Introduction to GraphsIntroduction to Graphs

Dependent variableDependent variable is on the vertical axis is on the vertical axis

(Y)(Y)

Independent variableIndependent variable is on the horizontal is on the horizontal axis (X)axis (X)

Y

X

(“Leppel”)

STUDYINGSTUDYINGGrade v. Hours of StudyGrade v. Hours of Study

Your course grade represents:

A = 4, B = 3, C = 2, D = 1, and F = 0.

(“Leppel”)

Upward Sloping Lines

grade

hrs. studied per week

4

3

2

1

0 2 4 6 8

study time grade 8 4 6 3 4 2 2 1 0 0

(“Introduction to Graphs”)

Your grade and the number of hours you study Your grade and the number of hours you study move in the same direction.move in the same direction.

When you look left to right, you notice the line When you look left to right, you notice the line slopes upward.slopes upward.

““This is called a This is called a positivepositive oror direct relationdirect relation.”.”

(“Introduction to Graphs”)

grade

hrs. studied per week

4

3

2

1

0 2 4 6 8

your grade hrs. studied your grade hrs. studied

your grade hrs. studied your grade hrs. studied

(“Leppel”)

What would your grade be, if you studied for two hours per week?

grade

hrs. studied per week

4

3

2

1

0 2 4 6 8

1 (D)1 (D)

(“Leppel”)

What would your grade be, if you studied for eight hours per week?

grade

hrs. studied per week

4

3

2

1

0 2 4 6 8

4 (A)4 (A)

(“Leppel”)

What would your grade be, if you studied for zero hours per week?

grade

hrs. studied per week

4

3

2

1

0 2 4 6 8

0 (F)0 (F)

(“Leppel”)

“At what number does the line intersect the vertical axis?”

grade

hrs. studied per week

4

3

2

1

0 2 4 6 8

0 0

(“Leppel”)

The number 0 is the value of the The number 0 is the value of the Y-interceptY-intercept..

y-intercept

grade

4

3

2

1

0 2 4 6 8

The Y-intercept tells you the value of the Y variable (grade), when the value of the X variable (hrs. of study) is zero.

(“Leppel”)

study time grade 8 4 6 3 4 2 2 1 0 0

If you normally study 2 hours per week, and decided to study an additional 2 hours per week. “By how much does your grade increase?” (2 – 1 = ?)

1If you study 6 hours per week. Then, you decide to study an additional 2 hours per week. “By how much does your grade increase?” (4 – 3 = ?)

1

What would be the change in the Y variable (grade) divided by the change in the X variable (study time)?

(“Leppel”)

The slope of the relation would be? (slope = ∆Y/ ∆X∆Y/ ∆X )

1/2

1/2 = .5

2 1

grade changes 1 hrs. studied changes 2grade changes 1 hrs. studied changes 2

change/change 1/2change/change 1/2

slope 1/2 = .5 slope 1/2 = .5

The The slopeslope = the change in the Y variable = the change in the Y variable divided by the change in the X variable divided by the change in the X variable

= = Y/ Y/ X = 1/2 = .5X = 1/2 = .5

The number .5 is The number .5 is the slopethe slope..(“Leppel”)

RECAP!

Also expressed as the "rise" over the Also expressed as the "rise" over the "run.”"run.”

““It is the distance the line “rises” in the It is the distance the line “rises” in the vertical direction divided by the distance it vertical direction divided by the distance it “runs” in the horizontal direction.”“runs” in the horizontal direction.”

The Slope FormulaThe Slope Formula

(“Leppel”)

slope = rise/run = 1/2

grade

hrs. studied per week

4

3

2

1

0 2 4 6 8

rise = 1run = 2

(“Leppel”)

RUNNING ILLUSTRATIONRUNNING ILLUSTRATION

Suppose the more rested you were the faster you Suppose the more rested you were the faster you could run.could run.

So, let’s use the relation between hours slept per So, let’s use the relation between hours slept per day and the number of minutes it takes you to day and the number of minutes it takes you to run a mile.run a mile.

