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Chapter 3 – Graphs of Linear Equations and Equalities; Functions Page 1 of 18 Section 1 – Reading Graphs; Linear Equations in two variables Notes: We sometimes abbreviate the condition x = 2 and y = 5 as (x, y) = (2, 5). Normally the variables are listed in alphabetical order. As we wrote it, it is clear which variable is which. If not explicitly indicated, the “ordered pair” is assumed to be in alphabetical order. Homework: 9-14, 21-23, 48-50, 55-59, 69-71 Additional Questions 1) Which quadrant is each of the following ordered pairs in? a) (1, 1) b) (1, -1) c) (-1, 1) d) (-1, -1) 2) For each of the following equations, is the ordered pair (2, 3) a solution? a) x + y = 5 b) 3x = 2y c) 2x = 3y 3) Plot the point below on the graph to the right. a) (4, 3) b) (3, -4) c) (-2, -2) d) (-3, 4) e) (2, 0) f) (0, 2) Complete each table of values, plot the ordered pairs and connect with a straight line. 4) 2x + 3y = 5 x y 1 1 4 3 Chapter 3 – Graphs of Linear Equations and Equalities; Functions Page 2 of 18 Section 2 – Graphing Linear Equations in two variables Note: The book notes than any equation of the form Ax + By = 0 goes through the origin (0, 0). You need another point to determine the line the equation satisfies. The points (B, -A) and (-B, A) also lie on the solution line. Homework: 4-6, 9-11, 19-21, 36 Additional Questions 1) For the equation y = 2x – 4 complete the ordered pairs below and plot them on the graph to the right and draw the line through them. a) (0, ___) b) (___, 0) 2) For the equation 3x + 2y = 12 a) What is the y-intercept? b) What is the x-intercept? 3) Graph the equation 2y = 3x on the right. Chapter 3 – Graphs of Linear Equations and Equalities; Functions Page 3 of 18 Section 3 – Slope of a Line Homework: 2-5, 13, 24-26, 33, 34, 36, 38, 46-50 Notes: We can use the two pictures below to help see slope. On the picture to the left is an 𝑎 𝑐 example of a positive slope of 𝑏 and the picture on the right has negative slope of − 𝑑. Later, you 𝑎 will learn that for the picture on the left, the lower left hand angle has tangent of 𝑏 so slope is just a measure of the angle the line forms (with the x axis). Similarly for the triangle on the right, 𝑐 the tangent of the lower right angle is 𝑑. However, the tangent of the angle the line makes with 𝑐 the x-axis is − 𝑑. Notes: For perpendicular lines, if one line is defined by the equation ax + by = c, all of its perpendicular lines look like bx – ay = d. where a, b, c, and d, are constants. The numerical coefficients are interchanged and one of the two numerical coefficients signs switches from positive to negative or vice versa. Since multiplying an equation by a constant does not really change the equation the perpendicular line may look like bnx – any = d where n is any non zero constant. For parallel lines to ax + by = c, all of its parallel lines look like anx + bny = d. Additional Questions Find the slope of the straight lines going through the two points 1) (4, 1) and (5, 2) 2) (5, 2) and (4, 1) 3) (1, -5) and (-4, 3) 4) (-3, -2) and (2, 5) Find the slope of the line defined by the equation 5) 4x + 5y = 7 6) 6x - 5y = 2 7) y = 4x + 5 Chapter 3 – Graphs of Linear Equations and Equalities; Functions Page 4 of 18 Section 4 – Equations of Lines Note: Below is how I prefer to see the standard forms of lines: A. Point-slope: Given that a straight line passes through the point (𝑥1 , 𝑦1 ) with the slope m, the equation for that line is: y - y1 = m(x - x1). B. Two points: Given that a straight line passes through the two points (𝑥1 , 𝑦1 ) and (𝑥2, 𝑦2 ) the equation for that line is: C. Slope-intercept: Given that a straight line has slope m, and when x = 0, y = a, the equation for that line is: y = mx + a. D. Parallel line - point: Given that a straight line passes through the point (𝑥1 , 𝑦1 ) and is parallel to the line ax + by = c, the new equation is 𝑎𝑥 + 𝑏𝑦 = 𝑎𝑥1 + 𝑏𝑦1. E. Perpendicular line - point: Given that a straight line passes through the point (𝑥1 , 𝑦1 ) and is perpendicular to the line ax + by = c, the new equation is 𝑏𝑥 − 𝑎𝑦 = 𝑏𝑥1 − 𝑎𝑦1 . NOTE: The “-” means to flip the sign between the x and y term. Homework: 3-6, 9-11, 13, 15-17, 22-24, 32-33, 37-40, 47-48 Additional Questions 2) Given: Slope = 2, y-intercept = (0,-3), write the equation of the line. 3) Change the equation 2x – 3y = 7 to slope intercept form and graph the equation on the right. 