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 FUTURELEARN DIRECTORS S. Botha | P.N. de Waal W. Pretorius | L. Ramiah Reg. No.: 2016/217646/07 CONTACT DETAILS 116 Witch-Hazel Ave, Highveld, Centurion 0169 PO Box 15132, Lyttelton, 0140 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. FUTURELEARN Reg. No.: 2016/217646/07 Page 2 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. FUTURELEARN Reg. No.: 2016/217646/07 Page 3 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. FUTURELEARN Reg. No.: 2016/217646/07 Page 4 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. FUTURELEARN Reg. No.: 2016/217646/07 Page 5 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? FUTURELEARN Reg. No.: 2016/217646/07 Page 6 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? FUTURELEARN Reg. No.: 2016/217646/07 Page 7 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 FUTURELEARN Reg. No.: 2016/217646/07 Page 8 2. B 3. A 4. B 5. A 6. B 7. B 8. A 9. A 10. C 11. B 12. B FUTURELEARN Reg. No.: 2016/217646/07 Page 9