Pre-Calculus Chapter 1

Functions and Their Graphs

1.3.1 Graphs of FunctionsObjectives:

Find the domains and ranges of functions & use the Vertical Line Test for functions.

Determine intervals on which functions are increasing, decreasing, or constant.

Determine relative maximum and relative minimum values of functions.

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VocabularyVertical Line Test

Increasing, Decreasing, and Constant

Functions

Relative Minimum and Relative Maximum

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Warm Up 1.3.1 A hand tool manufacturer produces a

product for which the variable cost is $5.35 per unit and the fixed costs are $16,000. The company sells the product for $8.20 and can sell all that it produces.

a. Write the total cost C as a function of x the number of units produced.

b. Write the profit P as a function of x.c. How many units need to be sold for the

company to be profitable? 4

x

yExample 1Use the graph of f to

find:

a.The domain of f.

b.The function values

f (–1) and f (2).

c.The range of f.

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(-1, -5)

(2, 4)

(4, 0)

How Do We Know It’s a Function?

Vertical Line Test

If any vertical line cuts the graph of a

relation in more than one place, then

the relation is not a function.

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Example 2Function or not?

a. b.

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Increasing, Decreasing, and Constant Functions

A function is increasing on an interval if, for any x1 and x2 in the interval,

x1 < x2 implies f (x1) < f (x2).

A function is decreasing on an interval if, for any x1 and x2 in the interval,

x1 < x2 implies f (x1) > f (x2).

A function is constant on an interval if, for any x1 and x2 in the interval,

f (x1) = f (x2).8

Picture = 103 Words

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Example 3aDetermine where the function is increasing,

decreasing, or constant.

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Example 3bDetermine where the function is increasing,

decreasing, or constant.

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Relative Minimum and Relative Maximum

A function value f (a) is a relative minimum of f

if there exists an interval (x1, x2) that contains a

such that

x1 < x < x2 implies f (a) ≤ f (x).

A function value f (a) is a relative maximum of

f if there exists an interval (x1, x2) that contains a

such that

x1 < x < x2 implies f (a) ≥ f (x).

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Picture = 103 More Words

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Example 4

Use your graphing calculator to approximate

the relative minimum of the function given

by:

f (x) = –x3 + x.

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Example 5During a 24-hour period, the temperature y (in

°F) of a certain city can be approximated

by the model

y = 0.0026x3 – 1.03x2 + 10.2x + 34, 0 ≤ x ≤ 24

where x represents the time of day,

with x = 0 corresponding to 6 A.M.

Approximate the maximum and minimum

temperatures during this 24-hour period.15

Homework 1.3.1Worksheet

1.3.1# 1 – 33 odd

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