Chapter 2: Linear Relations and Functions Algebra 2 Chapter 2: Linear Relations and Functions

Section 2.1 Relations and Functions

Objectives Analyze and graph relations. Find functional values.

Vocabulary Ordered Pair: A pair of coordinates, written in the form (x, y), used to locate any point on a coordinate plane. Cartesian Coordinate Plane: composed of the x-axis (horizontal) and y-axis (vertical), which meet at the origin (0, 0) and divide the plane into four quandrants.

Relation; Domain; Range Relation: is a set of ordered pairs. Domain (of a relation): the set of all first coordinates (x-coordinates) from the ordered pairs. Range (of a relation): the set of all second coordinates (y- coordinates) from the order pairs. Relation: { (12, 28), (15, 30), (8, 20), (12, 20), (20, 50)} Domain: {8, 12, 15, 20} Range: {20, 28, 30, 50}

Function Functions can be represented as 𝑓 𝑥 or 𝑔 𝑥 . When speaking, we say “F of x” or “G of x”. A function is a special type of relation. Each element of the domain is paired with exactly one element of the range. A mapping shows how the members are paired. An example is shown to the right. The example to the right is a function; each element of the domain is paired with exactly one element of the domain. This is called a one-to-one function. Relation: {(12, 28), (15, 30), (8, 20)} Domain Range 12 15 8 28 30 20

Function or not? Domain Range Domain Range -3 2 1 4 -1 1 4 3 5 2 1 4 -1 1 4 3 5 Function Function Domain Range -3 1 5 6 NOT a Function

Relations: Discrete or Continuous? Discrete graphs contain a set of points not connected. Continuous graphs contain a smooth line or curve. Note: You can draw the graph of a continuous relation Without lifting you pencil from the paper.

Vertical Line Test If no vertical line intersects a graph in more than one point, the graph represents a function. If some vertical line intersects a graph in two or more points, the graph DOES NOT represent a function.

Graphing Relations See examples on pages 60 and 61 in your textbook. When graphing, create a table of values.

Evaluate a function Given 𝑓 𝑥 = 𝑥 2 +2, find each value. f(-3) 𝑓 −3 = (−3) 2 +2 𝑓 −3 =9+2 𝑓 −3 =11 b. f(3z) 𝑓 𝑥 = 𝑥 2 +2 𝑓 3𝑧 = (3𝑧) 2 +2 𝑓 3𝑧 =9 𝑧 2 +2

HOMEWORK…..A#2.1 Assigned on Friday, 9/20/13 Due on Monday, 9/23/13 Pages 62-63 [#13-20 all, 24, 34, 36, 40]

Section 2.2 Linear Equations

Section Objectives Identify linear equations and functions. Write linear equations in standard form and graph them.

Identify Linear Equations and Functions A linear equation has no operations other than addition, subtraction, and multiplication of a variable by a constant. The variables may not be multiplied together or appear in a denominator. It does not contain variables with exponents other than 1. The graph of a linear equation is always a line. Linear Equations NOT Linear Equations 5𝑥−3𝑦=7 𝑥=9 6𝑠=−3𝑡−15 𝑦= 1 2 𝑥 7𝑎+4 𝑏 2 =−8 𝑦= 𝑥+5 𝑥+𝑥𝑦=1 𝑦= 1 𝑥

Identify Linear Equations State whether each function is a linear function. Explain. 𝑓 𝑥 =10−5𝑥 𝑔 𝑥 = 𝑥 4 −5 ℎ 𝑥,𝑦 =2𝑥𝑦 𝑓 𝑥 = 5 𝑥+6 𝑔 𝑥 =− 3 2 𝑥+ 1 3

Standard Form 𝐴𝑥+𝐵𝑦=𝐶 The standard form of a linear equation is… where A, B, and C are integers whose greatest common factor is 1, 𝐴≥0, and A and B are not both zero.

Write each equation in standard form. Identify A, B, and C. 𝑦=−2𝑥+3 2𝑦=4𝑥+5 − 3 5 𝑥=3𝑦−2 3𝑥−6𝑦−9=0

Graphing with Intercepts X-Intercept: the x-coordinate of the point at which it crosses the x-axis. y=0 Y-Intercept: the y-coordinate of the point at which it crosses the y-axis. x=0

Find the x-intercept and y-intercept of the graph of 3𝑥−4𝑦+12=0 Find the x-intercept and y-intercept of the graph of 3𝑥−4𝑦+12=0. Then graph the equation.

Find the x-intercept and y-intercept of the graph of 2𝑥+5𝑦−10=0 Find the x-intercept and y-intercept of the graph of 2𝑥+5𝑦−10=0. Then graph the equation.

HOMEWORK…..A#2.2 Assigned on Monday, 9/23/13 Due on Tuesday, 9/24/13 Page 107 [#16-22 all]

Section 2.3 Slope

Objectives for Section 2.3 Find and use the slope of a line. Graph parallel and perpendicular lines.

Vocabulary A rate of change measures how much a quantity changes, on average, relative to the change in another quantity, often time. The slope (m) of a line is the ratio of the change in y-coordinates to the corresponding change in x-coordinates. The slope m of the line passing through ( 𝑥 1 , 𝑦 1 ) and ( 𝑥 2 , 𝑦 2 ) is given by 𝑚= 𝑦 2 − 𝑦 1 𝑥 2 − 𝑥 1 , where 𝑥 1 ≠ 𝑥 2

Find the slope of the line that passes through (-1, 4) and (1, -2) Find the slope of the line that passes through (-1, 4) and (1, -2). Then graph the line.

Find the slope of the line that passes through (1, -3) and (3, 5) Find the slope of the line that passes through (1, -3) and (3, 5). Then graph the line.