4) Given: Slope = 2, point on line = (1,-1), write the equation of the line. Chapter 3 – Graphs of Linear Equations and Equalities; Functions Page 5 of 18 5) Given: Line passing through points (0, 4) and (-1, 3), find the equation of the line: 6) For the line 4x + 5y = 9: a) Find the equation of the line parallel to the equation going through the point (3, -6) b) Find the equation of the line perpendicular to the equation going through the point (3, -6) Chapter 3 – Graphs of Linear Equations and Equalities; Functions Page 6 of 18 Section 5 – Graphing Linear Inequalities in two variables Homework: 1, 5-7, 15-16, 19 Additional Questions 1) Graph 3x + 2y > 5. 2) Graph y > 2x – 2. 3) Graph 4x + 3y < 1. Chapter 3 – Graphs of Linear Equations and Equalities; Functions Page 7 of 18 4) Graph 3y – 2x < 4. 5) Graph x + 2y 7. 6) Graph x ≤ 3y. Chapter 3 – Graphs of Linear Equations and Equalities; Functions Page 8 of 18 Section 6 – Introduction to Functions Homework: 1, 3, 5-7, 17-19, 23-25, 44-45, 51-52, 63-65 Additional Questions For each of the following relations: a) Is the relation a single valued function? b) What is the range? c) What is the domain? 1) 𝑦 = 𝑥 2 + 1 a) b) c) 2) 𝑥 = 𝑦 2 + 1 a) b) c) 3) 2𝑥 − 5𝑦 = 11 a) b) c) For 𝑓(𝑥) = 3𝑥 − 2 and 𝑔(𝑥) = 2 − 3𝑥 4) Find 𝑓(2) 5) Find 𝑔(3) 6) Find 𝑓(𝑔(𝑥)) 7) Find 𝑔(𝑓(𝑥)) Chapter 3 – Graphs of Linear Equations and Equalities; Functions Page 9 of 18 Vocabulary 1) 2) 3) 4) 5) 6) 7) 8) 9) 10) 11) 12) 13) 14) 15) 16) 17) 18) 19) 20) 21) 22) 23) 24) 25) 26) 27) 28) 29) 30) 31) 32) 33) 34) 35) 36) 37) 38) 39) 40) axis boundary coincide component coordinate dependent diagram domain function graphing graphs horizontal independent inequalities intercept interpret notation origin parallel perpendicular plane plotting points quadrants range rectangular region relation rise run scatter shade slant slope solution subscript system table (of values) test vertical Chapter 3 – Graphs of Linear Equations and Equalities; FunctionsPage 10 of 18 Section 1 – Reading Graphs; Linear Equations in two variables Additional Answers 1) Which quadrant is each of the following ordered pairs in? a) (1, 1) I b) (1, -1) IV c) (-1, 1) II d) (-1, -1) III 2) For each of the following equations, is the ordered pair (2, 3) a solution? a) x + y = 5 2+3=5 Yes b) 3x = 2y 3(2) = 2(3) 6=6 Yes c) 2x = 3y Vocabulary axis coordinate diagram graphs interpret plane points quadrants rectangular scatter solution system table (of values) Chapter 3 – Graphs of Linear Equations and Equalities; FunctionsPage 11 of 18 Section 2 – Graphing Linear Equations in two variables Additional Answers 2) For the equation 3x + 2y = 12 c) What is the y-intercept? d) What is the x-intercept? Vocabulary coincide graphing horizontal intercept origin plotting slope vertical (0, 6) (4, 0) Chapter 3 – Graphs of Linear Equations and Equalities; FunctionsPage 12 of 18 Section 3 – Slope of a Line Additional Answers Find the slope of the straight lines going through the two points 1) (4, 1) and (5, 2) 2−1 1 Slope = 5−4 = 1 = 𝟏 2) (5, 2) and (4, 1) 1−2 −1 Slope = 4−5 = −1 = 𝟏 3) (1, -5) and (-4, 3) 3−(−5) 𝟖 𝟖 Slope = −4−1 = −𝟓 = − 𝟓 4) (-3, -2) and (2, 5) 5−(−2) 𝟕 Slope = 2−(−3) = 𝟓 𝟏 Find the slope of the line defined by the equation 5) 4x + 5y = 7 4 7 𝟒 𝑦 = − 5 𝑥 + 5 or slope = − 𝟓 6) 6x - 5y = 2 6 2 𝟔 𝑦 = 5 𝑥 − 5 or slope = 𝟓 7) y = 4x + 5 slope = 4 Vocabulary rise run slant slope subscript Chapter 3 – Graphs of Linear Equations and Equalities; FunctionsPage 13 of 18 Section 4 – Equations of Lines Additional Answers 2) Given: Slope = 2, y-intercept = (0,-3), write the equation of the line. y = 2x – 3. 4) Given: Slope = 2, point on line = (1,-1), write the equation of the line. y + 1 = 2(x – 1) or y = 2x – 3 5) Given: Line passing through points (0, 4) and (-1, 3), find the equation of the line: (y – 4)(-1 – 0) = (3 – 4)(x – 0) -y + 4 = -x or y=x+4 6) For the line 4x + 5y = 9: a) Find the equation of the line parallel to the equation going through the point (3, -6) 4x + 5y = 4(3) + 5(-6) = 12 – 30 or 4x + 5y = – 18 b) Find the equation of the line perpendicular to the equation going through the point (3, -6) 5x – 4y = 5(3) – 4(-6) = 15 + 24 or 5x – 4y = 39 Vocabulary parallel perpendicular Chapter 3 – Graphs of Linear Equations and Equalities; FunctionsPage 14 of 18 Section 5 – Graphing Linear Inequalities in two variables Additional Answers Chapter 3 – Graphs of Linear Equations and Equalities; FunctionsPage 15 of 18 Vocabulary boundary inequalities region shade test Chapter 3 – Graphs of Linear Equations and Equalities; FunctionsPage 16 of 18 Section 6 – Introduction to Functions Additional Answers For each of the following relations: a) Is the relation a single valued function? b) What is the range? c) What is the domain? 