Relations and
Functions
Lesson 2: Interpreting and
Sketching Graphs
Todays Objectives
 Describe
a possible situation for a given
graph
 Sketch a possible graph for a given
situation
 Sort a set of graphs as functions and nonfunctions
Interpreting and Sketching Graphs
 There
are many quantities encountered in
real life that can be represented on a
graph. In particular, we will look at cases
where one variable is responding to, or is
a function of, the other variable.
 Consider the following graph, which
illustrates height (h) in meters as a function
of time for an elevator in a five-floor
building. Assume that each floor is 4 m
high.
Example
Height of elevator as a function of Time
Height (m)
Time (s)
At point A, the elevator is on the 3rd floor (the ground floor is at 0m). It
then goes up at a steady speed to the 4th floor. It stays there for a few
seconds, as shown from point B to C. The elevator then continues up to
the 6th floor at a steady speed. It stays on the 6th floor for several
seconds, as shown from D to E, before going back to the 1st floor at a
steady speed without stopping.
Example
 *An
*Use the absolute values to determine which
slope is greatest
Δ𝑦
Δ𝑥
∆ℎ𝑒𝑖𝑔ℎ𝑡
∆𝑡𝑖𝑚𝑒
important thing
 𝑠𝑙𝑜𝑝𝑒 =
=
to notice on the
graph is the slope
of the line
segments. The
greater the slope,
the greater the rate
of change.
 In this case, the
slope is a measure
of the velocity of
20 − 12 4
0 − 20
𝑠𝑙𝑜𝑝𝑒 =
=
𝑠𝑙𝑜𝑝𝑒 =
= −2
the elevator.
20 − 10 5
50 − 40
(Greater rate of change)
Rate of change
 Horizontal
line
segments occur
when the value
on the vertical
axis is not
changing. So,
the flatter the
line segment, the
smaller the rate
of change
∆𝑦 = 0, 𝑠𝑙𝑜𝑝𝑒 = 0, 𝑟𝑎𝑡𝑒 𝑜𝑓 𝑐ℎ𝑎𝑛𝑔𝑒 = 0
(elevator is not moving)
Example
 The
following graph shows the distance
from Chias home as a function of time
during a walk that he took to the store
and back home.
 The distance from his home changes as
he walks, so Chias distance from his home
is a function of how long he has been
walking (time).
Example
Chias distance from home as a function of time
Distance from
home (m)
Time (min)
A. Chia met Francis along the way and stopped to talk to him for a
minute. Where is this on the graph?
B. What was happening from D to E?
C. Was Chia walking faster on the way to the store or on the way
back?
Example
 Guele
is driving straight down main street in his
Ferrari. When he starts, he is in a school zone. At
the end of the school zone, Guele has to stop for
a red light for a few seconds. He then drives
again, but this time at a faster speed. He then
reaches his destination after 4 minutes and stops
and parks.
 Draw a graph of Guele’s trip with distance
travelled as a function of time.
 *Time is ALWAYS plotted on the x-axis
(independent variable)
Example
Distance travelled as a function of time
Distance
travelled
Time (min)
Notice that the distance travelled does not have a
scale as there was not enough information given.
Example
 Lois
and Caroline drove their Audi A8 to
Pudong Airport to pick up their friend Bill. In
the first hour, the car travels 65 km/h to the
airport. The car then stops for 15 minutes as
Lois and Caroline stop to buy chicken feet for
a snack. The car then travels 65 km toward
the airport. After 2 hours, the car has traveled
130 km and has reached the airport where it
stops for 2 hours. The girls meet Bill and take 2
hours to travel 130 km back home.
 Draw a graph for this situation.
Example
Distance from home as a function of time
Distance
from home
(km)
Time (h)
Review Quiz
 One
line has a slope of 1/3, another line has a
slope of -4/3. Which line has the greater slope?

The second line (4/3 > 1/3)
 Which

line represents the greater rate of change?
The second line
A
horizontal line means a rate of change of zero.
What would a vertical line mean?


Slope is undefined. For example, lets say Guele
drives his Ferrari 30 km in 0 seconds. Is this possible?
𝑠𝑙𝑜𝑝𝑒 =
Δ𝑦
Δ𝑥
=
30𝑘𝑚
0𝑠
= 𝑢𝑛𝑑𝑒𝑓𝑖𝑛𝑒𝑑 (𝑑𝑒𝑛𝑜𝑚𝑖𝑛𝑎𝑡𝑜𝑟 𝑖𝑠 0)