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ADVANCED ALGEBRA II
CHAPTER 2 -- LINEAR EQUATIONS AND FUNCTIONS
ADVANCED ALGEBRA II
CHAPTER 2 – LINEAR EQUATIONS AND FUNCTIONS
2.1 REPRESENT RELATIONS AND FUNCTIONS
WARM-UP
1. Graph each ordered pair on the coordinate plane.
A. (-4. -3)
B. (3, 2)
C. (0, 0)
D. (1, -3)
E. (-1, 5)
2. Evaluate each expression for x = -2, 0, and 5
A. 𝑥 + 2
B. −2𝑥 + 3
C. 2𝑥 2 + 1
GRAPHING RELATIONS
(V) Relation -
Example: Graphing a Relation
Graph the relation {(−2, 4), (3, −2), (−1, 0), (1, 5)}
(V) Domain –
(V) Range Example: Finding Domain and Range
Write the ordered pairs for the relation shown to the right.
Find the domain and range.
D. |−2𝑥 + 5 |
(V) Mapping Diagram –
Example: Make a Mapping Diagram
Make a mapping diagram for the relation {(−2, 4), (3, −2), (3, 0), (1, 4), (−2, 7)}
IDENTIFYING FUNCTIONS
(V) Function -
Example: Identifying Functions
Determine whether each relation is a function.
(V) – Vertical-Line Test -
Example: Using the Vertical-Line Test
Use the vertical-line test to determine whether each represents a function.
(V) Function Notation –
Example: Using function notation
1
3
Find 𝑓(−3), 𝑓(0), 𝑎𝑛𝑑 𝑓(5) for the function: 𝑓(𝑦) = − 5 𝑦 + 5
EQUATIONS IN TWO VARIABLES
Example: Graph an equation in two variables
A. 𝑦 = −3𝑥 + 5
B. 𝑥 = 𝑦 2 + 1
ADVANCED ALGEBRA II
CHAPTER 2 – LINEAR EQUATIONS AND FUNCTIONS
SECTIONS 2.2 – 2.4 LINEAR EQUATIONS
WARM-UP
Evaluate if 𝑥 = −2, 0, 1 𝑎𝑛𝑑 4
A.
3
5
𝑥−7
B. −2𝑥 − 3
GRAPHING LINEAR EQUATIONS
(V) Linear Function –
(V) Linear Equation –
(V) Independent Variable –
(V) Dependent Variable –
Example: Graphing a Linear Equation
2
A. 𝑦 = 3 𝑥 + 3
B. 2𝑥 − 𝑦 = 8
(V) y-intercept –
(V) x-intercept –
(V) Standard Form of a Linear Equation
Example: Transportation
The equation 3𝑥 + 2𝑦 = 120 models the number of passengers who can sit in a train car where x is
the number of adults and y is the number of children. Graph the equation. Explain what the xand y-intercepts represent. Describe the domain and range.
(V) Slope –
Slope Formula
Example: Finding Slope
Find the slope of the line through the points
A. (-2, 7) and (4, -1)
B. (3, 7) and (3, -2)
C. (4, -6) and (-1, -6)
WRITING EQUATIONS OF LINES
There are several ways to write the equation of a line. We will discuss them here.
Point-Slope Form
Example: Writing the Equation Given the Slope and a Point
1
Write in point-slope form an equation of the line with the slope − 2 through the point (8, -1).
Example: Write an Equation Given Two Points
Write in point-slope form the equation of the line through (1, 5) and (4, -1).
The other forms of a line are Slope-Intercept and Standard Form.
Slope-Intercept –
Standard Form –
Example: Write an Equation in All Three Forms
Find the equation of the line through (-3, 6) and (4, 8). Please write in all three forms.
SPECIAL LINES AND RELATIONSHIPS
Horizontal Line
Vertical Line
Perpendicular Lines
Parallel Lines
Example: Writing the Equation of a Perpendicular Line
3
Write an equation of the line through each point and perpendicular to 𝑦 = 𝑥 + 2.
4
A. (0, 4)
B. (6, 1)
ADVANCED ALGEBRA II
CHAPTER 2 – LINEAR EQUATIONS AND FUNCTIONS
2.5 MODEL DIRECT VARIATION
WARM UP
1. Solve each equation for y.
A. 12𝑦 = 3𝑥
B. 12𝑦 = 5𝑥
C.
3
4
𝑦 = 15
WRITING AND INTERPRETING A DIRECT VARIATION
(V) Direct Variation –
(V) Constant of Variation –
Example: Identifying Direct Variation From a Table
For each function, determine whether y varies directly with x. If so, find the constant of variation
and write the equation.
A.
B.
Example: Identifying Direct Variation From an Equation
For each function, determine whether y varies directly with x. If so, find the constant of variation.
A. 3y = 2x
B. 7x + 4y = 10
Example: Water Conservation
A dripping faucet wastes a cup of water if it drips for three minutes. The amount of water wasted
varies directly with the amount of time the faucet drips.
A. Find the constant of variation k and write an equation to model the direct variation.
B. Find how long the faucet must drip to waste 4 ½ c of water.
Example: Using a Proportion
Suppose y varies directly with x, and x = 27 when y = -51. Find x, when y = -17.
