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Linear Equations and Linear Functions Chapter 1

Linear Equations and Linear Functions

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Chapter 1. Linear Equations and Linear Functions. Section 1.1. Using Qualitative Graphs to Describe Situations. Section 1.1. Slide 3. Reading a Qualitative Graphs. Reading Qualitative Graphs. Example. - PowerPoint PPT Presentation

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Page 1: Linear Equations and  Linear Functions

Linear Equations and

Linear Functions

Chapter 1

Page 2: Linear Equations and  Linear Functions

Using Qualitative Graphs to Describe Situations

Section 1.1

Page 3: Linear Equations and  Linear Functions

Lehmann, Intermediate Algebra, 3edSection 1.1 Slide 3

Reading a Qualitative Graphs

Let p be the retail price (in dollars) or Air Jordans and t be the number of years since 1985.

What does this graph (or curve) tell us?

As t increases, what happens to p?

Example

Reading Qualitative Graphs

Page 4: Linear Equations and  Linear Functions

Lehmann, Intermediate Algebra, 3edSection 1.1

Assume that an authentic situation can be described by using the variables t and p and that p depends on t:•We call t the independent variable.•We call p the dependent variable.

Slide 4

Identifying Independent and Dependent Variables

Definition

Independent and Dependant Variables

Page 5: Linear Equations and  Linear Functions

Lehmann, Intermediate Algebra, 3edSection 1.1 Slide 5

Identifying Independent and Dependent Variables

Identify the independent variable and the dependent variable.

You are filling a swimming pool. Let r be the rate (in gallons per hour) at which water is added to a swimming pool, and let t be the number of hours It takes to fill the pool.

Example

Independent and Dependant Variables

Page 6: Linear Equations and  Linear Functions

Lehmann, Intermediate Algebra, 3edSection 1.1 Slide 6

Identifying Independent and Dependent Variables

• Shorter amount of time to fill the pool, the higher the rate at which water is added to the pool• Amount of time, t, it takes to fill the pool,

depends on the rate at which water is added, r• t is the dependent variable and r is the

independent variable. • The rate at which water is added to the pool does

not depend on the amount of time it takes to fill the pool.

Solution

Independent and Dependant Variables

Page 7: Linear Equations and  Linear Functions

Lehmann, Intermediate Algebra, 3edSection 1.1 Slide 7

Reading a Qualitative Graph

Let A be the average age (in years) when men first marry, and let t be the number of years since 1900. The graph describes the relationship between the variables t and A.

What does the graph tell us?

Example

Independent and Dependant Variables

Page 8: Linear Equations and  Linear Functions

Lehmann, Intermediate Algebra, 3edSection 1.1 Slide 8

Reading a Qualitative Graph

• Graph tells us that the average age when men first marry decreased each year for a while and then increased each year after than

Solution

Independent and Dependant Variables

Page 9: Linear Equations and  Linear Functions

Lehmann, Intermediate Algebra, 3edSection 1.1 Slide 9

Definitions

An intercept of a curve is any point where the curve and an axis (or axes) intercept.

The shape of the top curve is a parabola.

The shape of the bottom curve is a linear line.

Definition

Independent and Dependant Variables

Page 10: Linear Equations and  Linear Functions

Lehmann, Intermediate Algebra, 3edSection 1.1 Slide 10

Sketching a Qualitative Graphs

Let C be the cost (in dollars) of a 30-second ad during the Super Bowl at t years since 1987. For most years the annual increase in cost is more than the previous annual increase in cost. Sketch a qualitative graph that describes the relationship between C and t.

Example

Sketching Qualitative Graphs

Page 11: Linear Equations and  Linear Functions

Lehmann, Intermediate Algebra, 3edSection 1.1 Slide 11

Sketching a Qualitative Graphs

• Cost of an ad varies according to year• C is the dependent variable • t is the independent variable• Ads were not free in 1987 (t=0)• C-intercept (or dependent variable intercept) is

positive.