Downward Sloping LinesDownward Sloping Lines

http://images.google.com/images?gbv=2&hl=en&q=a+runner

(“Leppel”)

min. per mile

hrs. slept/day 0 1 2 3 4 5 6 7 8 9 10

8 7 6 5 4

hrs min/mi 6 8 7 7 8 6 9 510 4

(“Leppel”)

““What is the slope of the What is the slope of the relation?”relation?”

slope = slope = Y/Y/X X = = min/min/hrs hrs = -1/+1 = -1 = -1/+1 = -1

*An *An increaseincrease denotes a denotes a positivepositive change. change.

*A *A decreasedecrease denotes a denotes a negativenegative change.change.

hrs min/mi 6 8 7 7 8 6 9 510 4

(“Leppel”)

Amount of sleep minutes needed to Amount of sleep minutes needed to run a mile . run a mile .

Amount of sleep minutes needed to Amount of sleep minutes needed to

run a mile .run a mile . ““The variables move in opposite directions.The variables move in opposite directions. This type of relation is called a This type of relation is called a negativenegative or or

inverseinverse relation relation.”.”

(“Leppel”)

Y

X

• “Negative or inverse relations are downward sloping from left to right.”

Y

X

(“Leppel”)

negative slope

positive slope• “Positive or direct relations are upward sloping from left to right.”

What is the Y-intercept for this relation?What is the Y-intercept for this relation?

(“Leppel”)

14 is the Y-intercept

hours slept min/mile

6 8

5 9

4 10

3 11

2 12

1 13

0 14

“We know it takes 8 minutes to run a mile when you have had 6 hours of sleep.”

Working down to zero for the number of hours slept, you will need 14 minutes to run the mile.

min./mile

hrs. slept/day 0 1 2 3 4 5 6 7 8 9 10

15 12 9 6 3

y-intercept

“ “The Y-intercept tells the value of theThe Y-intercept tells the value of the Y variable (minutes needed to Y variable (minutes needed to run a mile) when the value run a mile) when the value of the X variable (hours of the X variable (hours slept) is zero.”slept) is zero.”

“You can also find the intercept by extending the line in the graph to the vertical axis.”

(“Leppel”)

MEDICATION ILLUSTRATIONMEDICATION ILLUSTRATION

Suppose you are taking medication for a virus. “The medication has the effect on the number of heartbeats per minute as indicated in the following graph.”

Downward Sloping LinesDownward Sloping Lines

(“Leppel”)

beats/min.

medicine (mg.) 0 100 200 300 400 500

75 70 65 60 55 50

medication beats/min 0 75

100 70

200 65

300 60

400 55

500 50

(“Leppel”)

beats/min.

medicine (mg.) 0 100 200 300 400 500

75 70 65 60 55 50

At what number does the line intersect the vertical axis?

What would your heart rate be, if you didn’t take any medication?

(“Leppel”)

75

75

What is the Y-intercept? 75

- 5 (decreases by 5 beats/min.)- 5 (decreases by 5 beats/min.)

If you increase your medication from 400 to If you increase your medication from 400 to 500 milligrams, by how much does your heart 500 milligrams, by how much does your heart rate change?rate change?

- 5 (decreases by 5 beats/min.)- 5 (decreases by 5 beats/min.)

What is the change in the Y variable What is the change in the Y variable (beats/min) divided by the change in the X (beats/min) divided by the change in the X variable (medication)? variable (medication)?

- 5/100 or - .05- 5/100 or - .05 What is the slope of the relation?What is the slope of the relation? - .05- .05

If you increase your medication from 200 to 300 milligrams, by how much does your heart rate change?

(“Leppel”)

medication beats/min

0 75

100 70

200 65

300 60

400 55

500 50

The slope is The slope is negativenegative, because the , because the variables are inversely related.variables are inversely related.

When the amount of medication When the amount of medication

, the heart rate ., the heart rate .

When the amount of medication When the amount of medication , the heart rate ., the heart rate .

