Grade 11
MAKING SENSE OF GRAPHS
PART 1
Graph terminology
As you are still new to the subject it is important that you understand graphs and how they work. Yes,
in Grade 8 and 9 mathematics graphs was touched on but not as in depth as it will be in Mathematical
Literacy from now until the end of your Grade 12 year. It is imperative that you understand graphs,
because there are a lot of questions that you can get asked just by looking at the graph and depicting
all the information.
In this lesson, the basics of graphs will be focused on before moving on to the more detailed sections
about graphs and how they tell a story.
The basics of graph terminology that will be discussed is the following:
•
Shape of the graph
•
Direction of the graph
•
X-intercept
•
Y-intercept
•
Independent variable (x-axis)
•
Dependent variable (y-axis)
•
Maximum point
•
Minimum point
•
Break-even point (also referred to as the point of intersection or the equilibrium)
Graph shapes
The shape of the graph tells a lot about the type of graph that you are dealing with as well as the story
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that the graph is telling you.
Straight line graph:
When you see a straight-line graph in front of you,
you already need to know that you are dealing
with a constant difference. What a constant
difference means, is that as each point increases
the opposite point increases at the same rate.
A good example of this could be the following, 1
can of coke costs R8, 2 cans will cost R16, 3 cans
will cost R24. As you can see from this as each
coke increases by one, the price of the coke
increases by R8. This type of graph is also called
a direct proportion graph.
Curve graph:
When you see a curve graph it is an indication
that you are dealing with a constant product. This
type of graph shows that as one set of axes is
increasing, the other set of axes is decreasing
and vice versa. An example of this could be a
candle burning. You can all agree that when the
candle was just lit it was at its highest point, as
the candle starts burning it is going to become
shorter showing you a decrease in the size of the
candle, where the other axis will be showing you
the number of hours the candle was burning for,
which will be increasing. This type of graph is also
called an inverse proportion graph.
Direction of the graph
There are only two directions that a graph can be in at any given point, which is a positive direction,
which means that the gradient of that graph is positive, and a negative direction, which indicates the
gradient of the graph is negative.
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X - and y – intercept
The x-intercept is a point that intersects the x-axis and the
corresponding y-coordinate will always be zero.
The y-intercept is a point that intersects the y-axis and the
corresponding x-coordinate will always be zero.
Independent and Dependent variables
Independent and dependent variables are commonly asked in exams; however, this is normally the
question that learners get wrong because they swop the two variables around. This explanation will try
to make it easy for you to understand.
Let us look at the independent variable first. This variable will always be the variable that can be
controlled. This variable will always be your x-axis which is on your horizontal line. You can try to
understand it as follows: when you become an adult, you learn to do things on your own and that makes
you independent and that you can stand alone. This is the same for the independent variable; it can
stand alone and is independent.
The dependent variable is the total opposite; while the independent variable can stand on its own the
dependent variable is dependent on what the nature of the story is. The dependent variable is the
variable you have no control over at all. This variable will always be found on the y-axis which is the
vertical line on your Cartesian plane.
Questions will generally ask you what the label is on the x-axis or the which the dependent variable is.
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Maximum and minimum points
The maximum and minimum points on a graph are very important and could hold a lot of information
and may be asked in any question on graphs in papers that you are still going to write.
A maximum point should already be an indication that you are looking for the point that is the highest
on the graph. This point should also be the highest value on the dependent variable which is also known
as your y-axis. If you look at the graph below, it shows the number of people in the store at any given
time of the day. As you can see from the graph, the most number of people that were in the store was
22 people at 1p.m. This is an indication that the store never saw more than 22 people in the store at a
particular time during the day.
The minimum point should tell you that you are looking for the point on the graph that is the lowest. This
point should also be the lowest value on the dependent variable which as stated is your y-axis. If you
look at the graph below, it shows how many superheroes have been sighted in a particular
neighborhood. As you can see where the arrow is pointed, this shows the fewest sightings was in August
and had about 18 sightings.
Break-even point
A break-event point is also referred to as the point of intersection or an equilibrium point. This point is
normally a point on the graph where two (or more) graphs intersect or connect.
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What this can sometimes indicate, in the case of a business, is that no
profit or loss has been made and that you have broken even. This would
be an indication that your income is equal to your expenses at that point
of intersection.
Example 1: The story from a graph
If you look at the graph below, you will see that the left-hand side has a graph which shows the top
reasons as to why customers call the company. If you look at the right-hand side, this gives you 9 points
about the graph that you could have deduced yourself. Any of these 9 points could be asked in an
exam, but more in paper 2 as there are more reasoning skills behind this graph than mathematical
calculations.
If you look at the graph you can see that majority of the calls made to the company is mainly for
installation reasons, this is where 400 people called the company. However, even though it is still the
most reasons why people call, it increased by 5% in that month which means that it is more likely to
increase again in the next month.
The least amount of calls was made for cancellations and refunds. Even though cancellations had 30
people that called they were also the highest percentage increase. The reasons for cancellations could
vary depending on the services that the company offers which you will have to deduce from the
information given to you before the graph.
The other points can be looked at and are all self-explanatory.
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Example 2: Story from a graph
The second example is a line graph, but a story could also be deduced from this graph.
If you look at the graph, you should already deduce that the graph is representing coffee sales and each
month the amount of coffee sales increases. January had the least amount of sales while December
had the most amount of sales.
There was also a steep increase of coffee sales that took place from January to April, this could be due
to the weather or the season of the country where the sales were taking place. It could also be deduced
that the most coffee sales were made in December, either because it is winter time in the area or it
could also be the fact that it is holidays and some people enjoy bonding over a cup of coffee.
The other points of the graph can be looked at, feel free to make your own deductions and write them
down.
QUESTIONS
Question 1:
What type of graph does this graph represent?
a) Curve
b) Straight line
Question 2:
What type of slope does the following graph represent?
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a) Positive
b) Negative
Question 3:
The _____ intercept is when the y coordinate is equal to zero
a) X
b) Y
Question 4:
A point that intersects the y-axis is called the?
a) X-intercept
b) Y-intercept
Question 5:
What is known as the independent variable?
a) X-axis
b) Y-axis
Question 6:
The variable that you have no control over will be plotted on the?
a) X-axis
b) Y-axis
Question 7:
The highest point on the graph is called the?
a) Maximum point
b) Minimum point
Question 8:
The lowest point on the graph is called the?
a) Maximum point
b) Minimum point
Question 9:
What is the point of intersection between two graphs called?
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a) Break-even point
b) X-axis
c) Y-axis
d) Income and expenses
Question 10:
Break-even occurs when your income is ________ to your expenses.
a) More
b) Less
c) Equal
d) None of the above
Question 11:
The graph below shows different fruits and the amount of people that enjoy each fruit, answer the two
questions that follow.
11.1) Which fruit has 30 people that enjoy it?
a) Kiwi fruit
b) Orange
c) Apple
d) Blueberry
11.2) What is the difference in amount of people between the fruit that is enjoyed the most and the
fruit that is enjoyed the least?
a) 30
b) 35
c) 25
d) 40
ANSWERS
1. A
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2. B
3. A
4. B
5. A
6. B
7. B
8. A
9. A
10. C
11. B
12. B
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