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Chapter 2 Section 2.3

Prepared by Doron Shahar. Chapter 2 Section 2.3. Lines. Prepared by Doron Shahar. Warm-up: Class Notes Page 25. Today we are starting to learn about lines. In honor of that, here are several one-liners: I wondered why the Frisbee was getting bigger, and then it hit me.

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Chapter 2 Section 2.3

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  1. Prepared by DoronShahar Chapter 2 Section 2.3 Lines

  2. Prepared by DoronShahar Warm-up: Class Notes Page 25 Today we are starting to learn about lines. In honor of that, here are several one-liners: • I wondered why the Frisbee was getting bigger, and then it hit me. • I had a dream I was eating a giant marshmallow, and when I woke up my pillow was missing! • Right now I'm having amnesia and deja vu at the same time! I think I've forgotten this before? • How’s that new line I introduced you to? Ohh, she goes on forever.

  3. Prepared by DoronShahar Find points on a graph (−2,3) x coordinate first y coordinate second (2,1) (x , y)

  4. Prepared by DoronShahar Finding intercepts on a graph y-intercept where line touches y-axis (0,2) (4,0) x-intercept where line touches x-axis

  5. Prepared by DoronShahar Find slope from a graph Rise Run

  6. Prepared by DoronShahar Slope formula The slope of a line passing through two distinct points (x1, y1) and (x2, y2) is given by the formula____________ Exception: The slope is undefined if x1=x2. Extra 1: Find the slope of the line passing through the points (−2, −3) and (−4, 5). (x1, y1) and (x2, y2)

  7. Prepared by DoronShahar Slope-intercept form The slope-intercept form of the equation of a line with slope m and y-intercept (0, b) is given by the formula _____________ 2.3.4Find the slope-intercept form of the equation of the line with y-intercept (0, −8) and having slope m=2/5. Solution

  8. Prepared by DoronShahar Extra 2: Slope-intercept form Find the slope-intercept form of the equation of the line displayed in the graph below. y-intercept (2, 4) (0, 3) (x2, y2) (x1, y1) Solution

  9. Prepared by DoronShahar 2.3.4 Sketch a graph Sketch a graph of the line y=2/5x −8 . +( ) y-intercept: (0, −8) Slope: y-intercept Two points define a line. 2 5

  10. Prepared by DoronShahar 2.3.4 Identifying intercepts Find the y-intercept and thex-intercept of the liney=2/5x−8. To find the y-intercept, set x=0 and solve for y. To find the x-intercept, set y=0 and solve for x. Setting x equal to zero Setting y equal to zero Solution Solution x-intercept: y-intercept:

  11. Prepared by DoronShahar Point-slope form The point-slope form of the equation of a line passing through the point (x1, y1) and having slope m is given by the formula ______________________ • Extra 3: Find the point-slope form of the equation of the line • passingthrough the point (− 3,2) with slope m= −2. (x1, y1) Solution

  12. Prepared by DoronShahar Extra 4: Point-Slope form Find the point-slope form of the equation of the line passing through the points (−1, 3) and (1, 4). (x1, y1) and (x2, y2) First find the slope. Solution Solution

  13. Prepared by DoronShahar 2.3.2: Point-slope form Find the point-slope form of the equation of the line displayed in the graph below. First find the slope. (−4, 5) (x1, y1) (−2, −3) Solution (x2, y2)

  14. Prepared by DoronShahar Extra 3:Sketch a graph • Sketch a graph of the line y−2= −2(x+3). Point: Point (−3, 2) 1 Slope: 2 Two points define a line.

