Chapter 1 Linear Functions and

Mathematical Modeling Section 1.4

Section 1.4 Interpreting and Evaluating

Functions Using Graphs

- Graphs of Functions
- Identifying Domain and Range from Graphs of

Functions - Vertical Line Test
- Increasing, Decreasing, and Constant Functions

Graphs of Functions

- The graph of a function f is the set of all
- points on the plane of the form (x, f(x)).

- Example
- Construct the graph of f(x) 3x 4
- We can create a table of values, choosing several

values - for the input and calculating the corresponding

outputs. - Plotting and connecting the points, the graph of

the given - function follows

x 1 0 2 3

f(x) 7 4 2 5

- The graph of a function is shown below.
- a. Find C(100)
- C(100) means given the input (x-value) of 100,

find the output. - Therefore, C(100) 15
- b. Find x if C(x) 10
- If C(x) 10, we are looking for x-values where

the output is10. - Therefore, x 50 300

The graph of the function f(x) x2 x 5

is shown below. Use the graph to find the

following.

- a. Find f(2).
- We locate the output, y, when the input, x, is

2. f(2) 3, - which is equivalent to the point (2, 3) on the

graph. - b. Estimate and interpret f(0).
- f(0) 5. This is the vertical or y-intercept of

the function.

(Contd.) The graph of the function f(x)

x2 x 5 is shown below. Use the graph to find

the following.

- c. Find the input(s) when the output is 3.
- We look for any point(s) on the graph where y has

a value of - 3. Observe that f(2) 3 and f(1) 3. The

inputs are 2 - and 1.
- d. Find the value(s) of x for which f(x) 1.
- f(x) 1 is equivalent to y 1. Thus, f(x) 1

for x 3 and - x 2.

The graph of the function f(x) x2 x

6 is shown below. Use the graph to answer the

following. a. For what x-values is f(x) ? 4?

Write your answer using interval notation.

- f(x) ? 4 means the graph lies on or above y 4.
- This occurs when 1 ? x ? 2.
- In interval notation 1, 2

(Contd.) The graph of the function f(x)

x2 x 6 is shown below. Use the graph to

answer the following. b. For what x-values

is f(x) lt 0? Write your answer using interval

notation.

- f(x) lt 0 means the graph lies below y 0, that

is, below the - x-axis.
- This occurs when x lt 2 and x gt 3.
- In interval notation (?, 2) U (3, ?)

- The graph below illustrates the number of dogs in

an animal shelter after a tropical storm has

passed through a small - town. Let t represent time in days, and n

represent number - of dogs.
- a. Identify the input.
- The input is t, time in days.
- b. Identify the output.
- The output is n, number of dogs.

- (Contd.)
- c. How many dogs are in the shelter in 3 days?
- 24 dogs
- d. Find n(0) and interpret its meaning.
- n(0) 12. There were 12 dogs in the shelter

before the - tropical storm.
- e. Find the value of t for which n(t) 36.

Interpret this value. - t 6. Six days after the storm, there were 36

dogs in the shelter.

- Estimate the domain and range for the function

whose graph is shown below. - Domain (possible values of the input) From the

smallest - (leftmost) to the largest (rightmost).
- The domain consists of all x-values less than or

equal to 8, thus - the domain is given by the interval (?, 8.
- Range (possible values of the output) From the

lowest - (bottommost) to the highest (uppermost).
- The bottommost y-value is 6 and the uppermost

y-coordinate - is 6, therefore, the range is given by the

interval 6, 6.

- Estimate the domain and range for the function

whose graph is shown below. - Domain
- The domain consists of all x-values greater than

4, thus the - domain is given by the interval (4, ?).
- Range
- The range consists of all y-values greater than

or equal to 6, - thus the range is given by the interval 6, ?).

Vertical Line Test If a

vertical line meets a graph more than once, the

graph does not represent a function. For each

x-value, there can only be one y-value.

- True or False The following graph represents a

function. - True It passes the Vertical Line Test. Any

vertical line will intersect this graph only

once. - Notice that for each input there is only one

output.

Increasing, Decreasing, and Constant Functions

A function is increasing on an interval if it is

rising as it goes left to right on the interval.

(As x-coordinates increase, y-coordinates

increase.) A function is decreasing on an

interval if it is falling as it goes left to

right on the interval. (As x-coordinates

increase, y-coordinates decrease.) A function

is constant on an interval if the graph is a

horizontal line on the interval. (As

x-coordinates increase, y-coordinates remain

constant.)

- Notes
- ? We read a graph from left to right, and the

interval over - which a function is increasing, decreasing, or

constant is - given only in terms of the x-values. So, we use

interval - notation referring to the x-coordinates only.
- ? Use open intervals (parentheses, not brackets)

in the - interval notation, since the turning/ending

points are - neither increasing nor decreasing.

- Given the graph of y f(x), determine the

intervals for which the function is (a)

increasing, (b) decreasing, or (c) constant. - a. The graph rises (increases) from left to

right on the - interval (8, 4).
- b. The graph falls (decreases) from left to

right on the - interval (4, 12).
- c. The function is constant over the interval

(?, 8).

- The graph below shows the path of a model rocket

launched upward from the ground at an initial

velocity of 149 feet per - second. Its height, h, at t seconds, can be

modeled by - h(t) -16t2 149t.
- a. Use the graph to evaluate h(3) and explain its

meaning in the - context of the problem.
- h(3) 300.
- After 3 seconds, the model rocket's height going

up is 300 feet.

- (Contd.)
- The graph below shows the path of a model rocket

launched upward from the ground at an initial

velocity of 149 feet per - second. Its height, h, at t seconds, can be

modeled by - h(t) -16t2 149t.
- b. Estimate the location of the ordered pair

associated with - h(4.7) 346.9, and explain its meaning in terms

of the problem. - (4.7, 346.9). After 4.7 seconds, the model rocket

reaches a maximum height of approximately 346.9

feet.

- (Contd.)
- c. Estimate the increasing and decreasing

intervals. - Increasing (0, 4.7)
- Decreasing (4.7, 9.3)
- d. Estimate the domain and range from the graph.
- Domain 0, 9.3
- Range 0, 346.9

Using your textbook, practice the problems

assigned by your instructor to review the

concepts from Section 1.4.