ADVANCED ALGEBRA II

CHAPTER 2 -- LINEAR EQUATIONS AND FUNCTIONS

ADVANCED ALGEBRA IICHAPTER 2 – LINEAR EQUATIONS AND FUNCTIONS2.1 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. −2 x+3 C. 2 x2+1 D. |−2 x+5|

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.

(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

Find f (−3 ) , f (0 ) ,∧f (5 ) for the function: f ( y )=−15y+ 35

EQUATIONS IN TWO VARIABLES

Example: Graph an equation in two variables

A. y=−3x+5 B. x= y2+1

ADVANCED ALGEBRA IICHAPTER 2 – LINEAR EQUATIONS AND FUNCTIONSSECTIONS 2.2 – 2.4 LINEAR EQUATIONS

WARM-UP

Evaluate if x=−2 ,0 ,1∧4

A. 35 x−7 B. −2 x−3

GRAPHING LINEAR EQUATIONS

(V) Linear Function –

(V) Linear Equation –

(V) Independent Variable –

(V) Dependent Variable –

Example: Graphing a Linear Equation

A. y=23 x+3 B. 2 x− y=8

(V) y-intercept –

(V) x-intercept –

(V) Standard Form of a Linear Equation

Example: Transportation

The equation 3 x+2 y=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 x- and 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

Write in point-slope form an equation of the line with the slope −12 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

Write an equation of the line through each point and perpendicular to y=34 x+2.

A. (0, 4)

B. (6, 1)

ADVANCED ALGEBRA IICHAPTER 2 – LINEAR EQUATIONS AND FUNCTIONS2.5 MODEL DIRECT VARIATION

WARM UP

1. Solve each equation for y.

A. 12 y=3 x B. 12 y=5 x C. 34 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 IICHAPTER 2 – LINEAR EQUATIONS AND FUNCTIONS2.6 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.

A. f ( x )= 43x−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 1984 1988 1992 1994 1998 2002 2006 2010 201437 49 57 64 67 72 77 80 82 88

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 fittingline and use it to predict the number of participating countries in 2018.

ACTIVITY LAB – FINDING A LINE OF BEST FIT HANDOUT

ADVANCED ALGEBRA IICHAPTER 2 – LINEAR EQUATIONS AND FUNCTIONS2.7 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 IICHAPTER 2 – LINEAR EQUATIONS AND FUNCTIONSEXTENSION – PIECEWISE FUNCTIONS

Example: Writing a Piecewise Function

Write a piecewise function to represent the graphon the right.

Example: Graphing a Piecewise Function

Graph the function f ( x )= [x ] (The Greatest Integer Function)

Example: Graphing a Piecewise Function

Graph f ( x )={2 x−1 ,∧x<0−2 ,0≤ x<312x−1 , x>3

Example: Evaluating a Piecewise Function

If g ( x )={−12 x−1,∧x<23 x−7 ,∧x>2 , find g (−1 ) , g (2 ) ,∧g (4)

ADVANCED ALGEBRA IICHAPTER 2 – LINEAR EQUATIONS AND FUNCTIONS2.8 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. 12 p≤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 C. 2|3h−6|≥10

GRAPHING LINEAR INEQUALITIES

(V) Linear Inequality (in two variables) -

Example: Graphing a Linear Inequality

Graph the inequality y< 12 x−3

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>|2 x−1|

Example: Writing Inequalities

Write an inequality for each graph.

A. B.

ADVANCED ALGEBRA IICHAPTER 2 – LINEAR EQUATIONS AND FUNCTIONSCHAPTER 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: