W A T K I N S - J O H N S O N C O M P A N Y Semiconductor Equipment Group

1
Chabot Mathematics
2.4a Linesby Intercepts
Bruce Mayer, PE Licensed Electrical Mechanical
EngineerBMayer_at_ChabotCollege.edu
2
Review

• Any QUESTIONS About
• 2.3 ? Algebra of Funtions
• Any QUESTIONS About HomeWork
• 2.2 ? HW-05

3
Eqn of a Line ? Ax By C

• Determine whether each of the following pairs is
  a solution of eqn 4y 3x 18
• a) (2, 3) b) (1, 5).
• Soln-a) We substitute 2 for x and 3 for y

4y 3x 18 43 32 18 12 6
18 18 18 True

• Since 18 18 is true, the pair (2, 3) is a
  solution

4
Example ? Eqn of a Line

• Soln-b) We substitute 1 for x and 5 for y

• Since 23 18 is false, the pair (1, 5) is not a
  solution

4y 3x 18 45 31 18 20 3 18
23 18 ? False
5
To Graph a Linear Equation

1. Select a value for one coordinate and calculate
   the corresponding value of the other coordinate.
   Form an ordered pair. This pair is one solution
   of the equation.
2. Repeat step (1) to find a second ordered pair. A
   third ordered pair can be used as a check.
3. Plot the ordered pairs and draw a straight line
   passing through the points. The line represents
   ALL solutions of the equation

6
Example ? Graph y -4x 1

• Solution Select convenient values for x and
  compute y, and form an ordered pair.
• If x 2, then y -4(2) 1 -7 so (2,-7) is a
  solution
• If x 0, then y -4(0) 1 1 so (0, 1) is a
  solution
• If x 2, then y -4(-2) 1 9 so (-2, 9) is
  a solution.

7
Example ? Graph y -4x 1

• Results are often listed in a table.

x y (x, y)
2 7 (2, 7)
0 1 (0, 1)
2 9 (2, 9)

• Choose x
• Compute y.
• Form the pair (x, y).
• Plot the points.

8
Example ? Graph y -4x 1

• Note that all three points line up. If they
  didnt we would know that we had made a mistake

• Finally, use a ruler or other straightedge to
  draw a line

• Every point on the line represents a solution of
  y -4x 1

9
Example ? Graph x 2y 6
x y (x, y)
6 0 (6, 0)
0 3 (0, 3)
2 2 (2, 2)

• Solution Select some convenient x-values and
  compute y-values.
• If x 6, then 6 2y 6, so y 0
• If x 0, then 0 2y 6, so y 3
• If x 2, then 2 2y 6, so y 2
• In Table Form, Then Plotting

10
Example Graph 4y 3x

• Solution Begin by solving for y.

• To graph the last Equation we can select values
  of x that are multiples of 4
• This will allow us to avoid fractions when
  computing the corresponding y-values

• Or y is 75 of x

11
Example ? Graph 4y 3x
x y (x, y)
0 0 (0, 0)
4 3 (4, 3)
-4 -3 (?4 , ?3)

• Solution Select some convenient x-values and
  compute y-values.
• If x 0, then y ¾ (0) 0
• If x 4, then y ¾ (4) 3
• If x -4, then y ¾ (-4) -3
• In Table Form, Then Plotting

12
Example ? Application

• The cost c, in dollars, of shipping a FedEx
  Priority Overnight package weighing 1 lb or more
  a distance of 1001 to 1400 mi is given by c
  2.8w 21.05
• where w is the packages weight in lbs
• Graph the equation and then use the graph to
  estimate the cost of shipping a 10½ pound package

13
FedEx Soln c 2.8w 21.05

• Select values for w and then calculate c.
• c 2.8w 21.05
• If w 2, then c 2.8(2) 21.05 26.65
• If w 4, then c 2.8(4) 21.05 32.25
• If w 8, then c 2.8(8) 21.05 43.45
• Tabulatingthe Results

w c
2 26.65
4 32.25
8 43.45
14
FedEx Soln Graph Eqn

• Plot the points.

