Learning Objectives for Section Graphs and Lines. Cartesian coordinate system. Graphs

Transcription

1 Learning Objectives for Section Graphs and Lines After this lecture and the assigned homework, ou should be able to calculate the slope of a line. identif and work with the Cartesian coordinate sstem. graph lines using the slope-intercept method. graph lines using the graphing calculator. graph lines using the intercepts. graph special forms of equations of lines. write equations of lines given two points. solve applications of linear equations. 1 Cartesian coordinate sstem 2 Graphs 3

2 Eample Eplain the error in this graph 4 Reading Graphs 5 Reading Graphs 6

3 Reading Graphs 7 Linear Equations in Two Variables A linear equation in two variables is an equation that can be written in the standard form A + B = C, where A, B, and C are constants (A and B not both 0), and and are variables. Linear equations graph lines. Anthing not straight is called a curve. A solution of an equation in two variables is an ordered pair of real numbers that satisf the equation. For eample, (4,3) is a solution of 3-2 = 6. (plug into eq.) The solution set of an equation in two variables is the set of all solutions of the equation. The graph of an equation is the graph of its solution set. 8 Rate of Change The word slope (or rate of change) [gradient, incline, pitch] is used to describe the measurement of the steepness of a straight line. The higher the slope, the steeper the line. 9

4 Linear Equations How can ou tell a linear equation from other equations? 1. A linear equation can be written in the forms: GENERAL FORM: A+ B = C where A, B, and C are constants and and are variables. (numbers) 2. A linear equation graphs a straight line. 10 SLOPE-INTERCEPT FORM: = m+ b where m is the slope and b is the -intercept. POINT-SLOPE FORM: 1 = m( 1) where m is the slope and ( 1, 1 ) is a point. Slope of a Line Slope of a line: ( ) 1, 1 o 2 1 m = = 2 1 rise run rise run o (, ) 2 2 Note: The slope of a line is the SAME everwhere on the line!!! You ma use an two points on the line to find the slope. 11 Calculating Slope Eample 1: Calculate the slope between the given points. a) (1, 2) and (3, 2) 12

5 Calculating Slope Eample 1: Calculate the slope between the given points. (3, 7) and (1, 7)? 13 Slope Now consider the vertical line = 4: Is the vertical line going up on one end? Well, kind of. Is the vertical line going down on the other end? Well, kind of. Is there an number that is both positive and negative? Nope 14 Slope Verdict: vertical lines have NO SLOPE. In particular, the concept of slope simpl does not work for vertical lines. The slope doesn't eist! Let's do the calculations. I'll use the points (4, 5) and (4, 3); the slope is: 15

6 Calculating Slope Eample 1: Calculate the slope between the given points. c) (1, 2) and (5, 5) Slope = 3/4 16 Calculating Slope Eample 1: Calculate the slope between the given points. c) (4, 9) and (4, 3) 17 Determining Slope Eample 2: Estimate the slope of the line graphed below. Slope = -2 (2,0) and (1, 2) m = 0-2/2-1 = -2 18

7 Intercepts of a Line -intercept: where the graph crosses the -ais. The coordinates are (a, 0). To find the -intercept, i t t let = 0 and solve for. -intercept: where the graph crosses the -ais. The coordinates are (0, b). To find the -intercept, let = 0 and solve for. 19 Graphing Linear Equations To graph linear equations, ou ma use A table of values (aka t-chart,, chart) The - and - intercepts The graphing calculator The slope-intercept method 20 Graph equations (t-chart) First, ou draw what is called a "T-chart": it's a chart that looks a bit like the letter "T": The left column will contain the -values that ou will pick, and the right column will contain the corresponding -values that ou will compute.

8 Graphing linear equations Label the columns: The first column will be where ou choose our input () values; the second column is where ou find the resulting output () values. Together, these make a point, (, ). Graphing linear equations Pick some values for. It's best to pick at least three value, to verif (when ou're graphing) that ou're getting a straight line. Which -values ou pick is totall up to ou! And it's perfectl oka if ou pick values that are different from the book's choices, or different from our stud partner's choices, or different from m choices. Some values ma be more useful than others, but the choice is entirel up to ou. Then our -values will come from evaluating the equation at the -values ou've chosen. And the T-chart keeps the information all nice and neat. Graphing linear equations I'll pick the following -values:

9 Graphing linear equations You can pick whatever values ou like, but it's often best to "space them out" a bit. For instance, picking = 1, 2, 3 might not give ou as good a picture of our line as picking = 3, 0, 3. That's not a rule, but it's often a helpful method. Graphing linear equations You can pick whatever values ou like, but it's often best to "space them out" a bit. For instance, picking = 1, 2, 3 might not give ou as good a picture of our line as picking = 3, 0, 3. That's not a rule, but it's often a helpful method. Graphing linear equations Some people like to add a third column to their T-chart to give room for a clear listing of the points that the've found:

