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Chapter 3 Analytical Algebra 2 Rectangular Cooordinate System Rectangular Coordinate system Axes Ordered pairs x-coordinate y-coordiante Graph an ordered pair Origin Quadrants Homework Page 138 3, 4, 5, 6 . Distance and Midpoint Find the distabnce between the points: (2, 4) and (-5, 4) (2, -1) and (2, 3) 1 Chapter 3 Analytical Algebra 2 (3, 4) and (0, 0) (2 , 1) and (-1, 5) The Distance Formula Line – Line segment – Length of a line segment – Midpoint – Find the midpoint of the line segment whose indpoints are (2, 4) and (5, 4) 2 Chapter 3 Analytical Algebra 2 Find the midpoint of the line segment with endpoints ( -5, 4) and ( -3, 2) The Midpoint Formula – Find the midpoint of the line with endpoints (-3, -5) and (-2, 3) Homework Page 140 24, 27, 30 Graphs of Linear Equations Solve y + 1 = 2x How many solutions? What do they look like? How can we represent them? 3 Chapter 3 Analytical Algebra 2 Graph the equation y = -3x + 2 NOTE: When we graph an equation we are Why is y = -3x + 2 called a linear equation? Graph 4x – y = 4 Graph 6 + 2y = -2 Homework Page 174 5, 6, 8, 15, 16, 17 4 Chapter 3 Analytical Algebra 2 5 Graph the equation y = 4 Graph the equation x = -2 Homework Page 175 18 - 22 Intercepts – x-intercept – ( , ) y-intercept – ( , ) Find the x and y intercepts, if they exist, for: To find intercepts: y = 2x + 1 x-intercept yintercept 1. 1. 2. 2. 3. 3. Chapter 3 Analytical Algebra 2 Find, if they exist, the x-intercept and y-intercept for 3x + 4y = -12. Homework Page 175 26, 30, 33, 36 Slope Slope of a line – Slope Formula – Remember our fraction rules. Find the slope of line which passese through the points: (2, 4) and (4, 5) (-1, 2) and (3, -4) 6 Chapter 3 Analytical Algebra 2 Find the slope of the line which passes through thepoints: (3, 2) and (-4, 2) (2, 3) and (2, -4) Every horizontal line has: An equation of: A slope of _______. Every vertical line: An equation of: ______ slope. A line which slants downward has __________________ slope. A line which slants upwards has ____________________ slope. Homework Page 186 3, 5, 7, 11, 15 Draw the graph of the line which passes through the point (-4, 2) and which has a slope of 3. 7 Chapter 3 Note : 3 = ----- Analytical Algebra 2 -4 = ---- 3 4 = Draw the graph of the line which passes through: the point (0, 4) and which has a slope of 2/3. the point (-3,0) and which has a slope of -3/4. Graph the equation y = 5/3x – 4 We can graph by finding three ____________ / solutions. Or We can graph by recognizing that... Graph the equation x + 2y = 4 3 Graph the equation y x 4 2 Homework Page 188 31, 33, 37, 38, 42, 44 8 Chapter 3 Analytical Algebra 2 9 More with Slopes Horizonatl lines have _________________ slope. Vertical lines have ____________ slope Upward slanting lines have __________________ slopes. Downward slating lines have _________________ slopes. “Steep” lines have ______________________ slopes, greater than _______( in absolute value ) Slow rising lines have _____________________ slopes, less than _______ ( in absolute value ) Sketch a line with the given slopes: 3 ½ -2/5 1 -1 -6 -7/2 9/4 A positive slopes means that as x increases, y ________________________, or as x decresases, y ___________________ A negative slope means that as x increase, y _________________________, or as x decreases, y ______________________. Solve for x: 4 x 5 2 2 x 1 3 5 Chapter 3 Analytical Algebra 2 A line with a slope of 2/5 passes through the points (4, 3) and (-1, y). Find y. A line with slope -3, passes though the points (x, 3) and (0, 6). Find x. a line passes through the points ( -4, 7) , (4 , 1) and ( x , 4). Find x. Homework Page 189 47 – 53 10 Chapter 3 Analytical Algebra 2 11 Writing Equations of Lines y = 2x + 3 2x – 3y = 9 x = 4y – 7 2x – 3 + 3y = 4x – 7y + 10 y – 3 = 2(x + 4) x2 – y = 9 There are a few different forms of a line that we can use, we will focus on tow of them. Slope-intercpt form Point-slope form Standard form Write an equation of the line: Passing through the point (0, 3) with a slope of 4. Passing through the point (2, 0) with a slope of -2/3 Chapter 3 Analytical Algebra 2 Passing through the point (-2, 4) with m = -4 Passing through the points (0, 4) and (3, 5). Passing through the points (-1, 1) and (4, 4) Homework Page 194 3, 5, 7, 11, 17, 29, 37, 38 Graph the line y = 4 Wirte the equation of the line which passes through the point (2, 4) and has slope = 0. 12 Chapter 3 Analytical Algebra 2 Write the equation of the line which passes through the point (-2 , 4) and has no slope. Write the equation of the line which passes through the points (-2, 3) and (4, 3). Write the equation of the line which passes through the points (-2, 3) and (-2, 3). All horizontal lines, lines with a slope of ________________, have equations: All vertical lines, lines with ____________ slope, have equations : Homework Page 195 19, 21, 25, 43, 47, 49 13 Chapter 3 Analytical Algebra 2 Parallel and Perpendicular Lines In geometry: Parallel lines Perpendicular lines In algebra: Parallel lines Perpenduclar lines Are the lines, one passing though the points (-2, 1) and (-5, -1) and the other passing through the points (1, 0) and (4, 2), parallel, perpendicular, neither or both? 14 Chapter 3 Analytical Algebra 2 Are the graphs of the lines, y= 3/2x – 5 and 2x + 3y = 6, parallel, perpendicular or netiher? Write an equation of a line perpendicular to the line y = 4x + 5, which passes through (0, 2). Write a n equation of the line which passes through the point (2, 3) and which is parallel to the line y = 1/2x – 4. Write the equation of the line which passes through the point (3, -4) and which is perpendicular to the line 2x – y = -3. 15 Chapter 3 Analytical Algebra 2 What slope does a line have, which is perpendicular to a line with slope of 0 have? Write the equation of the line which is parallel to the line x = 5 and which passes thorugh the point (3, 1). Write the equation of the line which is perpendicular, to the line x = 5 and which passes through the point (5, 2). Homework Page 203 3, 5, 11, 13, 19, 21, 23, 27, 29 16 Chapter 3 Analytical Algebra 2 How many solutions does y + 4 = 3 have? How many solutions does y = x – 1 have? How many solutions does y > x – 1 have? The graph of the solutions to y = x – 1 creates a _____________. The graph of the solutions to y > x – 1 creates a ______________. Graph y > x – 1 Graph x – 2y ≤ 4 17 Chapter 3 Graph y < -x + 2 Analytical Algebra 2 Graph x > -1 Graph y ≤ 3 Homework Page 209 11, 13, 14, 19, 20 18