  15. Prepared by DoronShahar 2.3.2 Identifying Intercepts • Find the y-intercept and thex-intercept of the liney−5= −4(x+4). To find the y-intercept, set x=0 and solve for y. To find the x-intercept, set y=0 and solve for x. Setting x equal to zero Setting y equal to zero Solution Solution x-intercept: y-intercept:

  16. Prepared by DoronShahar Point-slope to Slope-intercept form • 2.3.2 Write the liney−5=−4(x+4)in slope-intercept form. Starting Equation Distribute right hand side Slope-intercept form

  17. Prepared by DoronShahar Standard form of a line The standard from of the equation of a line is given by the formula __________________ , where A and B are not both zero. • 2.3.1Write x/3 − 4y +1 =0in standard form such that • A, B, C are integers. Subtract 1 from both sides Multiply both sides by 3

  18. Prepared by DoronShahar 2.3.1 Identifying Intercepts • Find the y-intercept and thex-intercept of the linex−12y= −3. To find the y-intercept, set x=0 and solve for y. To find the x-intercept, set y=0 and solve for x. Setting x equal to zero Setting y equal to zero Solution Solution x-intercept: y-intercept:

  19. Prepared by DoronShahar 2.3.1 Sketch a graph • Sketch a graph of the line x−12y= −3. x-intercept: (−3, 0) y-intercept: (0, 1/4) x-intercept y-intercept Two points define a line.

  20. Prepared by DoronShahar Standard form Every line can be written in standard form. This is not true of the other two forms. In particular, verticals lines can only be written in standard form, because their slope is undefined. It is difficult to get the equation of a line directly into standard form given a graph or “usual” information. Instead, you will need to be able to get the equations of lines into and out of this form.

  21. Prepared by DoronShahar Point-slope to Standard form To get the equation of a line from point-slope form into standard form, first convert it into slope-intercept form as we have already learned. Then proceed as on the next slide to convert the equation into standard form.

  22. Prepared by DoronShahar Slope-intercept to Standard form 2.3.4 Write the liney=2/5x−8 in standard form such that A, B, and C are integers. Starting Equation Multiply both sides by 5 Distribute the 5 Alternative Standard form

  23. Prepared by DoronShahar Standard to Slope-intercept to form • 2.3.1 Write the linex−12y =−3 in slope-intercept form. Starting Equation Subtract x from both sides Divide by −12 Slope-intercept form

  24. Prepared by DoronShahar Standard to Slope-intercept form One reason for converting the equation of a line from Standard form to slope-intercept form is to find the slope of the line. It also helps if you are trying to graph the line. • 2.3.1 Find the slope of the x−12y =−3. Slope-intercept form Slope:

  25. Prepared by DoronShahar Turtle Sign language for turtle

  26. Prepared by DoronShahar Horizontal lines A horizontal line passing through the point (a, b) is given by the equation ________. The slope of a horizontal line is _________. • 2.3.5Write the equation of the horizontal line passing • through the point (−9/2, 15/2). Solution

  27. Prepared by DoronShahar Extra 5: Horizontal lines Find the equation of the horizontal line displayed below. (2, 3) (a, b) Solution

  28. Prepared by DoronShahar 2.3.5 Sketch a graph Sketch a graph of the line y=15/2. (−9/2, 15/2) Slope:

  29. Prepared by DoronShahar 2.3.5 Identifying intercepts Find the y-intercept and thex-intercept of the liney=15/2. To find the y-intercept, set x=0 and solve for y. To find the x-intercept, set y=0 and solve for x. Setting x equal to zero Setting y equal to zero Solution ¡PROBLEMA! There is no x-intercept! y-intercept:

  30. Prepared by DoronShahar Vertical lines A vertical line passing through the point (a, b) is given by the equation ________. The slope of a vertical line is _____________. • 2.3.3Write the equation of the vertical line • with x-intercept (1/3, 0). Solution

  31. Prepared by DoronShahar Extra 6: Vertical lines Find the equation of the verticalline displayed below. (2, 3) (a, b) Solution

  32. Prepared by DoronShahar 2.3.3 Sketch a graph Sketch a graph of the line x=1/3. Slope: (1/3, 0)

  33. Prepared by DoronShahar 2.3.3 Identifying intercepts Find the y-intercept and thex-intercept of the linex=1/3. To find the y-intercept, set x=0 and solve for y. To find the x-intercept, set y=0 and solve for x. Setting x equal to zero Setting y equal to zero Solution ¡PROBLEMA! There is no y-intercept! x-intercept:

  34. Prepared by DoronShahar Horizontal lines vs. Vertical lines Exception: y=0 Exception: y=0

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