Chapter 3: Straight Lines and Linear Functions

E. Smith

MAT 1020

E. Smith Chapter 3: Straight Lines and Linear Functions MAT 1020 1 / 10

Section 3.1: The Geometry of Lines

Section 3.1 Key Ideas

The definition of a linear function

The meaning of the slope of a line

The relationship between slope and ARC

Solving geometric problems involving straight lines via slope

Horizontal and Vertical Intercepts

E. Smith Chapter 3: Straight Lines and Linear Functions MAT 1020 2 / 10

Section 3.1: The Geometry of Lines

Recall that the formula for the average rate of change is

f (b) − f (a)

b − a

Example 1: Suppose f (x) measures the population of a protectedspecies (in number of animals) t years after that species has beenintroduced to a wildlife preserve. What would the average rate ofchange of f (x) from x = 5 to x = 10 tell you in terms of thepopulation?

E. Smith Chapter 3: Straight Lines and Linear Functions MAT 1020 3 / 10

Section 3.1: The Geometry of Lines

Recall that the formula for the average rate of change is

f (b) − f (a)

b − a

Example 1: Suppose f (x) measures the population of a protectedspecies (in number of animals) t years after that species has beenintroduced to a wildlife preserve. What would the average rate ofchange of f (x) from x = 5 to x = 10 tell you in terms of thepopulation?

E. Smith Chapter 3: Straight Lines and Linear Functions MAT 1020 3 / 10

Section 3.1: The Geometry of Lines

Linear Function

: Any function who’s average rate of changebetween any two points is always equal and when graphed makes astraight line.1

Some characteristics of a linear function:

Can be written in the form y = mx + b.2

The slope of a linear function is m = f (b)−f (a)b−a = ARC

The slope, or rate of change, m, of a line shows how steeply it isincreasing or decreasing. It tells the vertical change along the line whenthere is a horizontal change of 1 unit.If m = 0, the line is horizontal.

1Although we call these linear functions, they are actually one variable affinefunctions.

2This is often called Y-Intercept Form, where b isthe y-intercept.

E. Smith Chapter 3: Straight Lines and Linear Functions MAT 1020 4 / 10

Section 3.1: The Geometry of Lines

Linear Function: Any function who’s average rate of changebetween any two points is always equal and when graphed makes astraight line.1

Some characteristics of a linear function:

Can be written in the form y = mx + b.2

The slope of a linear function is m = f (b)−f (a)b−a = ARC

The slope, or rate of change, m, of a line shows how steeply it isincreasing or decreasing. It tells the vertical change along the line whenthere is a horizontal change of 1 unit.If m = 0, the line is horizontal.

1Although we call these linear functions, they are actually one variable affinefunctions.

2This is often called Y-Intercept Form, where b isthe y-intercept.

E. Smith Chapter 3: Straight Lines and Linear Functions MAT 1020 4 / 10

Section 3.1: The Geometry of Lines

Linear Function: Any function who’s average rate of changebetween any two points is always equal and when graphed makes astraight line.1

Some characteristics of a linear function:

Can be written in the form y = mx + b.2

The slope of a linear function is m = f (b)−f (a)b−a = ARC

The slope, or rate of change, m, of a line shows how steeply it isincreasing or decreasing. It tells the vertical change along the line whenthere is a horizontal change of 1 unit.If m = 0, the line is horizontal.

1Although we call these linear functions, they are actually one variable affinefunctions.

2This is often called Y-Intercept Form, where b isthe y-intercept.

E. Smith Chapter 3: Straight Lines and Linear Functions MAT 1020 4 / 10

Section 3.1: The Geometry of Lines

Linear Function: Any function who’s average rate of changebetween any two points is always equal and when graphed makes astraight line.1

Some characteristics of a linear function:

Can be written in the form y = mx + b.2

The slope of a linear function is m = f (b)−f (a)b−a = ARC

The slope, or rate of change, m, of a line shows how steeply it isincreasing or decreasing. It tells the vertical change along the line whenthere is a horizontal change of 1 unit.If m = 0, the line is horizontal.

1Although we call these linear functions, they are actually one variable affinefunctions.

2This is often called Y-Intercept Form, where b isthe y-intercept.

E. Smith Chapter 3: Straight Lines and Linear Functions MAT 1020 4 / 10

Section 3.1: The Geometry of Lines

Linear Function: Any function who’s average rate of changebetween any two points is always equal and when graphed makes astraight line.1

Some characteristics of a linear function:

Can be written in the form y = mx + b.2

The slope of a linear function is m = f (b)−f (a)b−a = ARC

The slope, or rate of change, m, of a line shows how steeply it isincreasing or decreasing. It tells the vertical change along the line whenthere is a horizontal change of 1 unit.If m = 0, the line is horizontal.

1Although we call these linear functions, they are actually one variable affinefunctions.

2This is often called Y-Intercept Form, where b isthe y-intercept.

