chapter 1 relations and linear functions. cartesian coordinate plane
TRANSCRIPT
CHAPTER 1
RELATIONS AND LINEAR FUNCTIONS
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).
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.
Function Examples
If f(x) = x³ - 3 , evaluate: – f(-2)– f(3a)
If g(x) = 5x2 - 3x+7 , evaluate: – g(4-2a)– g(-3c)
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?
Find the domain of each function
F(x) = x3+5x x2-4x
The denominator can’t be zero.G(x) = 1
√x-4Radicandcan’t be negative.
CHAPTER 1.2
Composition of Functions
Sum: (f+g)(x)=f(x) + g(x)Difference: (f-g)(x)=f(x) - g(x)Product: (f*g)(x)=f(x) * g(x)Quotient: (f/g)(x)=f(x) / g(x)
Operations with Functions
[fog](x)=f[g(x)]f[g(x)] means to substitute the function g(x) wherever you see an x in f(x)
Composition of Functions
CHAPTER 1.3
Graphing Linear Equations
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
SLOPE
CHANGE IN Y OVER CHANGE IN X RISE OVER RUN RATIO STEEPNESS RATE OF CHANGE
FORMULA12
12
xx
yym
Slope
Positive Slope Negative Slope 0 Slope Undefined Slope
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
SLOPE-INTERCEPT FORM
y = mx + b m IS SLOPE b IS Y-INTERCEPT
CHAPTER 1.4/1.5
Writing Linear Equations & Writing Equations of Parallel and Perpendicular Lines
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?
LINES
PARALLEL SAME SLOPE VERTICAL LINES ARE
PARALLEL
PERPENDICULAR OPPOSITE
RECIPROCALS (flip it and change sign)
VERTICAL AND HORIZONTAL LINES
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 1.7
Piecewise Functions
Piecewise Functions
Different equations used for different intervals of the domain
Piecewise Functions
F(x)= { 1 if x≤-2
2+x if -2<x ≤3
2x if x>3
Step Function
Looks like a set of stairs Breaks in the graph of the function
Greatest Integer Function
Type of step function Means the greatest integer
not greater than x.
Example: [[8.9]]=8
ABSOLUTE VALUE FUNCTION
V-shaped PARENT GRAPH (Basic graph)
xxf
Examples
F(x)=2│x │-6 F(x) = { 2x+1 if x<0
2x-1 if x≥0 F(x) = [[x-1]] F(x) = { x+3 if x≤0
3-x if 1<x ≤3
3x if x>3
CHAPTER 1.8
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 POINT
FALSE-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
x y 6
Review
f xx x
x x( )
RST5 2 2
2 2
if
if 2
f x
x
x x
x x
( )
RS|T|4 5
2 5 8
24 8
if
if
if
f x xbg 4
f x xbg 3 ?
2 3 6x y
y x 2 2
Quiz
f xx x
x x( )
RST
4 0
4 0
if
if x y 2 –2
y x 4 3
f x( ) = 3 3x
F(x)=│x+3│
F(x) = { x+3 if x≤03-x if 1<x ≤33x if x>3
CHAPTER 1.6
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?
Regression
Regression Line Line of best fitLinear correlation coefficient (r)
– The closer the value of r is to 1 or -1, the closer the data points are to the line.
