Chapter 2 Graphing Linear Relations and Functions By

Chapter 2 Graphing Linear Relations and Functions By Kathryn Valle

2 -1 Relations and Functions • A set of ordered pairs forms a relation. – Example: {(2, 4) (0, 3) (4, -2) (-1, -8)} • The domain is the set of all the first coordinates (x-coordinate) and the range is the set of all the second coordinates (y-coordinate). – Example: domain: {2, 0, 4, -1} and range: {4, 3, -2, -8} • Mapping shows how each member of the domain and range are paired. – Example: 1 -3 7 9 2 -5 4 1 -2 3 0 -6 7

2 -1 Relations and Functions (cont. ) • A function is a relation where an element from the domain is paired with only one element from the range. – Example (from mapping example): The first is a function, but the second is not because the 1 is paired with both the 3 and the 0. • If you can draw a vertical line everywhere through the graph of a relation and that line only intersects the graph at one point, then you have a function.

2 -1 Relations and Functions (cont. ) • A discrete function consists of individual points that are not connected. • When the domain of a function can be graphed with a smooth line or curve, then the function is called continuous.

2 -1 Practice 1. Find the domain and range of the following: a. b. {(3, 6) (-1, 5) (0, -2)} {(4, 1) (1, 0) (3, 1) (1, -2)} 2. Are the following functions? If yes, are they discrete or continuous? a. b. {(2, 2) (3, 6) (-2, 0) (0, 5)} {(9, 3) (8, -1) (9, 0) (9, 1) (0, -4)} c. y = 8 x 2 + 4 Answers: 1)a) domain: {3, -1, 0} range: {6, 5, -2} b) domain: {4, 1, 3} range: {1, 0, -2} 2)a) function; discrete b) not a function c) function; continuous

2 -2 Linear Equations • A linear equation is an equation whose graph is a straight line. The standard form of a linear equation is: Ax + By = C, where A, B, and C are all integers and A and B cannot both be 0. • Linear functions have the form f(x) = mx + b, where m and b are real numbers. • A constant function has a graph that is a straight, horizontal line. The equation has the form f(x) = b

2 -2 Linear Equations (cont. ) • The point on the graph where the line crosses the y-axis is called the y-intercept. – Example: find the y-intercept of 4 x – 3 y = 6 4(0) – 3 y = 6 substitute 0 for x y = -2, so the graph crosses the y-axis at the point (0, -2) • The point on the graph where the line crosses the x-axis is called the x-intercept. – Example: find the x-intercept of 3 x + 5 y = 9 3 x + 5(0) = 9 substitute 0 for y x = 3, so the graph crosses the x-axis at the point (3, 0)

2 -2 Practice 1. Determine if the following are linear equations. If so, write the equation in standard form and determine A, B, and C. a. 4 x + 3 y = 10 b. x 2 + y = 2 c. 5 – 3 y = 8 x d. 1/x + 4 y = -5 2. Find the x- and y-intercepts of the following: a. 4 x – 3 y = -12 b. ½ y + 2 = ½ x 2)a) x-intercept: -3 y-intercept: 4 b) x-intercept: 4 y-intercept: -4 Answers: 1)a) yes; A = 4, B = 3, C = 10 b) no c) yes; A = 8, B = 3, C = 5 d) no

2 -3 Slope • The slope of a line is the change in y over the change in x. • If a line passes through the points (x 1, y 1) and (x 2, y 2), then the slope is given by m = y 2 – y 1 , where x 1 ≠ x 2 – x 1 • In an equation with the from y = mx + b, m is the slope and b is the y-intercept. • Two lines with the same slope are parallel. • If the product of the slopes of two lines is -1, then the lines are perpendicular.

2 -3 Practice 1. Find the slope of the following: a. (-2, 4) (3, -6) b. (3. 5, -2) (0, -16) c. y = 3 x + b d. (-1, 8) (14, 8) e. 12 x + 3 y – 6 = 0 f. y = -7 2. Determine whether the following lines are perpendicular or parallel by finding the slope. a. (4, -2) (6, 0), (7, 3) (6, 2) b. y = 2 x – 3, (6, 6) (4, 7) Answers: 1)a) -2 b) 4 c) 3 d) 0 e) -4 f) 0 2)a) 1; parallel b) -1; perpendicular

2 -4 Writing Linear Equations • The form y = mx + b is called slopeintercept form, where m is the slope and b is the y-intercept. • The point-slope form of the equation of a line is y – y 1 = m(x – x 1). Here (x 1, y 1) are the coordinates of any point found on that line.