(“Leppel”)

medication beats/min

0 75

100 70

200 65

300 60

400 55

500 50

“ “The negative slope is evident in the The negative slope is evident in the graph by the fact that the line slopes graph by the fact that the line slopes downward toward the right.”downward toward the right.”

beats

mg.

(“Leppel”)

Horizontal LinesHorizontal Lines

DIETINGILLUSTRATION

Let us suppose that no matter how hard you tried or how few calories you consumed, your weight remained the same.

http://images.google.com/images?hl=en&q=scales&gbv=2

(“Leppel”)

weight

180

170

160

150

140

1000 1100 1200 1300 1400 calories

calories weight

1000 180 1100 180 1200 180 1300 180 1400 180

(“Leppel”)

Y never changes, it Y never changes, it stays constantstays constant

slope = slope = ∆∆Y/ Y/ ∆∆X = 0/X = 0/∆∆X = 0X = 0 ““The slope of a horizontal line is zero.The slope of a horizontal line is zero.

Your weight would remain at 180, even if you consumed zero calories.Your weight would remain at 180, even if you consumed zero calories.

So, the Y-intercept is 180.”So, the Y-intercept is 180.”

Y

180

X

(“Leppel”)

Vertical LinesVertical Lines

Now let us suppose you consumed the same number of calories and your weight varied with exercise and stress.

http://images.google.com/images?hl=en&q=scales&gbv=2

(“Leppel”)

weight

180

170

160

150

140

40

0 1000 1100 1200 1300 1400 calories

calories weight 1100 140 1100 150 1100 160 1100 170 1100 180

(“Leppel”)

The Y variable, The Y variable, (weight) changes, but (weight) changes, but the X variable the X variable (calories) remains (calories) remains constant.constant.

The slope = The slope = Y/Y/XX

In this case, a non-zero number divided by zero. In this case, a non-zero number divided by zero.

The slope is The slope is infinityinfinity or or undefinedundefined..

The slope of a vertical line is infinity or undefined, because there is no Y-intercept.

wgt

1100 calories

(“Leppel”)

Let’s look at the vertical line graph using the study time and grade concept.

Vertical LinesVertical Lines

grade

hrs. studied per week

4

3

2

1

0 2 4 6 8

study timestudy time gradegrade 6 46 4 6 36 3 6 26 2 6 16 1 6 06 0

(“Leppel”)

grade

hrs. studied per week

4

3

2

1

0 2 4 6 8

6 6

How many hours did you study to get a grade

of 2 (C)?

(“Leppel”)

grade

hrs. studied per week

4

3

2

1

0 2 4 6 8

6 6

How many hours did you study to get a grade

of 3 (B)?

(“Leppel”)

grade

hrs. studied per week

4

3

2

1

0 2 4 6 8

6 6

How many hours did you study to get a grade

of 4 (A)?

(“Leppel”)

You always studied the same amount.

So, why did your grade vary?

The only reason had to be other factors, such as the amount of sleep you had and/or your diet.

(“Leppel”)

study timestudy time gradegrade 6 4 6 4 6 3 6 3 6 2 6 2 6 1 6 1 6 0 6 0

1/01/0

What is the slope of the What is the slope of the relation?relation?

undefined or infinityundefined or infinity

“What is the change in the Y variable (grade) divided by the change in the X variable (study time)?”

(“Leppel”)

Nonlinear RelationsNonlinear RelationsConvexConvex

“ “If a curve looks If a curve looks like the letter U or like the letter U or part of a U, it is part of a U, it is convexconvex (from (from below).”below).”

(“Leppel”)

““This curve is This curve is downwarddownward slopingsloping and and convexconvex from below.”from below.”

min. per mile

hrs. slept per day

(“Leppel”)

““This curve is This curve is upward upward slopingsloping and and convexconvex from from below.below.

(It bulges toward some (It bulges toward some reference point, usually the reference point, usually the horizontal axis or the origin horizontal axis or the origin of a diagram.)of a diagram.)

