Name: _________________________________ Date: ____________________
Linear Functions
Sections Covered: 1.1 Solving Linear & Absolute Value Equations
2.2 Intro to FUNctions 2.3 Linear Functions
2.7 Linear Regressions
◄ August ~ September 2014 ~ October ►
Sun Mon Tue Wed Thu Fri Sat 1
2 A Introductions 1.1
3 B Introductions 1.1
4 A 1.1 Solving Linear and Absolute Value Equations HW 1.1
5 B 1.1 Solving Linear and Absolute Value Equations HW 1.1
6
7
8 A 2.2 Intro to FUNctions HW 2.2
9 B 2.2 Intro to FUNctions HW 2.2
10 A 2.3 Linear Functions HW 2.3A
11 B 2.3 Linear Functions HW 2.3A
12 A 2.3 Linear Functions/ Piecewise Functions HW 2.3B
13
14
15 B 2.3 Linear Functions/ Piecewise Functions HW 2.3B
16 A QUIZ: Graph/ Write Lin Eqs 2.7 Linear Regressions HW 2.7
17 B QUIZ: Graph/ Write Lin Eqs 2.7 Linear Regressions HW 2.7
18 A Spaghetti Bridge Lab HW Quiz Review
19 B Spaghetti Bridge Lab HW Quiz Review
20
21
22 A Finish Lab Review for Quiz Study
23 B Finish Lab Review for Quiz Study
24 A QUIZ: 1.1, 2.2, 2.3, 2.7, Lab
25 B QUIZ: 1.1, 2.2, 2.3, 2.7, Lab
26
27
Solving Linear Equations
1. 3 4 9x + =
2. 7 9 12 15x x− = −
3. 8(7 3) 5(10 4)x x− = +
4. 3 4 7 8 9 3 2x x x+ + − = − + 5. 5( 2) 8 4 2 3 4x x x x+ − + = + −
Fractions If an equation has fractions, GET RID OF THEM! Multiply the entire equation by the _________________ ___________________ Example:
6. 2 36103 5 5
xx + − =
You Try:
7. 7 14 3 2x x ++ =
Solving Absolute Value Equations Absolute value means___________________________________________________ Using this definition, we can simplify expressions with absolute value symbols: |−8| −|8| |−3 + 8| |−3| + |8|
1. Distribute 2. Collect like terms 3. Gather variables on one side of equation 4. Gather constants on opposite side 5. Add or subtract 6. Multiply or divide
To solve an absolute value equation, we once again use the absolute value definition. If |x| = 3, then x can equal 3 or −3! Think of inside the absolute as a location. Where do I need to be to be 3 steps from 0??? Examples: Solve for x. Check for Extraneous Solutions. 1) |x| = 5 2) |2x + 3| + 2 = 7 3) |3x + 1| = x − 3 You must get the absolute by itself before asking, “Where is my location?”
4) 2|4x - 6| - 8 = -4 5) 31 |2x - 1| - 3 = 8
6) Think about this one… |2x + 3| = −7, what would your answer be?
7) 3 64
x += 8) 4 1 17x − = −
1. Isolate absolute value 2. Create 2 equations w/o abs val
• One with positive on other side • One with negative
3. Solve each as linear equations 4. Check for extraneous solutions
DEFN] ordered pair – ( )yx, or ( )ordinateabscissa , DEFN] Domain – set of all first elements of an ordered pair (usually all the x-values) DEFN] Range – set of all second elements of an ordered pair (usually all the y-values) DEFN] Relation – set of ordered pairs Relations may be expressed as:
a set of ordered pairs a mapping
a table of values
a graph an equation
DEFN] Function – a relation in which each element of the domain is paired with EXACTLY one
element of the range A FUNction cannot repeat the DOMAIN! All FUNctions are relations. However, all relations are not FUNctions!
Testing for FUNctions:
• Mapping – each element in the first set must be matched with an element in the second set (without repeating the domain)
• Vertical Line Test – a vertical line drawn on the graph of a relation cannot pass through more than one point on the graph
EX 1] State the domain and range of each relation. Then state whether the relation is a function.
a) ( ) ( ) ( ){ }1,2,2,4,0,3 −−−
b) ( ) ( ) ( ) ( ){ }3,9,3,9,2,4,2,4 −−
x y
x
y
EX 2] State whether the relation is a function.
a) b) FUNction Notation:
A function is commonly denoted by _____________ .