Slope – tells the direction in which it rises or falls.

Negative Slope

Zero slope

Family of graphs A family of graphs is a group of graphs that displays one or more similar characteristics. The parent graph is the simplest of the graphs in a family. Parent: y = x Family: y = 3x + 2 y = x + 2

Parallel Lines In a plane, nonvertical lines with the same slope are parallel. All vertical lines are parallel.

Graph the line through (-1, 3) that is parallel to the line with equation 𝑥+4𝑦=−4.

Graph the line through (-2, 4) that is parallel to the line with equation 𝑥−3𝑦=3.

Perpendicular Lines Two lines are perpendicular if the product of their slopes = −1. When you have two perpendicular lines, their slopes are opposite reciprocals of each other. Slope of line AB: Slope of line CD: C(-3,2) A(2,1) D(1,-4) B(-4,-3)

Graph the line through (-3, 1) that is perpendicular to the line with equation 2𝑥+5𝑦=10.

Graph the line through (-6, 2) that is perpendicular to the line with equation 3𝑥−2𝑦=6.

HOMEWORK…..A#2.3 Assigned on Due on Page 108 [#23-29 all]

Section 2.4 Writing Linear Equations

Objectives After this section, you will be able to… Write an equation of a line given the slope and a point on the line. Write an equation of a line parallel or perpendicular to a given line.

Slope-Intercept Form of a Linear Equation 𝑦=𝑚𝑥+𝑏 slope y-intercept

Write an Equation Given Slope and a Point Write an equation in slope-intercept form for the lines that has a slope of 4 3 and passes through the point (3, 2).

Practice Write and equation in slope-intercept form for the line that has a slope of −4 and passes through (−2, −2).

Graph an Equation in Slope-Intercept Form Graph the following equations: 𝑦= 4 3 𝑥+2 𝑦=−3𝑥−4

Point-Slope Form of a Linear Equation 𝑦− 𝑦 1 =𝑚(𝑥− 𝑥 1 ) Given point

Write an Equation Given Two Points What is the equation of the line through 2, 3 and −4, −5 ? Procedure: Find the slope. Write an equation using slope and one of the given points.

Write an Equation of a Perpendicular Line Write an equation for the line that passes through (3, 7) and is perpendicular to the line whose equation is 𝑦= 3 4 𝑥−5.

HOMEWORK…..A#2.4 Assigned on Thursday 9/26/13 Due on Friday 9/27/13 Page 108 [#30-34 all]

Section 2.5 Statistics: Using Scatter Plots

Objectives After this section, you will be able to… Draw scatter plots. Find and use prediction equations.

Vocabulary Bivariate Data: Scatter Plot: Speed (mph) Calories 5 508 6 Bivariate Data: data with two variables Scatter Plot: a set of bivariate data graphed as ordered pairs in a coordinate plane. This table shows the number of Calories burned per hour by a 140-pound person running at various speeds. We can use a linear function to model these data. Scaffolding questions: Will a person that weighs less than 140 pounds burn more or fewer calories per hour than shown in the table at the given speeds? Fewer What is a reasonable estimate of the number of calories a 140-pound person burns in one hour running at a speed of 6.5 miles per hour? About 685 calories Speed (mph) Calories 5 508 6 636 7 731 8 858

Scatter Plot Correlations

Prediction Equations Line of Fit: Prediction Equation: To find a line of fit and prediction equation: Line of fit: when you find a line that closely approximates a set of data. Prediction Equation: an equation of a line of best fit. To find a line of fit and a prediction equation for a set of data, select two points that appear to represent the data well. Matter of personal judgment, so your line and prediction equation may be different from someone else’s.

Find and Use a Prediction Equation HOUSING: The table below shows the median selling price of new, privately- owned, one-family houses for some recent years. Year 1994 1996 1998 2000 2002 2004 Price ($1000) 130.0 140.0 152.5 169.0 187.6 219.6

Draw a Scatter Plot and a line of fit for the data Draw a Scatter Plot and a line of fit for the data. How well does the line fit the data? 250 230 210 190 170 150 130 110 Price ($1000) 0 2 4 6 8 10 Years since 1994 Year 1994 1996 1998 2000 2002 2004 Price ($1000) 130.0 140.0 152.5 169.0 187.6 219.6

Find a prediction equation. What do the slope and y-intercept indicate? Slope formula: (187.6 – 140.0) / (8 – 2) = approx. 7.93 Pt-slope form y – 140.0 = 7.93(x – 2) y-140.0 = 7.93x – 15.86 y = 7.93x + 124.14 One prediction equation is this. The slope indicates that the median price is increasing at a rate of $7930 per year. The y-intercept inidicates that, according to the trend of the rest of the data, the median price in 1994 should have been about 124,140.

Predict the median price in 2014. How accurate does the prediction appear to be? The year 2014 is 20 years after 1994, so use the prediction equation to find the value of y when x = 20. Y = 7.93x + 124.14 = 7.93(20) + 124.14 = 282.74 The model predicts that the median price in 2014 will be about $282,740.

PRACTICE The table shows the mean selling price of new, privately owned one-family homes for some recent years. Draw a scatter plot and line of fit for the data. Then find a prediction equation and predict the mean price in 2014. Year 1994 1996 1998 2000 2002 2004 Price ($1000) 154.5 166.4 181.9 207.0 228.7 273.5

Year 1994 1996 1998 2000 2002 2004 Price ($1000) 154.5 166.4 181.9 207.0 228.7 273.5 Price ($1000) Years since 1994

Practice workspace

HOMEWORK…..A#2.5 Assigned on Monday 9/30/13 Due on Tuesday 10/1/13 Page 89 [#3-9 all]