " or "a function of x." Note that f (x) does not mean "f times x." The expression y = f (x) indicates that for every value that replaces x, the function assigns only one replacement value for y.
f (x) = 3x + 5, Let x = 4 also written f(4)– This indicates that the ordered pair (4, 17) is a solution of the function.
Vocab.
Independent Variable– In a function, the variable whose values make up
the domain– Usually X
Dependent Variable– In a function, the variable whose values depend
on the independent variable– Usually Y
Function Examples
If f(x) = x³ - 3 , evaluate: – f(-2)– f(3a)
If g(x) = 5x2 - 3x+7 , evaluate: – g(4)– g(-3c)
2.2 Linear Relations and Functions
LINEAR EQUATION One or two variables Highest exponent is 1
NOT LINEAR EQUATION Exponent greater than
1 Variable x variable Square root Variable in denominator
Linear/not linear
2x + 3y = -5 f (x) = 2x – 5 g (x) = x³ + 2 h (x,y) =- 1 + xy
EVALUATING A LINEAR FUNCTION
The linear function f(C) = 1.8C + 32 can be used to find the number of degrees Fahrenheit, f, that are equivalent to a given number of degrees Celsius, C
On the Celsius scale, normal body temperature is 37°C. What is the normal body temp in degrees Fahrenheit?
STANDARD FORM
Ax + By = C 1. X POSITIVE 2. A & B BOTH NOT ZERO 3. NO FRACTIONS 4. GCF OF A, B, C = 1
EX: Y = 3X – 9 8X – 6Y + 4 = 0
-2/3X = 2Y – 1
USE INTERCEPTS TO GRAPH A LINE
X – INTERCEPT SET Y=0
Y – INTERCEPT SET X=0
PLOT POINTS AND DRAW LINEEX: - 2X + Y – 4 = 0
CHAPTER 2.3
SLOPE
SLOPE
CHANGE IN Y OVER CHANGE IN X RISE OVER RUN RATIO STEEPNESS RATE OF CHANGE
FORMULA12
12
xxyy
m
USE SLOPE TO GRAPH A LINE
1. PLOT A GIVEN POINT 2. USE SLOPE TO FIND ANOTHER POINT 3. DRAW LINE
EX: DRAW A LINE THRU (-1, 2) WITH SLOPE -2
RATE OF CHANGE
OFTEN ASSOCIATED WITH SLOPE MEASURES ONE QUANTITY’S CHANGE
TO ANOTHER
LINES
PARALLEL SAME SLOPE VERTICAL LINES ARE
PARALLEL
PERPENDICULAR OPPOSITE
RECIPROCALS (flip it and change sign)
VERTICAL AND HORIZONTAL LINES
CHAPTER 2.4
WRITING LINEAR EQUATIONS
SLOPE-INTERCEPT FORM
y = mx + b m IS SLOPE b IS Y-INTERCEPT
POINT-SLOPE FORM
FIND SLOPE PLUG IN ARRANGE IN SLOPE INTERCEPT FORM
11 xxmyy
11 ,, yxm
EX: WRITE AN EQUATION OF A LINE THRU (5, -2) WITH SLOPE -3/5
EX: WRITE AN EQUATION FOR A LINE THRU (2, -3) AND (-3, 7)
INTERPRETING GRAPHS
WRITE AN EQUATION IN SLOPE-INTERCEPT FORM FOR THE GRAPH
REAL WORLD EXAMPLE
As a part time salesperson, Dwight K. Schrute is paid a daily salary plus commission. When his sales are $100, he makes $58. When his sales are $300, he makes $78.
Write a linear equation to model this. What are Dwight’s daily salary and commission rate? How much would he make in a day if his sales were
$500?
Write an equation for the line that passes through (3, -2) and is perpendicular to the line whose equation is y = -5x + 1
Write an equation for the line that passes through (3, -2) and is parallel to the line whose equation is y = -5x + 1
CHAPTER 2.5
LINEAR MODELS
Prediction line
SCATTER PLOT – GRAPH WITH MANY ORDERS PAIRS
LINE OF BEST FIT – LINE DRAWN THROUGH DATA THAT BEST REPRESENTS IT
MAKE A SCATTER PLOT
APPROXIMATE PERCENTAGE OF STUENTS WHO SENT APPLICATIONS TO TWO COLLEGES IN VARIOUS YEARS SINCE 1985
YEARSSINCE1985
0 3
6
9
12 15
% 20 18 15 15 14 13
LINE OF BEST FIT
SELECT TWO POINTS THAT APPEAR TO BEST FIT THE DATA
IGNORE OUTLIERS
DRAW LINE
PREDICTION LINE
FIND SLOPE
WRITE EQUATION IN SLOPE-INTERCEPT FORM
INTERPRET
WHAT DOES THE SLOPE INDICATE?
WHAT DOES THE Y-INT INDICATE?
PREDICT % IN THE YEAR 2010
HOW ACCURATE ARE PREDICTIONS?
CHAPTER 2.6/2.7
SPECIAL FUNCTIONS and Transformations
ABSOLUTE VALUE FUNCTION
V-shaped PARENT GRAPH (Basic graph) FAMILIES OF GRAPHS (SHIFTS)
xxf
EXAMPLESmake table and graph
1 xxf 2 xxg
3 xxh 2 xxp
xxq 2 x21
EFFECTS
+,- OUTSIDESHIFTS UP AND DOWN
+,- INSIDESHIFTS LEFT AND RIGHT
X,÷ NARROWS AND WIDENS
Be able to use calculator to find graphs and interpret shifts
Be able to identify domain and range
CHAPTER 2.8
GRAPHING INEQUALITIES
BOUNDARY
EX: y ≤ 3x + 1 THE LINE y = 3x + 1 IS THE BOUNDARY
OF EACH REGION SOLID LINE INCLUDES BOUNDARY
____________________________ DASHED LINE DOESN’T INCLUDE
BOUNDARY----------------------------------------------
GRAPHING INEQUALITIES
1. GRAPH BOUNDARY (SOLID OR DASHED)
2. CHOOSE POINT NOT ON BOUNDARY AND TEST IT IN ORIGIONAL INEQUALITY
3. TRUE-SHADE REGION WITH POINTFALSE-SHADE REGION W/O POINT
On calculator
Enter slope-int form under “y=“ Scroll to the left to select above or below Zoom 6
GRAPH THE FOLLOWING INEQUALITIES
x – 2y < 4
2 xy