Solution

Sketching Qualitative Graphs

Page 12: Linear Equations and  Linear Functions

Lehmann, Intermediate Algebra, 3edSection 1.1

The shape of this curve is said to be exponential.

Slide 12

Sketching a Qualitative Graphs

• Cost increase, so the curve goes up from left to right• Most increases are more than the

previous year. • Curve should “bend” upward

from left to right.

Solution Continued

Sketching Qualitative Graphs

Page 13: Linear Equations and  Linear Functions

Lehmann, Intermediate Algebra, 3edSection 1.1

If a curve goes…

… upwards from left to right it is saidto be increasing.

… downward from left to right it is saidto be decreasing.

Slide 13

Definitions

Definition

Sketching Qualitative Graphs

Page 14: Linear Equations and  Linear Functions

Lehmann, Intermediate Algebra, 3edSection 1.1

Which curve is increasing

Which is decreasing?

Slide 14

Increasing Decreasing

Example

Sketching Qualitative Graphs

Page 15: Linear Equations and  Linear Functions

Lehmann, Intermediate Algebra, 3edSection 1.1 Slide 15

Increasing Decreasing

Decreasing curve

From left to right the first curve is increasing and the second is decreasing.

Increasing curve

Solution

Sketching Qualitative Graphs

Page 16: Linear Equations and  Linear Functions

Lehmann, Intermediate Algebra, 3edSection 1.1

The percentage of flight attendants laid-off by top United States airlines increased from 2001 to 2004 and decreased thereafter.

Let P be the percentage of flight attendants laid-off by top United States airlines, and t be years since 2001.

Sketch a quantitative graph that describes the relationship between the variable.

Slide 16

Sketching a Qualitative Graph

Example

Sketching Qualitative Graphs

Page 17: Linear Equations and  Linear Functions

Lehmann, Intermediate Algebra, 3edSection 1.1

• The percentage of flight attendants laid off is determined by the time after 2000• The curve increases then decreases• P is the dependant variable and t is

the independent variable• Both P and t are nonnegative.

Slide 17

Sketching a Qualitative Graph

Solution

Sketching Qualitative Graphs

Page 18: Linear Equations and  Linear Functions

Lehmann, Intermediate Algebra, 3edSection 1.1

P and t are both nonnegative, so the curve is in Quadrant I.

•What quadrant would the curve be in if P and t were both negative?

• What would the values of P and t be if the curve was only in Quadrant II, or IV>

Slide 18

QuadrantsSketching Qualitative Graphs

Page 19: Linear Equations and  Linear Functions

Lehmann, Intermediate Algebra, 3edSection 1.1

Guidelines for Responding to Concept Questions: •Create example• Illustrate concept• Outline the procedures• Help jump starts creating example

Slide 19

Describing a Concept or Procedure

Guidelines

Sketching Qualitative Graphs

Page 20: Linear Equations and  Linear Functions

Lehmann, Intermediate Algebra, 3edSection 1.1

• Describe key ideas/steps for examples

• Use complete sentences

• Use correct terminology

• Review book for ideas

• Write your own ideas

Slide 20

Describing a Concept or Procedure

Guidelines Continued

Sketching Qualitative Graphs

Page 21: Linear Equations and  Linear Functions

Lehmann, Intermediate Algebra, 3edSection 1.1

• Describe key ideas in general terms

• Reflect on several examples

• Find commonalities in examples

• Describe without the use of an example

• Similarities and differences

• Find similarities and differences in concepts

• Find similarities and differences in procedures

Slide 21

Describing a Concept or Procedure

Guidelines Continued

Sketching Qualitative Graphs

Page 22: Linear Equations and  Linear Functions

Lehmann, Intermediate Algebra, 3edSection 1.1

• Know the benefits

• Describe benefits of knowing a concepts/procedure

• If steps are used

• Why can they can be used

Slide 22

Describing a Concept or Procedure

Guidelines Continued

Sketching Qualitative Graphs