1) 𝑦 = 𝑥 2 + 1 a) Yes b) [1, ∞) c) (-∞, ∞) 2) 𝑥 = 𝑦 2 + 1 a) No b) (-∞, ∞) c) [1, ∞) 3) 2𝑥 − 5𝑦 = 11 a) Yes b) (-∞, ∞) c) (-∞, ∞) For 𝑓(𝑥) = 3𝑥 − 2 and 𝑔(𝑥) = 2 − 3𝑥 4) Find 𝑓(2) = 3(2) − 2 = 6 − 2 = 𝟒 5) Find 𝑔(3) = 2 − 3(3) = 2 − 9 = −𝟕 6) Find 𝑓(𝑔(𝑥)) = 3(2 − 3𝑥) − 2 = 6 − 9𝑥 − 2 = 𝟒 − 𝟗𝒙 7) Find 𝑔(𝑓(𝑥)) = 2 − 3(3𝑥 − 2) = 2 − 9𝑥 + 6 = 𝟖 − 𝟗𝒙 Vocabulary component dependant domain function independent notation range relation Chapter 3 – Graphs of Linear Equations and Equalities; FunctionsPage 17 of 18 Glossary Axis: is a straight line where you measure one of the values from (on). x-axis: is normally measures the x values from left (negative) to right (positive) y-axis: is normally measures the y values from back (negative) to front (positive) z-axis: is normally measures the z values from bottom (negative) to top (positive) Boundary: of a figure is normally what separates the inside of a figure from the outside of the figure. For an inequality, you might also consider it the line that separates where the inequality is true from where the inequality is false. Coincide: means at the same place. Examples: Two straight lines coincide (or meet or intersect) at a point. Two straight lines that overlap each other are really the same line. Component: Technically means part of the whole. In an ordered pair (or triplet, etc.) it is one of the numbers that make up the ordered pair (or triplet, etc.). Coordinate: When plotting a point it is one of the values used to place the point. Warning: For a two dimensional graph, the x-coordinate is sometimes called the ordinate and the y-ordinate is referred to as the coordinate. Dependent (variable): A function of variables is called a dependent variable. (Scatter) Diagram: Shows points on a Cartesian graph where data is been observed to occur. Domain: is where a function is defined. May also be defined as the set of values where a function is defined. Horizontal: lines extend to your left and right. Independent (variables): are used to define a function. Inequalities: are used to limit something is true. Intercept: is where a graph crosses a given axis (see axis above). Notation: is a way of representing something. Origin: of a Cartesian coordinate system is where all the coordinates are zero. Parallel: straight lines are two lines that are at a constant distance from each other. (Pick a point on one line and find the point closest to it on the other line. The distance between those two points is always the same). Perpendicular: lines meet at right angles to each other. Plane: is a flat two-dimensional surface. Plotting a point is determining where it goes on a graph. Point: is a single location on a graph. Quadrants: In the Cartesian plane, the x-axis and y-axis divide the plane into four quadrants. Quadrant I: is the set of points where both the x-coordinate and y-coordinate are positive Quadrant II: is the set of points where the x-coordinate is negative and the y-coordinate is positive Quadrant III: is the set of points where both the x-coordinate and y-coordinate are negative Quadrant IV: is the set of points where the x-coordinate is positive and the y-coordinate is negative Range: of a function is the set of values the function actually take on. Rectangular coordinate system: Gives the location of points in the system by telling how to get to that point by measuring several perpendicular distances from the origin. Relation: Describes the constraints put upon several variables. Rise: (see slope) Run: (see slope) Scatter: (See diagram) Shade: is to color a figure to show where something is true. Chapter 3 – Graphs of Linear Equations and Equalities; FunctionsPage 18 of 18 Slant: (see slope) Slope: For a two dimensional graph is a measure of the slant of a line (or curve) at a particular point on the curve. Basically it is the tangent of the angle that a straight line with the same slant makes with the x-axis. This is sometimes referred to the rise over run (how much increase or decrease the y-coordinate changes per change in the x-coordinate). Solution: is one place where a condition or equation is true. Table of Values: lists points on a graph. The table is used to help draw what the graph looks like. Test: Seeing if a particular condition is true or not. Vertical: lines extend up and down.