2 -4 Writing Linear Equations • Example: Find the slope-intercept form of the equation passing through the point (-3, 5) with a slope of 2. y = mx + b 5 = (2)(-3) + b 5 = -6 + b b = 11 y = 2 x + 11

2 -4 Writing Linear Equations • Example: Find the point-slope form of the equation of a line that passes through the points (1, -5) and (0, 4). m = y 2 – y 1 x 2 – x 1 m=4+5 0– 1 m=9 -1 m = -1 y – y 1 = m(x – x 1) y – (-5) = (-1)(x – 1) y + 5 = -x + 1 y = -x – 4

2 -4 Practice 1. Find the slope-intercept form of the following: a. a line passing through the point (0, 5) with a slope of -7 b. a line passing through the points (-2, 4) and (3, 14) 2. Find the point-slope form of the following: a. a line passing through the point (-2, 6) with a slope of 3 b. a line passing through the points (0, -9) and (-2, 1) Answers: 1)a) y = -7 x + 5 b) y = 2 x + 8 2)a) y = 3 x + 12 b) y = -5 x -9

2 -5 Modeling Real-World Data Using Scatter Plots • Plotting points that do not form a straight line forms a scatter plot. • The line that best represents the points is the best-fit line. • A prediction equation uses points on the scatter plot to approximate through calculation the equation of the best-fit line.

2 -5 Practice Plot the following data. Approximate the best-fit line by creating a prediction equation. Person ACT Score 1 15 2 19 3 21 4 28 5 30 6 35 Answers: 1) y = 4 x + 11 1.

2 -6 Special Functions • Whenever a linear function has the form y = mx + b and b = 0 and m ≠ 0, it is called a direction variation. • A constant function is a linear function in the form y = mx + b where m = 0. • An identity function is a linear function in the form y = mx + b where m = 1 and b = 0.

2 -6 Special Functions • Step functions are functions depicted in graphs with open circles which mean that the particular point is not included. – Example:

2 -6 Special Functions • A type of step function is the greatest integer function which is symbolized as [x] and means “the greatest integer not greater than x. ” – Examples: [8. 2] = 8 [5. 0] = 5 [3. 9] = 3 [7. 6] = 7 • An absolute value function is the graph of the function that represents an absolute value. – Examples: |-4| = 4 |-9| = 9

2 -6 Practice 1. Identify each of the following as constant, identity, direct variation, absolute value, or greatest integer function a. b. c. d. h(x) = [x – 6] f(x) = -½ x g(x) = |2 x| h(x) = 7 e. f. g. h. f(x) = 3|-x + 1| g(x) = x h(x) = [2 + 5 x] f(x) = 9 x 2. Graph the equation y = |x – 6|

2 -6 Answers • Answers: 1)a) greatest integer function b) direct variation c) absolute value d) constant e) absolute value f) identity g) greatest integer function h) direct variation 2)

2 -7 Linear Inequalities • Example: Graph 2 y – 8 x ≥ 4 – Graph the “equals” part of the equation. 2 y – 8 x = 4 2 y = 8 x + 4 y = 4 x + 2 x-intercept 0 = 4 x + 2 -2 = 4 x -1/2 = x y-intercept y = 4(0) +2 y=2

2 -7 Linear Inequalities – Use “test points” to determine which side of the line should be shaded. (2 y – 8 x ≥ 4) (-2, 2) 2(2) – 8(-2) ≥ 4 4 – (-16) ≥ 4 20 ≥ 4 true (0, 0) 2(0) – 8(0) ≥ 4 0– 0≥ 4 0 ≥ 4 false – So we shade the side of the line that includes the “true” point, (-2, 2)

2 -7 Linear Inequalities • Example: Graph 12 < -3 y – 9 x – Graph the line. 12 ≠ -3 y – 9 x 3 y ≠ -9 x – 12 y ≠ -3 x – 4 x-intercept 0 = -3 x – 4 4 = -3 x -4/3 = x y-intercept y = 3(0) – 4 y = -4

2 -7 Linear Inequalities – Use “test points” to determine which side of the line should be shaded. (12 < -3 y – 9 x) (-3, -3) 12 < -3(-3) – 9(-3) 12 < 9 + 27 12 < 36 true (0, 0) 12 < -3(0) – 9(0) 12 < 0 – 0 12 < 0 false – So we shade the side of the line that includes the “true” point, (-3, -3)