ADVANCED ALGEBRA II
CHAPTER 2 – LINEAR EQUATIONS AND FUNCTIONS
2.6 DRAW SCATTER PLOTS AND BEST-FITTING LINES
WARM UP
1. Find the change in x and the change in y between each pair of points.
A. (10, 17) and (11.5, 13.5)
B. (0, 3/10) and (-1, 2/5)
2. Evaluate each function for the given values.
4
A. 𝑓(𝑥) = 3 𝑥 − 2 for x = -3, 0, ½
B. 𝑔(𝑥) = 3(2 − 𝑥) for x = 0, 1/6, a
MODELING REAL-WORLD DATA
Example: Transportation
Jacksonville, Florida has an elevation of 12 ft above sea level. A hot-air balloon taking off from
Jacksonville rises at a rate of 50 ft/min. Write an equation to model the balloon’s elevation as a
function of time. Interpret the intercept at which the graph intersects the vertical axis.
Example: Science
A candle is 6 in. tall after burning for 1 hr. After 3 hr., it is 5 ½ in. tall. Write a linear equation to
model the height y of the candle after burning x hours.
PREDICTING WITH LINEAR MODELS
(V) Scatter Plot –
(V) Correlation –
(V) Correlation Coefficient -
(V) Trend Line (aka Line of Best Fit)
Example: Winter Olympics
The table shows the numbers of countries that participated in the Winter Olympics from 1980 to
2014.
1980
37
1984
49
1988
57
1992
64
1994
67
1998
72
A. Create a scatter plot of the data pairs. Use the
graph to the right.
B. Describe the correlation between the two variables.
C. Write an equation that approximates the best fitting
line and use it to predict the number of participating
countries in 2018.
ACTIVITY LAB – FINDING A LINE OF BEST FIT HANDOUT
2002
77
2006
80
2010
82
2014
88
ADVANCED ALGEBRA II
CHAPTER 2 – LINEAR EQUATIONS AND FUNCTIONS
2.7 USE ABSOLUTE VALUE FUNCTIONS AND TRANSFORMATIONS
WARM UP
Graph each equation for the given domain and range.
1. 𝑦 = 𝑥 for real numbers x and 𝑦 ≥ 0
2. 𝑦 = 2𝑥 − 4 for real numbers x and 𝑦 ≥ 0
3. 𝑦 = −𝑥 + 6 for 𝑥 ≤ 5
GRAPHING ABSOLUTE VALUE FUNCTIONS
(V) Absolute Value Function –
Parent Function for Absolute Value Function:
Translations:
Stretches, Shrinks, and Reflections:
Graph a Function of the Form y = | x – h | + k:
ADVANCED ALGEBRA II
CHAPTER 2 – LINEAR EQUATIONS AND FUNCTIONS
EXTENSION – PIECEWISE FUNCTIONS
Example: Writing a Piecewise Function
Write a piecewise function to represent the graph
on the right.
Example: Graphing a Piecewise Function
Graph the function 𝑓(𝑥) = [𝑥]
(The Greatest Integer Function)
Example: Graphing a Piecewise Function
2𝑥 − 1, 𝑥 < 0
Graph 𝑓(𝑥) = {−2, 0 ≤ 𝑥 < 3
1
𝑥 − 1, 𝑥 > 3
2
Example: Evaluating a Piecewise Function
1
− 𝑥 − 1, 𝑥 < 2
If 𝑔(𝑥) = { 2
, find 𝑔(−1), 𝑔(2), 𝑎𝑛𝑑 𝑔(4)
3𝑥 − 7, 𝑥 > 2
ADVANCED ALGEBRA II
CHAPTER 2 – LINEAR EQUATIONS AND FUNCTIONS
2.8 GRAPH LINEAR INEQUALITIES IN TWO VARIABLES
WARM UP
1. Solve each inequality. Graph the solution on a number line and write in interval notation.
A. 12𝑝 ≤ 15
B. 5 − 2𝑡 ≥ 11
2. Solve and graph each absolute value equation or inequality. Then write in interval notation.
A. |4𝑐| = 18
B. |5 − 2𝑏| = −2
GRAPHING LINEAR INEQUALITIES
(V) Linear Inequality (in two variables) -
C. 2|3ℎ − 6| ≥ 10
Example: Graphing a Linear Inequality
1
Graph the inequality 𝑦 < 𝑥 − 3
2
Example: At least 35 performers of the Big Tent Circus are in the grand finale. Some pile into cars,
while others balance on bicycles. Seven performers are in each car, and five performers are on
each bicycle. Draw a graph showing all the combinations of cars and bicycles possible for the
finale.
GRAPHING TWO-VARIABLE ABSOLUTE VALUE INEQUALITIES
Example: Graphing Absolute Value Inequalities
A. 𝑦 ≤ |𝑥 − 4| + 5
B. – 𝑦 + 3 > |2𝑥 − 1|
Example: Writing Inequalities
Write an inequality for each graph.
A.
B.
ADVANCED ALGEBRA II
CHAPTER 2 – LINEAR EQUATIONS AND FUNCTIONS
CHAPTER 2 REVIEW
In preparation for the Chapter 2 Examination, you may do the following things:
1. Study / Review Notes
2. Chapter 2 Review: p. 141 - 144 # 1 – 34
3. Chapter 2 Test: p. 145 # 1 – 27
4. Chapter 1 Extra Practice: p. 1011 # 1 – 42
Supplemental Notes / Problems: