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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. Relation: { (12, 28), (15, 30), (8, 20), (12, 20), (20, 50)} 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 (ycoordinates) from the order pairs. Domain: {8, 12, 15, 20} Range: {20, 28, 30, 50} Function 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. Functions can be represented as 𝑓 𝑥 or 𝑔 𝑥 . When speaking, we say “F of x” or “G of x”. Relation: {(12, 28), (15, 30), (8, 20)} Domain Range 12 28 15 30 8 20 Function or not? Domain Range Range Domain -3 1 -1 0 2 1 3 2 4 4 5 Function Function Domain Range -3 0 1 1 5 6 NOT a Function Relations: Discrete or Continuous? Discrete Discrete graphs contain a set of points not connected. Continuous 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. a. f(-3) 𝑓 𝑥 = 𝑥2 +2 𝑓 −3 = (−3)2 +2 𝑓 −3 = 9 + 2 𝑓 −3 =11 b. f(3z) 𝑓 𝑥 = 𝑥2 + 2 𝑓 3𝑧 = (3𝑧)2 +2 𝑓 3𝑧 = 9𝑧 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 7𝑎 + 4𝑏2 = −8 𝑥=9 𝑦 = 𝑥+5 6𝑠 = −3𝑡 − 15 𝑥 + 𝑥𝑦 = 1 𝑦= 𝑦= 1 𝑥 2 1 𝑥 Identify Linear Equations State whether each function is a linear function. Explain. a. 𝑓 𝑥 = 10 − 5𝑥 b. 𝑔 𝑥 = 𝑥4 − 5 c. ℎ 𝑥, 𝑦 = 2𝑥𝑦 5 d. 𝑓 𝑥 = 𝑥+6 e. 𝑔 𝑥 = −2𝑥 + 3 3 1 Standard Form The standard form of a linear equation is… 𝐴𝑥 + 𝐵𝑦 = 𝐶 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. a. 𝑦 = −2𝑥 + 3 b. − 𝑥 = 3𝑦 − 2 3 5 c. 2𝑦 = 4𝑥 + 5 d. 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. Then graph the equation. 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 (𝑥1 , 𝑦1 ) and (𝑥2 , 𝑦2 ) is given by 𝑦 −𝑦 𝑚 = 𝑥2−𝑥1, where 𝑥1 ≠ 𝑥2 2 1 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). 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: C(-3,2) A(2,1) Slope of line CD: 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 and passes through the point (3, 2). 3 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 𝑥 3 +2 𝑦 = −3𝑥 − 4 Point-Slope Form of a Linear Equation Slope 𝑦 − 𝑦1 = 𝑚(𝑥 − 𝑥1 ) Given point Write an Equation Given Two Points What is the equation of the line through 2, 3 and −4, −5 ? Procedure: 1. Find the slope. 2. 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 3 to the line whose equation is 𝑦 = 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 6 636 7 731 8 858 Scatter Plot Correlations Prediction Equations Line of Fit: Prediction Equation: To find a line of fit and prediction equation: Find and Use a Prediction Equation HOUSING: The table below shows the median selling price of new, privatelyowned, 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. How well does the line fit the data? 250 Price ($1000) 230 210 190 170 150 130 110 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? Predict the median price in 2014. How accurate does the prediction appear to be? 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 1994 1996 1998 2000 2002 2004 Price ($1000) 154.5 166.4 181.9 207.0 228.7 273.5 Price ($1000) Year 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]