?51

• To estimate costs for a 10½ pound package, we
  locate the point on the line that is above 10½
  lbs and then find the value on the c-axis that
  corresponds to that point

Mail cost (in dollars)

• The cost of shipping an 10½ pound package is
  about 51.00

10 ½ pounds
Weight (in pounds)
15
Finding Intercepts of Lines

• An Intercept is the point at which a line or
  curve, crosses either the X or Y Axes
• A line with eqn Ax By C (A B ? 0) will
  cross BOTH the x-axis and y-axis
• The x-CoOrd of the point where the line
  intersects the x-axis is called the x-intercept
• The y-CoOrd of the point where the line
  intersects the y-axis is called the y-intercept

16
Example ? Axes Intercepts

• For the graph shown
• a) find the coordinates of any x-intercepts
• b) find the coordinates of any y-intercepts

• Solution
• a) The x-intercepts are (-2, 0) and (2, 0)
• b) The y-intercept is (0,-4)

17
Graph Ax By C Using Intercepts

1. Find the x-Intercept ? Let y 0, then solve for
   x
2. Find the y-Intercept ? Let x 0, then solve for
   y
3. Construct a CheckPoint using any convenient value
   for x or y
4. Graph the Equation by drawing a line thru the
   3-points (i.e., connect the dots)

18
To FIND the Intercepts

• To find the y-intercept(s) of an equations
  graph, replace x with 0 and solve for y.
• To find the x-intercept(s) of an equations
  graph, replace y with 0 and solve for x.

19
Example ? Find Intercepts

• Find the y-intercept and the x-intercept of the
  graph of 5x 2y 10
• SOLUTION To find the y-intercept, we let x 0
  and solve for y
• 5 0 2y 10
• 2y 10
• y 5
• Thus The y-intercept is (0, 5)

20
Example ? Find Intercepts cont.

• Find the y-intercept and the x-intercept of the
  graph of 5x 2y 10
• SOLUTION To find the x-intercept, we let y 0
  and solve for x
• 5x 2 0 10
• 5x 10
• x 2
• Thus The x-intercept is (2, 0)

21
Example ? Graph w/ Intercepts

• Graph 5x 2y 10 using intercepts
• SOLUTION
• We found the intercepts in the previous example.
  Before drawing the line, we plot a third point
  as a check. If we let x 4, then
• 5 4 2y 10
• 20 2y 10
• 2y -10
• y - 5
• We plot Intercepts (0, 5) (2, 0), and also (4
  ,-5)

5x 2y 10
y-intercept (0, 5)
x-intercept (2, 0)
Chk-Pt (4,-5)
22
Example ? Graph w/ Intercepts

• Graph 3x - 4y 8 using intercepts
• SOLUTION To find the y-intercept, we let x 0.
  This amounts to ignoring the x-term and then
  solving. -4y 8
• y -2
• Thus The y-intercept is (0, -2)

23
Example ? Graph w/ Intercepts

• Graph 3x 4y 8 using intercepts
• SOLUTION To find the x-intercept, we let y 0.
  This amounts to ignoring the y-term and then
  solving 3x 8 x 8/3
• Thus The x-intercept is (8/3, 0)

24
Example ? Graph w/ Intercepts

• Construct Graph for 3x 4y 8
• Find a third point. If we let x 4, then
• 34 4y 8
• 12 4y 8
• 4y 4
• y 1
• We plot (0, -2), (8/3, 0), and (4, 1)and
  Connect the Dots

Chk-Pt Charlie
x-intercept
y-intercept
3x ? 4y 8
25
Example ? Graph y 2

• SOLUTION We regard the equation y 2 as the
  equivalent eqn 0x y 2.
• No matter what number we choose for x, we find
  that y must equal 2.

