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Transcript of Warm Up To help guide this chapter, a project (which will be explained after the warm up) will...
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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
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CHAPTER 2.1, 2.2
RELATIONS AND LINEAR FUNCTIONS
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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).
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Cartesian Coordinate Plane
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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).
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Mapping-shows how each member of the domain is paired with each member of the range.
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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.
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One-to-one Mapping
– example: {(4,5), (–2,1), (5,6), (0,2)}
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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
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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.
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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
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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.
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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
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Function Examples
If f(x) = x³ - 3 , evaluate: – f(-2)– f(3a)
If g(x) = 5x2 - 3x+7 , evaluate: – g(4)– g(-3c)
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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
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Linear/not linear
2x + 3y = -5 f (x) = 2x – 5 g (x) = x³ + 2 h (x,y) =- 1 + xy
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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?
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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
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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
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CHAPTER 2.3
SLOPE
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SLOPE
CHANGE IN Y OVER CHANGE IN X RISE OVER RUN RATIO STEEPNESS RATE OF CHANGE
FORMULA12
12
xxyy
m
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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
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RATE OF CHANGE
OFTEN ASSOCIATED WITH SLOPE MEASURES ONE QUANTITY’S CHANGE
TO ANOTHER
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LINES
PARALLEL SAME SLOPE VERTICAL LINES ARE
PARALLEL
PERPENDICULAR OPPOSITE
RECIPROCALS (flip it and change sign)
VERTICAL AND HORIZONTAL LINES
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CHAPTER 2.4
WRITING LINEAR EQUATIONS
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SLOPE-INTERCEPT FORM
y = mx + b m IS SLOPE b IS Y-INTERCEPT
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POINT-SLOPE FORM
FIND SLOPE PLUG IN ARRANGE IN SLOPE INTERCEPT FORM
11 xxmyy
11 ,, yxm
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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)
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INTERPRETING GRAPHS
WRITE AN EQUATION IN SLOPE-INTERCEPT FORM FOR THE GRAPH
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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?
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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
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CHAPTER 2.5
LINEAR MODELS
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Prediction line
SCATTER PLOT – GRAPH WITH MANY ORDERS PAIRS
LINE OF BEST FIT – LINE DRAWN THROUGH DATA THAT BEST REPRESENTS IT
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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
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LINE OF BEST FIT
SELECT TWO POINTS THAT APPEAR TO BEST FIT THE DATA
IGNORE OUTLIERS
DRAW LINE
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PREDICTION LINE
FIND SLOPE
WRITE EQUATION IN SLOPE-INTERCEPT FORM
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INTERPRET
WHAT DOES THE SLOPE INDICATE?
WHAT DOES THE Y-INT INDICATE?
PREDICT % IN THE YEAR 2010
HOW ACCURATE ARE PREDICTIONS?
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CHAPTER 2.6/2.7
SPECIAL FUNCTIONS and Transformations
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ABSOLUTE VALUE FUNCTION
V-shaped PARENT GRAPH (Basic graph) FAMILIES OF GRAPHS (SHIFTS)
xxf
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EXAMPLESmake table and graph
1 xxf 2 xxg
3 xxh 2 xxp
xxq 2 x21
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EFFECTS
+,- OUTSIDESHIFTS UP AND DOWN
+,- INSIDESHIFTS LEFT AND RIGHT
X,÷ NARROWS AND WIDENS
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Be able to use calculator to find graphs and interpret shifts
Be able to identify domain and range
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CHAPTER 2.8
GRAPHING INEQUALITIES
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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----------------------------------------------
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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
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On calculator
Enter slope-int form under “y=“ Scroll to the left to select above or below Zoom 6
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GRAPH THE FOLLOWING INEQUALITIES
x – 2y < 4
2 xy