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1.5

GEOMETRIC PROPERTIES OF LINEAR FUNCTIONS

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Interpreting the Parameters of a Linear Function

Example 1 With time, t, in years, the populations of four towns, PA , PB , PC , and PD , are given by the following formulas:PA = 20,000 + 1600t, PB = 50,000 − 300t, PC = 650t + 45,000, PD = 15,000(1.07)t.

(a) Which populations are represented by linear functions?(b) Describe in words what each linear model tells you about that town’s

population. Which town starts out with the most people? Which town is growing fastest?

Solution(a) The populations of towns A, B, and C are represented by linear

functions because they can be written in the form P = mt + b. Town D’s population does not grow linearly since its formula PD = 15,000(1.07)t, cannot be expressed in the form PD = b + m t.

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Interpreting the Parameters of a Linear Function

Example 1 continuedPA = 20,000 + 1600t, PB = 50,000 − 300t, PC = 650t + 45,000, PD = 15,000(1.07)t.

(b) Describe in words what each linear model (towns A, B and C) tells you about that town’s population. Which town starts out with the most people? Which town is growing fastest?

SolutionFor town A, b = 20,000 and m = 1600. This means that in year t = 0, town A

has 20,000 people. It grows by 1600 people per year.

For town B, b = 50,000 and m = −300. This means that town B starts with 50,000 people. The negative slope indicates that the population is decreasing at the rate of 300 people per year.

For town C, b = 45,000 and m = 650. This means that town C begins with 45,000 people and grows by 650 people per year.

Town B starts with the most. Town A grows the fastest.

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The Effect of the Parameters on the Graph of a Linear Function

Let y = mx + b. Then the graph of y against x is a line.• The y-intercept, b, tells us where the line crosses the y-

axis.• If the slope, m, is positive, the line climbs from left to

right. If the slope, m, is negative, the line falls from left to right.

• The slope, m, tells us how fast the line is climbing or falling.

• The larger the magnitude of m (either positive or negative), the steeper the graph of f.

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Intersection of Two LinesExample 3 The cost in dollars of renting a car for a day from three different

rental agencies and driving it d miles is given by the following functions:

C1 = 50 + 0.10d C2 = 30 + 0.20d C3 = 0.50d.(a) Describe in words the daily rental arrangements made by each of

these three agencies.(b) Which agency is cheapest? Solution(a) Agency 1 charges $50 plus $0.10 per mile driven. Agency 2

charges $30 plus $0.20 per mile. Agency 3 charges $0.50 per mile driven.

(b) To determine which agency is cheapest, we will graph all three.

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0 5 0 1 0 0 1 5 0 2 0 0d m i l e s 2 0

4 0

6 0

8 0

1 0 0

1 2 0C $ Intersection of Two Lines

Example 3 - continuedC1 = 50 + 0.10d C2 = 30 + 0.20d C3 = 0.50d.

(100,50)

(125, 62.50)(200,70)

Based on the points of intersection and minimizing C, Agency 3 is cheapest for d < 100Agency 2 is cheapest for 100 < d < 200Agency 1 is cheapest for d > 200

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Equations of Horizontal and Vertical Lines

For any constant k:• The graph of the equation y = k is a

horizontal line and its slope is zero.• The graph of the equation x = k is a

vertical line and its slope is undefined.

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Slopes of Parallel and Perpendicular Lines

Let l1 and l2 be two lines having slopes m1 and m2, respectively. Then:• These lines are parallel if and only if m1 = m2.• These lines are perpendicular if and only if m1

= −1/m2 . (or ) 1 2 1m m