E. Smith Chapter 3: Straight Lines and Linear Functions MAT 1020 4 / 10

Section 3.1: The Geometry of Lines

Linear Function: Any function who’s average rate of changebetween any two points is always equal and when graphed makes astraight line.1

Some characteristics of a linear function:

Can be written in the form y = mx + b.2

The slope of a linear function is m = f (b)−f (a)b−a = ARC

The slope, or rate of change, m, of a line shows how steeply it isincreasing or decreasing. It tells the vertical change along the line whenthere is a horizontal change of 1 unit.

If m = 0, the line is horizontal.

1Although we call these linear functions, they are actually one variable affinefunctions.

2This is often called Y-Intercept Form, where b isthe y-intercept.

E. Smith Chapter 3: Straight Lines and Linear Functions MAT 1020 4 / 10

Section 3.1: The Geometry of Lines

Linear Function: Any function who’s average rate of changebetween any two points is always equal and when graphed makes astraight line.1

Some characteristics of a linear function:

Can be written in the form y = mx + b.2

The slope of a linear function is m = f (b)−f (a)b−a = ARC

The slope, or rate of change, m, of a line shows how steeply it isincreasing or decreasing. It tells the vertical change along the line whenthere is a horizontal change of 1 unit.If m = 0, the line is horizontal.

1Although we call these linear functions, they are actually one variable affinefunctions.

2This is often called Y-Intercept Form, where b isthe y-intercept.

E. Smith Chapter 3: Straight Lines and Linear Functions MAT 1020 4 / 10

Section 3.1: The Geometry of Lines

Example 2:1 If a linear function has a positive slope what do we know about the

graph of the function?

2 If a linear function has a negative slope what do we know about thegraph of the function?

3 If f (x) has a larger slope in absolute value than g(x) what do we knowabout how the shapes of the graphs of f (x) and g(x) compare?

E. Smith Chapter 3: Straight Lines and Linear Functions MAT 1020 5 / 10

Section 3.1: The Geometry of Lines

Example 2:1 If a linear function has a positive slope what do we know about the

graph of the function?2 If a linear function has a negative slope what do we know about the

graph of the function?

3 If f (x) has a larger slope in absolute value than g(x) what do we knowabout how the shapes of the graphs of f (x) and g(x) compare?

E. Smith Chapter 3: Straight Lines and Linear Functions MAT 1020 5 / 10

Section 3.1: The Geometry of Lines

Example 2:1 If a linear function has a positive slope what do we know about the

graph of the function?2 If a linear function has a negative slope what do we know about the

graph of the function?3 If f (x) has a larger slope in absolute value than g(x) what do we know

about how the shapes of the graphs of f (x) and g(x) compare?

E. Smith Chapter 3: Straight Lines and Linear Functions MAT 1020 5 / 10

Section 3.1: The Geometry of Lines

We can also use lines and their properties to solve problems.

Example 3: Three horizontal feet north of a 10-foot tall maple tree isa 4-foot tall forsythia. We wish to place a spotlight north of theforsythia so that the light just hits the top of both the forsythia andthe maple tree. How many feet north of the forsythia should we placethe light. Note: We wish for the spotlight to be on the groundpointing upwards.

E. Smith Chapter 3: Straight Lines and Linear Functions MAT 1020 6 / 10

Section 3.1: The Geometry of Lines

We can also use lines and their properties to solve problems.

Example 3: Three horizontal feet north of a 10-foot tall maple tree isa 4-foot tall forsythia. We wish to place a spotlight north of theforsythia so that the light just hits the top of both the forsythia andthe maple tree. How many feet north of the forsythia should we placethe light. Note: We wish for the spotlight to be on the groundpointing upwards.

E. Smith Chapter 3: Straight Lines and Linear Functions MAT 1020 6 / 10

Section 3.1: The Geometry of Lines

Example 4: Plywood siding is to be used to cover the exterior wall ofa house. Plywood siding comes in sheets 4 feet wide and 8 feet high.The piece on the far right is 1 foot high on the shorter side and 2 feet6 inches high on the longer side (towards the peak of the roof). Thefinal cut for the plywood would be the length k. What is the length?

E. Smith Chapter 3: Straight Lines and Linear Functions MAT 1020 7 / 10

Section 3.2: Linear Functions

Section 3.2 Key Ideas

Understanding the meaning of slope, vertical intercept, and horizontalintercept and begging able to give practical information of them.

Finding equations of lines given the slope (or enough information tofind the slope) and a point.

Finding equations of lines given two points.

E. Smith Chapter 3: Straight Lines and Linear Functions MAT 1020 8 / 10

Section 3.2: Linear Functions

Review

What makes a function linear?

Its slope is constant (always the same).Its graph is a straight line.

The slope of a linear function is give by

m =f (b) − f (a)

b − a=

y2 − y1x2 − x1

Positive slope mean: increasing from left to right (uphill).

Negative slope means: decreasing from left to right (downhill).

X-Intercepts: When the y-value is equal to zero.

