Any questions on the Section 3.2 homework?Section 3.1 Homework Issues Graphing non-linear equations:PowerPoint PresentationSlide 4Slide 5Section 3.3 Graphing Linear EquationsSlide 7Slide 8Slide 9Slide 10Slide 11Slide 12Slide 13Slide 14Slide 15Slide 16Slide 17Slide 18Slide 19Slide 20Slide 21Slide 22Example from today’s homework:Slide 24Slide 25Slide 26Slide 27Slide 28Reminder:Slide 30Any questions on the Section 3.2 homework?Section 3.1 Homework IssuesGraphing non-linear equations:•Graphs of absolute value equations•Graphs of quadratic equationsCompare to y = |x| - 4Compare to y = x3 + 4Now pleaseCLOSE YOUR LAPTOPSand turn off and put away your cell phones.Sample Problems Page Link(Dr. Bruce Johnston)Section 3.3Graphing Linear EquationsA linear function is a function that can be written in the form f(x) = mx + b.Graph the following functions on the same pair of axes.Examplexy• f(x) = -2x• g(x) = -2x + 3• h(x) = -2x - 5Where do each of these graphs intersect the y-axis?• At (0,0), (0,3) and (0,-5) respectively.• These are referred to as the y-intercepts.• When a linear function is written in the form of f(x) = mx + b or y = mx + b, the y-intercept is (0, b).How would we find the y-intercept if the line is not written in the form f(x) = mx + b or y = mx + b?• Since all points on the y-axis have an x-coordinate of 0, substitute x = 0 into the linear equation and find the corresponding y-coordinate.Find the y-intercept of 4 = x – 3y.Let x = 0.Then 4 = x – 3y becomes 4 = 0 – 3y (replace x with 0) 4 = -3y (simplify the right side)Example = y (divide both sides by -3)34So the y-intercept is (0, ).34• Correspondingly, all points on the x-axis would have a y-coordinate of 0.• So to find an x-intercept of a linear function, we would substitute y = 0 into the equation and find the corresponding x-coordinate.Find the x-intercept of 4 = x – 3y.Let y = 0.Then 4 = x – 3y becomes 4 = x - 3(0) (replace y with 0) 4 = x (simplify the right side) So the x-intercept is (4,0).ExampleGraph the linear equation 4 = x – 3y by plotting intercepts.Example34We previously found that the y-intercept is (0, ) and the x-intercept is (4, 0).Plot both of these points and then draw the line through the 2 points.Note: You should still find a 3rd solution to check your computations.Example (cont.)Graph the linear equation 4 = x – 3y.Along with the intercepts, for the third solution, let y = 1.(You can use any value for x or y that you want to find a third point, so pick something that will make your calculation easy.) Then 4 = x – 3y becomes 4 = x – 3(1) (replace y with 1) 4 = x – 3 (simplify the right side)4 + 3 = x (add 3 to both sides) 7 = x (simplify the left side)So the third solution is (7, 1).xyAnd then we draw the line that contains the three points.Now we plot the two intercepts (0, ) and (4, 0) along with the third solution (7, 1).34(4, 0)(7, 1)(0, )34Graph 2x = y by plotting intercepts.To find the y-intercept, let x = 0. 2(0) = y 0 = y, so the y-intercept is (0,0).To find the x-intercept, let y = 0. 2x = 0 x = 0, so the x-intercept is (0,0).Oops! It’s the same point. What do we do?ExampleSince we need at least 2 points to graph a line, we will have to find at least one more point.Let x = 3 (Again, you can pick any value for x that you want.) 2(3) = y 6 = y, so another point is (3, 6).Let y = 4 2x = 4 x = 2, so another point is (2, 4).Example (cont.)Now we plot all three of the solutions (0, 0), (3, 6) and (2, 4).xy(3, 6)(0, 0)(2, 4)And then we draw the line that contains the three points.Graph y = 3.Note that this line can be written as y = 0•x + 3.The y-intercept is (0, 3), but there is no x-intercept! (Since an x-intercept would be found by letting y = 0, and 0 can’t equal 0•x + 3, there is no x-intercept.)Every value we substitute for x gives a y-coordinate of 3.The graph will be a horizontal line through the point (0,3) on the y-axis.Examplexy(0, 3)Example (cont.)Graph x = -3.This equation can be written x = 0•y – 3.When y = 0, x = -3, so the x-intercept is (-3,0), but there is no y-intercept.Any value we substitute for y gives an x-coordinate of –3.So the graph will be a vertical line through the point (-3,0) on the x-axis.Examplexy(-3, 0)Example (cont.)Example from today’s homework:Vertical lines•Appear in the form of x = c, where c is a real number.•x-intercept is at (c, 0), no y-intercept unless c = 0 (y-axis).Horizontal lines•Appear in the form of y = c, where c is a real number.•y-intercept is at (0, c), no x-intercept unless c = 0 (x-axis).Using the online graphing tool:When the problem asks you to graph x- and y-intercepts, you MUST graph these two points.For example, in this problem, you would have to graph (0, 1) and (-4, 0). If you used (0, 1) and (4, 2), you’d get the same line, but the computer would mark your answer wrong.Solving for y gives: y= 1/5 x + 3/5Choose x = 2 to get y = 2/5 + 3/5 = 5/5 = 1Choose x = 7 to get y = 7/5 + 3/5 = 10/5 = 2Or choose x = -3 to get y = -3/5 + 3/5 = 0/5 = 0Note: The homework for this section shouldn’t take too long, so you might want to look ahead at section 3.4 on slopes of lines and section 3.5 on slope-intercept equations of lines.Reminder:This homework assignment on Section 3.3 is due at the start of next class period.(This assignment is entirely online, with no worksheet. However, you should do the work for each problem in your notebook and keep these notes to study for quizzes and tests.)You may now OPEN your LAPTOPSand begin working on the homework