10 Graphing linear equations - eamples Graph the following 1. = = -/3-1 Graphing linear equations - eamples Graph the following 1. = = Slope-Intercept Form The equation = m + b is called the slope-intercept form of an equation of a line. m represents the slope b represents the -coordinate of the -intercept 30

11 Y-intercept of a Line As mentioned before -intercept: where the graph crosses the -ais. The coordinates are (0, b). For instance, if b = 6, the line has a -intercept at (0, 6). Eample 3: Give the coordinates of the -intercept of the graphs of the following equations: a) = 2 4 b) = intercept = -4 -intercept = 9 31 Find the Slope and Intercept from the Equation of a Line Eample 4: Find the slope and - intercept of the line whose equation is 3 4 = = : added -3 to both sides = /-4 : Divided both sides b 4 = 3/4-3 Slope = ¾ and -intercept = Using Slope-Intercept to Graph a Line Eample 5: Now graph the equation 3 4 = 12 using the slope-intercept method. 1. Write equation in slope-intercept form. = 2/ Plot - intercept. 3. Plot other points b counting the slope from the -intercept. 4. Draw the line Label the line. and intercepts

12 Using Slope-Intercept to Graph a Line Eample 6: Graph the equation = 18 using the slopeintercept method. = -2/3 + 9 (9,0), (0,6) 34 Using Intercepts to Graph a Line Eample 7: Graph 2-6 = 12 b first finding the intercepts. 1. Compute the - and - intercepts. 2. Plot the intercepts. 3. Compute a 3 rd point as a check. 4. Draw the line. 5. Label the line. 35 Using Intercepts to Graph a Line Eample 7: Graph 2-6 = 12 b finding the intercepts

13 Using a Graphing Calculator Eample 8: Graph 2-6 = 12 on a graphing calculator and find the intercepts. 1. Get the equation in = m + b form. (Solve for.) 2. Tpe the equation into the calculator. Hit (=) and enter in the right side of the equation. 37 Using a Graphing Calculator (continued) Eample 8 continued: Graph 2-6 = 12 on a graphing calculator and find the intercepts. 3. Hit (graph). Adjust the window settings, if necessar. The standard window settings work well for this eample, but other problems will require adjusting the window. 38 Using a Graphing Calculator (continued) Eample 8 continued: Graph 2-6 = 12 on a graphing calculator and find the intercepts. 4. Find the intercepts. To find the -intercept, Hit then (CALC menu). Choose 2: Zero. Then answer the following questions that appear b using the left and right arrows: Left bound? pick a point to the left of the - intercept, then hit Right Bound? pick a point to the right of the - intercept, then hit Guess? Place the cursor on our guess of the - intercept, then hit 39

14 Using a Graphing Calculator Eample Eample 9: Given = find the intercepts to one decimal place 1) algebraicall and 2)using the graphing calculator. 1) Algebraicall 40 Using a Graphing Calculator Eample (continued) Eample 9: Given = find the intercepts to one decimal place 2)using the graphing calculator. 1. Solve for, if not alread done so. 2. Hit (=) then tpe in the equation. 41 Using a Graphing Calculator Eample (continued) Eample 9: Given = find the intercepts to one decimal place 2)using the graphing calculator. 3. Hit (graph) and decide whether or not to adjust the viewing window. Looks like we ll need to adjust that window! 42

15 Using a Graphing Calculator Eample (continued) Eample 9: Given = find the intercepts to one decimal place 2)using the graphing calculator. 4. Hit (windows ke) and choose appropriate values for Xmin, Xma, Ymin, Yma and the scale. 43 Standard window These settings do allow us to view the intercepts. Using a Graphing Calculator Eample (continued) Eample 9: Given = find the intercepts to one decimal place 2)using the graphing calculator. 5. For the -intercept: 44 Eample Eample 10: Graph a) = -4 b) = 2 45

16 Point-Slope Form The point-slope form of the equation of a line is = m( ) 1 1 where m is the slope and ( 1, 1 ) is a given point. It is derived from the definition of the slope of a line: = m Cross-multipl and substitute the more general for 2 46 Eample Eample 11: Write the equation of a line that passes through the point (-4, 3) with a slope of 1/2. Give our answer in slopeintercept form. 47 Eample Eample 12: Find the equation of the line through the points (-5, 7) and (4, 16). The equation of the line that passes through the points (-5,7) and (4,16) is =

17 Interpreting Slope Lines that increase from left to right have a slope. Lines that decrease from left to right have a slope. 49 Interpreting Slope Lines that are horizontal have a slope of. Lines that are vertical have an slope. 50 Application: Cost analsis A small business mfg s picnic tables. The weekl fied cost is $1,200 and the variable cost is $45 per table. Find the total dail cost of producing picnic tables. How man picnic tables can be produced for a total weekl cost of $4,800? Step1: Let C be the total dail cost of producing picnic tables. Step2: C = $1, Step3: For TC = $4,800 Step4: $1, = $4,800, ten solve for Step4: = $4,800 - $1,200 / 45 = 80 51