Learning Objectives for Section 1.2 Graphs and Lines. Linear Equations in Two Variables. Linear Equations

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1 Learning Objectives for Section 1.2 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 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. 2 Linear Equations How can ou tell a linear equation from other equations? 1. A linear equation can be written in the forms: STANDARD FORM: A+ B = C where A, B, and C are constants and and are variables. (numbers) 2. A linear equation graphs a straight line. 3 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.

2 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. 4 Calculating Slope Eample 1: Calculate the slope between the given points. a) (1, 2) and (3, 2) 5 Calculating Slope Eample 2: Calculate the slope between the given points. (3, 7) and (1, 7)? 6

3 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: 7 Calculating Slope Eample 3: Calculate the slope between the given points. c) (1, 2) and (5, 5) Slope = 3/4 8 Calculating Slope Eample 4: Calculate the slope between the given points. c) (4, 9) and (4, 3) 9

4 Determining Slope Eample 5: Estimate the slope of the line graphed below. Slope = -2 (2,0) and (1, 2) m = 0-2/2-1 = 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. 11 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 (Personall, I usuall find this method most useful) 12

5 Slope-Intercept Form The equation = m+b is called the slope-intercept form of an equation of a line. intercept m represents the slope b represents the -coordinate of the - 13 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 14 Find the Slope and Intercept from the Equation of a Line Eample 6: 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 =

6 Using Slope-Intercept to Graph a Line Eample 7: Now graph the equation 3 4 = 12 using the slope-intercept method. 1. Write equation in slope-intercept form. = 3/ Plot - intercept. 3. Plot other points b counting the slope from the -intercept. 4. Draw the line Label the line. and intercepts Using Slope-Intercept to Graph a Line Eample 8: Graph the equation = 18 using the slopeintercept method. = -2/3 + 9 (9,0), (0,6) 17 Using Intercepts to Graph a Line Eample 9: 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. 18

7 Using Intercepts to Graph a Line Eample 10: Graph 2 6 = 12 b finding the intercepts Using a Graphing Calculator Eample 11: 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. 20 Using a Graphing Calculator (continued) Eample 11 continued: Graph 2-6 = 12 on a graphing calculator and find the intercepts. 3. Hit. Adjust the window settings, if necessar. The standard window settings work well for this eample, but other problems will require adjusting the window. 21

8 Using a Graphing Calculator (continued) Eample 11 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 22 Using a Graphing Calculator Eample Eample 12: Given = find the intercepts to one decimal place 1) algebraicall and 2) using the graphing calculator. 1) Algebraicall 23 Using a Graphing Calculator Eample (continued) Eample 13: 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. 24

9 Using a Graphing Calculator Eample (continued) Eample 13: Given = find the intercepts to one decimal place 2)using the graphing calculator. 3. Hit and decide whether or not to adjust the viewing window. Looks like we ll need to adjust that window! 25 Using a Graphing Calculator Eample (continued) Eample 13: Given = find the intercepts to one decimal place 2)using the graphing calculator. 4. Hit and choose appropriate values for Xmin, Xma, Ymin, Yma and the scale. 26 Standard window These settings do allow us to view the intercepts. Using a Graphing Calculator Eample (continued) Eample 13: Given = find the intercepts to one decimal place 2)using the graphing calculator. 5. For the -intercept: 27

10 Eample Eample 14: Graph a) = -4 b) = 2 28 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 29 Eample Eample 15: 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. 30

11 Eample Eample 16: Find the equation of the line through the points (-5, 7) and (4, 16). Step 1 Find slope with point slope form: m = 1 Step 2 Use points (4, 16) with point intercept form: = m + b 16 = b b = 16 (1)(4) so b = 12 Step 3: = Interpreting Slope Lines that increase from left to right have a slope. Lines that decrease from left to right have a slope. 32 Interpreting Slope Lines that are horizontal have a slope of. Lines that are vertical have an slope. 33

12 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 C = $4,800 Step4: $1, = $4,800, then solve for Step5: = $4,800 - $1,200 / 45 = Eample: Given the points (4,3) and (4,1) a) Find the slope of the line connecting the points. b) Write the equation of the line connecting the points. Given the equation, 2 = 5, find a) The -intercept. Write our answer as an ordered pair. b) The -intercept. Write our answer as an ordered pair. Write the equation of the line, in slope intercept form, that passes through the point (2, -4) with a slope of 1/3. 35 Scenario So, assume we have a product we want to sell for $10.00 and we want to sell 1,000 of them. For this eample our total fied costs are going to be $7,700 and our total variable costs are $4.50/unit. Our formula would look like this: P=1,000 ($ $4.50) - $7, = $5, $7, = -$2, What happened? Instead of making mone we have just lost $2,200. At break even the $2,200 number should be $0. We can't make mone at 1000 units so how man must we reall sell to break even? 36 BE = 1,400

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