[email protected] MTH55_Lec-07_sec_2-3a_Lines_by_Intercepts.ppt 1 Bruce Mayer, PE Chabot...

[email protected] • MTH55_Lec-07_sec_2-3a_Lines_by_Intercepts.ppt1

Bruce Mayer, PE Chabot College Mathematics

Bruce Mayer, PELicensed Electrical & Mechanical Engineer

[email protected]

Chabot Mathematics

§2.4a Lines§2.4a Linesby by

InterceptsIntercepts



Review §Review §

Any QUESTIONS About• § 2.3 → Algebra of Funtions

Any QUESTIONS About HomeWork• § 2.2 → HW-05

2.3 MTH 55



Eqn of a Line Eqn of a Line AxAx + + ByBy = = CC

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 4•3 + 3•2 | 18 12 + 6 | 18

18 = 18 True

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



Example Example Eqn of a Line 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 4•5 + 3•1 | 18 20 + 3 | 18

23 = 18 False



To Graph a Linear EquationTo 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



Example Example Graph Graph yy = = −−44xx + 1 + 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.



Example Example Graph Graph yy = = −−44xx + 1 + 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.



Example Example Graph Graph yy = = −−44xx + 1 + 1 Note that all three

points line up. If they didn’t 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



Example Example Graph Graph xx + 2 + 2yy = 6 = 6 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

x y (x, y)

6 0 (6, 0)

0 3 (0, 3)

2 2 (2, 2)



Example Graph 4Example Graph 4yy = 3 = 3xx Solution: Begin by

solving for y.

xy 34

xy 34

14

4

1

xxy 75.04

3

Or y is 75% of x

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



Example Example Graph Graph 44yy = 3 = 3xx 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

x y (x, y)

0 0 (0, 0)

4 3 (4, 3)

−4 −3 (4 , 3)

3

4y x



Example Example Application 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 package’s weight in lbs

Graph the equation and then use the graph to estimate the cost of shipping a 10½ pound package



FedEx Soln: FedEx Soln: cc = 2.8 = 2.8ww + 21.05 + 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



FedEx Soln: Graph EqnFedEx Soln: Graph Eqn Plot the points.

Weight (in pounds)

Mai

l co

st (

in d

olla

rs) 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

10 ½ pounds The cost of shipping an 10½ pound package is about $51.00

$5

1



Finding Intercepts of Lines 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



Example Example Axes Intercepts 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)



Graph Graph AxAx + + ByBy = = CC Using Intercepts 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)



To FIND the InterceptsTo FIND the Intercepts

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

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



Example Example Find Intercepts 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)



Example Example Find Intercepts Find Intercepts cont. 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)



Example Example Graph w/ Graph w/ InterceptsIntercepts

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

x-intercept (2, 0)

y-intercept (0, 5)

Chk-Pt (4,-5)




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)




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)



Example Example Graph w/ Graph w/ InterceptsIntercepts Construct Graph for 3x – 4y = 8

• Find a third point. If we let x = 4, then – 3•4 – 4y = 8

– 12 – 4y = 8

– –4y = –4

– y = 1

• We plot (0, −2), (8/3, 0), and (4, 1)and Connect the Dots

x-intercept

y-intercept

3x 4y = 8

Chk-Pt Charlie



Example Example Graph Graph yy = 2 = 2

SOLUTION: We regard the equation y = 2 as the equivalent eqn: 0•x + y = 2. • No matter what number we choose for x,

we find that y must equal 2.

(−4 , 2)2−4 (4, 2)24(0, 2)20(x, y)yxChoose any number for x.

y must be 2.

y=2



Example Example Graph Graph yy = 2 = 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

(4, 2)

(0, 2)

(4, 2)



Example Example Graph Graph xx = = −−22

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

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.

x = −2



Example Example Graph Graph xx = = −−22

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, 4)

(2, 1)



Linear Eqns of ONE VariableLinear 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)



Example Example Horiz or Vert Line 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



Example Example Horiz or Vert Line 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



SLOPE DefinedSLOPE Defined

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

12

12

run

rise

in x Change

yin Change

xx

yy

m



Example Example Slope City Slope City Graph the line

containing the points (−4, 5) and (4, −1) & find the slope, m

SOLUTION

Thus Slopem = −3/4

Ch

ange

in y

= −

6

Change in x = 8

12

12

run

rise

in x Change

yin Change

xx

yy

m

8

6

44

51

m



Example Example ZERO Slope ZERO Slope Find the slope of the

line y = 3

32

33

run

rise

m

05

0m

(3, 3) (2, 3) SOLUTION: Find Two Pts on the Line

• Then the Slope, m

A Horizontal Line has ZERO Slope



Example Example UNdefined Slope UNdefined Slope Find the slope of

the line x = 2

22

24

run

rise

m

??0

6m

SOLUTION: Find Two Pts on the Line

• Then the Slope, m

A Vertical Line has an UNDEFINED Slope

(2, 4)

(2, 2)



Applications of Slope = GradeApplications 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 road’s steepness. • That is, a 3% grade means

that for every horizontal distance of 100 ft, the road rises or falls 3 ft.



Grade ExampleGrade Example Find the slope

(or grade) of the treadmill

0.42

ft

5.5 ft

SOLUTION: Noting the Rise & Run

0764.05.5

42.0

run

rise

ft

ftm

In %-Grade for Treadmill

%64.71

%1000764.0Grade m



Slope SymmetrySlope Symmetry We can Call

EITHER Point No.1 or No.2 and Get the Same Slope

Example, LET• (x1,y1) = (−4,5)

Moving L→R

12

12

run

rise

xx

yym

4

3

8

6

44

51

m

(−4,5) Pt1

(4,−1)



Slope SymmetrySlope Symmetry cont cont

Now LET• (x1,y1) = (4,−1)

12

12

run

rise

xx

yym

4

3

8

6

44

15

m

(−4,5)

(4,−1)Pt1 Moving R→L

Thus

21

21

12

12

in x Chg

yin Chg

xx

yy

xx

yym

12

21

21

12 or xx

yy

xx

yy



Slopes SummarizedSlopes Summarized POSITIVE Slope NEGATIVE Slope



Slopes SummarizedSlopes Summarized ZERO Slope UNDEFINED Slope

slope = 0

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

slope = undefined

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



WhiteBoard WorkWhiteBoard Work

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

More Lines



P2.4-26 P2.4-26 Find Slope for Lines Find Slope for Lines Recall

12

12

run

rise

xx

yym

12

2

run

rise

1

11

m

323

2

run

rise

2

22

m

22

4

run

rise

3

33

m



All Done for TodayAll Done for Today

SomeSlopeCalcs

x

y

xx

yym

12

12

run

rise



20x20 Grid

20x20 Grid

-10

-9

-8

-7

-6

-5

-4

-3

-2

-1

0

1

2

3

4

5

6

7

8

9

10

-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10

file =XY_Plot_0211.xlsfile =XY_Plot_0211.xls



Bruce Mayer, PELicensed Electrical & Mechanical Engineer

[email protected]

Chabot Mathematics

AppendiAppendixx

–

srsrsr 22

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