)(xf is read ___________________________________________________________ .
)(xf should be interpreted as the ____________________________________________.
• x is called the ___________________ variable. (input or variable that represents the domain)
• y is called the ___________________ variable. (output or variable that represents the range)
)(xfy = indicates that for each element of the domain that replaces x,
Ordered pairs may be written as ____________________ or _______________________ . EX 3] Find )4(f and )1( +xf if 27)( xxf −= . DEFN] Implied Domain Interval Notation: ( or )
[ or ]
If … Monomial/Polynomial in denominator Radical Radical in Denominator
x
y
x
y
EX 4] State the implied domain of f.
a) x
xxxf4
5)(3 +
=
b) 4
1)(−
=x
xf
c) xxf −= 5)( EX 5] Use a TI-83 to sketch each graph. Determine the domain, range, and whether each equation is
a relation or function.
a) 13 2 += xy b) xy −= 4 Increasing, Decreasing, and Constant FUNctions:
f is increasing on I if )()( bfaf < whenever ba < . f is decreasing on I if )()( bfaf > whenever ba < . f is constant on I if )()( bfaf = for all a and b.
x
y
x
y
EX 6] Label the increasing, decreasing and constant intervals in terms of the x-axis.
a) b) One-to-One FUNction: EX 7] State whether each of the following is a One-to-One FUNction.
a) b)
x
y
5
8
y
x 1
1
y
x 1
1
y
x
Slope of a nonvertical line that passes through the points ( )111 , yxP and ( )222 , yxP is
xinchangeyinchange
runrise
xxyym ==
−−
=12
12
EX 1] Find the slope of the line that passes through the point ( )4,3 and ( )10,5 . EX 2] Find the slope of the line that passes through the point ( )3,4 −− and ( )2,1 − . Forms of Equations of Lines:
Point-slope form: ( )11 xxmyy −=− Slope-intercept form: bmxy += Standard form: CByAx =+ (A, B, and C are integers and A is not negative.) Horizontal line: by = Horizontal lines have zero slope and pass through ( )b,0 . Vertical line: ax = Vertical lines have undefined slope and pass through ( )0,a .
Write the equation of a line in slope intercept form: Steps: 1. Ask yourself “What two letters do I need to write the equation of a line?”__________________
2. Identify which letters you need to still find. 3. If you need m, plug the points into the slope formula. 4. If you need b, plug m and an ordered pair (x, y) into the slope intercept formula and solve for b. 5. Write the equation of a line with the new m and b.
EX 3] Write an equation of the line that passes through the point ( )1,2 − with slope 31 .
EX 4] Write an equation of the line with slope 54 and y-intercept 3− .
EX 5] Write an equation of the vertical line that passes through the point ( )1,2 − . EX 6] Write an equation of the line passing through ( )2,6 − and ( )4,2 − Parallel Lines – lines that have the same slope or both have undefined slope
Perpendicular Lines – lines whose slopes are negative reciprocals of each other
NOTE: Horizontal lines (with zero slope) are perpendicular to vertical lines (with undefined slope).
EX 7] State whether the lines are parallel, perpendicular, or neither.
HINT: Write each equation in slope-intercept form and determine the slope of each line.
762 =+ yx and 93 =+ yx EX 8] Write an equation of the line passing through ( )1,2 − and parallel to 10=+ yx . EX 9] Write an equation of the line passing through ( )4,3− and perpendicular to 72 =− yx .
Given ____ and ____: G
iven ____ and ____: G
iven ____ and _______________:
Given ____ and ____: If the new
line must
be ____ to…
If the new line m
ust be ____ to…
Zeros of function f are the values of x for which _____________________.
x-intercepts of a graph (written as ordered pairs) are the values of x for which ___________________.
The solution of a linear equation is the ______________________________________________.
NOTE: Constant Function – the graph of a ____________________ line EX 10] Graph the line to find the solution.
a. 62 +−= xy b. 1236 −=− yx
Solution: _________ Solution: _________ EX 11] Find the solution 0)( =xf . Verify the solution of 0)( =xf is the same as the x-intercept of the graph of )(xfy = .