Y-Intercepts: When the x-value is equal to zero.

E. Smith Chapter 3: Straight Lines and Linear Functions MAT 1020 9 / 10

Section 3.2: Linear Functions

Review

What makes a function linear?

Its slope is constant (always the same).Its graph is a straight line.

The slope of a linear function is give by

m =f (b) − f (a)

b − a=

y2 − y1x2 − x1

Positive slope mean: increasing from left to right (uphill).

Negative slope means: decreasing from left to right (downhill).

X-Intercepts: When the y-value is equal to zero.

Y-Intercepts: When the x-value is equal to zero.

E. Smith Chapter 3: Straight Lines and Linear Functions MAT 1020 9 / 10

Section 3.2: Linear Functions

Review

What makes a function linear?

Its slope is constant (always the same).

Its graph is a straight line.

The slope of a linear function is give by

m =f (b) − f (a)

b − a=

y2 − y1x2 − x1

Positive slope mean: increasing from left to right (uphill).

Negative slope means: decreasing from left to right (downhill).

X-Intercepts: When the y-value is equal to zero.

Y-Intercepts: When the x-value is equal to zero.

E. Smith Chapter 3: Straight Lines and Linear Functions MAT 1020 9 / 10

Section 3.2: Linear Functions

Review

What makes a function linear?

Its slope is constant (always the same).Its graph is a straight line.

The slope of a linear function is give by

m =f (b) − f (a)

b − a=

y2 − y1x2 − x1

Positive slope mean: increasing from left to right (uphill).

Negative slope means: decreasing from left to right (downhill).

X-Intercepts: When the y-value is equal to zero.

Y-Intercepts: When the x-value is equal to zero.

E. Smith Chapter 3: Straight Lines and Linear Functions MAT 1020 9 / 10

Section 3.2: Linear Functions

Review

What makes a function linear?

Its slope is constant (always the same).Its graph is a straight line.

The slope of a linear function is give by

m =f (b) − f (a)

b − a=

y2 − y1x2 − x1

Positive slope mean: increasing from left to right (uphill).

Negative slope means: decreasing from left to right (downhill).

X-Intercepts: When the y-value is equal to zero.

Y-Intercepts: When the x-value is equal to zero.

E. Smith Chapter 3: Straight Lines and Linear Functions MAT 1020 9 / 10

Section 3.2: Linear Functions

Review

What makes a function linear?

Its slope is constant (always the same).Its graph is a straight line.

The slope of a linear function is give by

m =f (b) − f (a)

b − a=

y2 − y1x2 − x1

Positive slope mean: increasing from left to right (uphill).

Negative slope means: decreasing from left to right (downhill).

X-Intercepts: When the y-value is equal to zero.

Y-Intercepts: When the x-value is equal to zero.

E. Smith Chapter 3: Straight Lines and Linear Functions MAT 1020 9 / 10

Section 3.2: Linear Functions

Review

What makes a function linear?

Its slope is constant (always the same).Its graph is a straight line.

The slope of a linear function is give by

m =f (b) − f (a)

b − a=

y2 − y1x2 − x1

Positive slope mean: increasing from left to right (uphill).

Negative slope means: decreasing from left to right (downhill).

X-Intercepts: When the y-value is equal to zero.

Y-Intercepts: When the x-value is equal to zero.

E. Smith Chapter 3: Straight Lines and Linear Functions MAT 1020 9 / 10

Section 3.2: Linear Functions

Review

What makes a function linear?

Its slope is constant (always the same).Its graph is a straight line.

The slope of a linear function is give by

m =f (b) − f (a)

b − a=

y2 − y1x2 − x1

Positive slope mean: increasing from left to right (uphill).

Negative slope means: decreasing from left to right (downhill).

X-Intercepts: When the y-value is equal to zero.

Y-Intercepts: When the x-value is equal to zero.

E. Smith Chapter 3: Straight Lines and Linear Functions MAT 1020 9 / 10

Section 3.2: Linear Functions

Review

What makes a function linear?

Its slope is constant (always the same).Its graph is a straight line.

The slope of a linear function is give by

m =f (b) − f (a)

b − a=

y2 − y1x2 − x1

Positive slope mean: increasing from left to right (uphill).

Negative slope means: decreasing from left to right (downhill).

X-Intercepts: When the y-value is equal to zero.

Y-Intercepts: When the x-value is equal to zero.

E. Smith Chapter 3: Straight Lines and Linear Functions MAT 1020 9 / 10

Section 3.2: Linear Functions

Example 1: The value of a car in dollars t years after it is purchasedis given by V (t) = 18700 − 1700t.

What is the slope of V (t) and what does it mean in practical terms?What is the vertical intercept of V (t) and what does it mean inpractical terms?Show that t = 11 is the horizontal intercept of V (t) and explain whatthis means in practical terms.

E. Smith Chapter 3: Straight Lines and Linear Functions MAT 1020 10 / 10