y2
(x, y)
y
x
(0, 2)
2
0
(4, 2)
2
4
(-4 , 2)
2
-4
26
Example ? Graph y 2

• Next plot the ordered pairs (0, 2), (4, 2) (-4,
  2) and connect the points to obtain a horizontal
  line.
• Any ordered pair of the form (x, 2) is a
  solution, so the line is parallel to the x-axis
  withy-intercept (0, 2)

y 2
(0, 2)
(?4, 2)
(4, 2)
27
Example ? Graph x -2

• SOLUTION We regard the equation x -2 as x
  0y -2. We build a table with all -2s in the
  x-column.

x -2
x y (x, y)
-2 4 (-2, 4)
-2 1 (-2, 1)
-2 -4 (-2, -4)
x must be ?2.
Any number can be used for y.
28
Example ? Graph x -2

• When we plot the ordered pairs (-2,4), (-2,1)
  (-2, -4) and connect them, we obtain a vertical
  line
• Any ordered pair of the form (-2,y) is a
  solution. The line is parallel to the y-axis
  with x-intercept (-2,0)

x ?2
(?2, 4)
(?2, 1)
(?2, ?4)
29
Linear Eqns of ONE Variable

• The Graph of y b is a Horizontal Line, with
  y-intercept (0,b)

• The Graph of x a is a Vertical Line, with
  x-intercept (a,0)

30
Example ? Horiz or Vert Line

• Write an equation for the graph

• SOLUTION Note that every point on the horizontal
  line passing through (0,-3) has -3 as the
  y-coordinate.
• Thus The equation of the line is y -3

31
Example ? Horiz or Vert Line

• Write an equation for the graph

• SOLUTION Note that every point on the vertical
  line passing through (4, 0) has 4 as the
  x-coordinate.
• Thus The equation of the line is x 4

32
SLOPE Defined

• The SLOPE, m, of the line containing points (x1,
  y1) and (x2, y2) is given by

33
Example ? Slope City

• Graph the line containing the points (-4, 5) and
  (4, -1) find the slope, m
• SOLUTION

Change in y -6
Change in x 8

• Thus Slopem -3/4

34
Example ? ZERO Slope

• Find the slope of the line y 3

• SOLUTION Find Two Pts on the Line

(?3, 3)
(2, 3)

• Then the Slope, m

• A Horizontal Line has ZERO Slope

35
Example ? UNdefined Slope

• Find the slope of the line x 2

(2, 4)

• SOLUTION Find Two Pts on the Line

• Then the Slope, m

(2, ?2)

• A Vertical Line has an UNDEFINED Slope

36
Applications of Slope Grade

• Some applications use slope to measure the
  steepness.
• For example, numbers like 2, 3, and 6 are
  often used to represent the grade of a road, a
  measure of a roads steepness.
• That is, a 3 grade means that for every
  horizontal distance of 100 ft, the road rises
  or falls 3 ft.

37
Grade Example

• Find the slope (or grade) of the treadmill

• SOLUTION Noting the Rise Run

0.42 ft
5.5 ft

• In -Grade for Treadmill

38
Slope Symmetry

• We can Call EITHER Point No.1 or No.2 and Get the
  Same Slope
• Example, LET
• (x1,y1) (-4,5)

(-4,5) Pt1
(4,-1)

• Moving L?R

39
Slope Symmetry cont

• Now LET
• (x1,y1) (4,-1)

(-4,5)
(4,-1)Pt1

• Moving R?L
• Thus

40
Slopes Summarized

• POSITIVE Slope

• NEGATIVE Slope

41
Slopes Summarized

• ZERO Slope

• UNDEFINED Slope

         slope 0
slope undefined

• Note that when a line is horizontal the slope is 0

• Note that when the line is vertical the slope is
  undefined

42
WhiteBoard Work

• Problems From 2.4 Exercise Set
• 26 (PPT), 12, 24, 52, 56

• More Lines

43
P2.4-26 ? Find Slope for Lines

• Recall

44
All Done for Today
SomeSlopeCalcs
45

• 20x20 Grid

46
Chabot Mathematics
Appendix
Bruce Mayer, PE Licensed Electrical Mechanical
EngineerBMayer_at_ChabotCollege.edu