42)( −−= xxf
EX 12] Solve )()( 21 xfxf = by an algebraic method and by graphing.
623)(1 += xxf
12)(2 −−= xxf
x
y
x
y
x
y
x
y
2.2 “Piecewise and Greatest Integer Functions” Functions can be represented by a combination of functions, each corresponding to a part of the domain. These functions are called ________________________________________________. 1. Evaluate.
3, 5
Evaluate ( ) for thegiven valueof x.2 1, 5x x
f xx x+ <
= + ≥
Which equation to use?
x + 3 or 2x + 1
Explain why… Substitute the value of x and solve.
Value of f (x)
1. f (3)
2. f (7)
3. f (5)
2. Graph.
EX 1]
>−
≤−=
0,121
0,2)(
xx
xxxf EX 2]
<≤<≤<≤<≤−
=
43,232,121,010,1
)(
xxxx
xf
x
y
x
y
3. Write the equation. Write the equation for the piecewise function shown below. Assume all points are integers. Final Answer
Steps:
1. Divide the graph into _________________________________________
2. Find the ___________________________________________________
3. Write your _________________________________________________
Find the slope. 2 1
2 1
y y risemx x run−
= =−
Find the y-intercept. y mx b= + . (Substitute x, y, and m. Solve for b) 4. Real Life Uses EX] Parking Rates $3 for the 1st hour $2 for each hour thereafter Maximum charged per day $9
≤<≤<≤<≤<
=
243____,32____,
21____,10____,
)(
tiftiftiftif
xf
x
y
t
y
)(xf
DEFN] Greatest Integer Function is also call the “floor function” denoted by
_________________________________________________________________. The value of the greatest integer function is
_________________________________________________________________. The symbol [ ][ ]x means the greatest integer that is less than or equal to x. EX 1] Find each of the following.
a) int
414 b) [ ]8.6 c) [ ][ ]1.3−
d) [ ]π e)
52 f) int ( )2.53−
EX 2] If [ ] 4)( += xxf , find each of the following.
a) )(πf b) )4(−f c) )3.6(−f To Graph Greatest Integer Function… First, realize that the value of the floor function is constant (the same answer) between any
2 consecutive integers. For instance, between 1 and 2:
int
34 = [ ]8.1 = [ ][ ]6789.1 = int ( )999999.1 = int ( )1 =
Next, create a table of values for [ ][ ]xy = and graph.
x [ ][ ]xy = 45 −<≤− x
34 −<≤− x
23 −<≤− x
12 −<≤− x
01 <≤− x
10 <≤ x
21 <≤ x
32 <≤ x
43 <≤ x
x
y
Find the line of best fit or the least-squares regression line given the data set ( ) ( ) ( ) ( ) ( ){ }1,2 , 2,3 , 3,3 , 4,4 , 5,7S = . Find the least-squares regression (or Line of Best Fit) rounded to the nearest tenth on your calculator ________________________
Correlation Coefficient and Coefficient of Determination (for all data sets 1 1r− ≤ ≤ ) Linear correlation Coefficient (r): If r = -1: If r = 1: Coefficient of Determination (r2): Note: If a population/sports team data is r2 = 0.90, then 90% of the total variation in the dependent variable (number of teams) can be attributed to the state population. This also means that population alone does not predict with certainty the number of sports teams. Other factors, such as climate, are also involved in the number of sports teams.
Unit 1A Homework Completion Sheet Name: _______________________ Block: _______ Prerequisite Skills:
1. Solving Linear Equations 2. Graphing Coordinates 3. Writing and Graphing Linear Equations
Date Assignment Score (out of 20)
Learning Targets Notes
What am I confident about? Specifically, what was difficult or confusing?
Am I weak on a prerequisite skill?
HW 1.1 Solving Linear
and Absolute Value Equations
HW 2.2 Intro to
FUNctions
HW 2.3A Linear
Functions
HW 2.3A Linear and Piecewise Functions
HW 2.7 Linear
Regression
Learning Targets: 1.) Use linear and absolute value equations to solve problems.
2.) Use properties of functions to describe and classify relations. 3.) Create graphs of linear equations.
4.) Create equations of lines. 5.) Create equations for a linear regression and analyze its accuracy in describing the data.