A curve is convex from below A curve is convex from below (or convex to something (or convex to something below it) if all straight lines below it) if all straight lines connecting points on it lie connecting points on it lie on or above it.”on or above it.”

calories

wgt

Convex Curve

(“Leppel”)

“ “Suppose that you're trying to lose weight.” Suppose that you're trying to lose weight.” The chart below “represents your weight The chart below “represents your weight and the number of calories you consume and the number of calories you consume per day.” per day.”

caloriescalories weightweight 1000 142 1000 142 1100 1431100 143

1200 1451200 145 1300 1501300 150 1400 1601400 160

http://images.google.com/images?hl=en&q=scales&gbv=2

(“Leppel”)

caloriescalories weightweight 1000 142 1000 142 1100 1431100 143

1200 1451200 145 1300 1501300 150 1400 1601400 160

“If you reduce your intake from 1400 to 1300 calories, your weight drops 10 pounds.

When you reduce your intake from 1300 to 1200 calories, your weight only drops 5 pounds.”

(“Leppel”)

caloriescalories weightweight 1000 142 1000 142

1100 143 1100 143 1200 1451200 145 1300 1501300 150 1400 1601400 160

“When your reduce your intake from 1200 to 1100 calories, your weight drops just 2 pounds.”

(“Leppel”)

weight

160

155

150

145

140

1000 1100 1200 1300 1400 calories

The line is no longer a straight line (linear) The line is no longer a straight line (linear)

relationship. Instead the relation is now curved. relationship. Instead the relation is now curved.

Reflecting a changing slope.

“The slope is the change in the Y-variable (wgt) divided by the change in the X-variable (calories).” (“Leppel”)

calories weight 1000 140 900 130 800 120 700 110 600 100 500 90 400 80 300 70 200 60 100 50 0 40

What would you weigh if your calories were zero?

40 pounds

(“Leppel”)

Concave Concave

“ “Picture the opening Picture the opening of a cave. If a curve of a cave. If a curve looks like this or part looks like this or part of this, it is conof this, it is concavecave (from below).”(from below).”

(“Leppel”)

Nonlinear RelationsNonlinear Relations

“ “This curve is This curve is upwardupward slopingsloping and and concaveconcave from below.”from below.”

wgt

calories

The thin person's perspective!The thin person's perspective!

Suppose you were trying to gain weight. Suppose you were trying to gain weight.

http://images.google.com/images?hl=en&q=scales&gbv=2

(“Leppel”)

caloriescalories weightweight 1000 100 1000 100 1100 1101100 110 1200 1151200 115 1300 1181300 118 1400 1191400 119

By increasing your intake from 1000 to 1100 calories, your weight increased 10 pounds.

But, when you increased your intake from 1100 to 1200 calories, your weight only increased 5 pounds.

(“Leppel”)

weight

120

115

110

105

100

1000 1100 1200 1300 1400 calories

caloriescalories weightweight 1000 100 1000 100 1100 1101100 110 1200 1151200 115 1300 1181300 118 1400 1191400 119

(“Leppel”)

caloriescalories weightweight calories calories wgt wgt slope=slope=wt/wt/calcal 1000 1001000 100 100100 10 .10 10 .10 1100 1101100 110 100 100 5 .05 5 .05 1200 1151200 115 100 100 3 .03 3 .03 1300 1181300 118 100100 1 .01 1 .01 1400 1191400 119

“As calories increase, the slope decreases; the curve gets flatter.”

(“Leppel”)

Recap Graphs!Recap Graphs!

Constant Opportunity Cost GraphConstant Opportunity Cost Graph

Y

X

Zero Opportunity Cost GraphZero Opportunity Cost Graph

Y

X

Decreasing Opportunity Cost or Decreasing Opportunity Cost or Convex GraphConvex Graph

Y

X

Increasing Opportunity Cost or Increasing Opportunity Cost or Concave GraphConcave Graph

Y

X

Works CitedWorks Cited

Leppel, Professor Karen. “Introduction to Graphs.” Widener University. 25 Jul 2008. http://www.muse.widener.edu/~kleppel/EC202_ppt/GRAPHS.PPT