 Why do we have graphical models?  To be able to model situations and data  To be able to compare, interpret and analyze graphs in order to recognize trends and patterns and better understand the situation being modeled.  This will enable predictions to be made about future behaviour and outcomes  Which in turn will enable better decision making

 GRAPHS: a visual representation of the relationship between 2 quantities and how one quantity changes with respect to the other.  TRENDS: are patterns of change - in graphs, they are used to justify decisions and make predictions.

 Trends occur in these 3 broad groups  Using the graph below, divide it into intervals of time when number of births is increasing, constant or decreasing

The following graph illustrates info about college enrollment in Ont. by representing the data in a graph as opposed to a table of numbers, it is much easier to discover trends and make comparisons and predictions. How would you predict enrollment for the year? Why might this information be important?

From 1990 to 2005, the enrollment increased by approx. 57,000. This is, on average, 3800 students per year. If this trend continues until next year, the enrolment will increase by another 38,000 (3800 x 10) to about 197,000 students. Colleges must make predictions like this so they can build facilities and hire staff accordingly.

 Activity: investigate how rates of change and initial conditions of a relationship affect the corresponding graph. This will allow you to make comparisons between relations.

EX. Suppose you need to ship a package and you are trying to decide between two different delivery companies. The shipping charges for each company are represented by the following equations. C: Shipping Cost ($); m: mass of package (kg) Donny’s DeliveriesSuzanne’s Shipping C = mC = m Step 1: Graph each situation.

1.What is the ‘starting’ price for each delivery company? 2.How are these initial conditions represented on the graph? 3.The lines representing the delivery costs for each company have different slopes. Explain the significance of this in the context of the problem. 1.Donny’s Deliveries C = m Starting price: $3.50 Suzanne’s Shipping C = m Starting price: $ These initial conditions are represented in the graph using the y-intercept, i.e. when mass = 0kg (no package). 3. If you write the Cost functions in the form of y = mx + b, you get: y = 4.75x for DD and y = 4x + 5 for SS. The slopes are respectively, 4.75 and The significance is this is the cost per kg rate.

EX. Two large containers are filled with water. A hole is punched in the bottom of each container, and water drains out. The relationship between time and volume of water in each container is shown by the following equations. CONTAINER A: V = 24t 2 – 240t +600 CONTAINER B: V = 10t 2 – 140t +490 Step 1: Graph each situation.

1.What type of function is represented by each equation? 2.Explain why the domain of each function must be restricted? 3.Explain how to determine the initial value of each function using both the equation and the graph. What is the meaning of this value? 4.Which container empties at a greater rate? How can you tell using: (a) the equation (b) graph 1.Quadratic because highest power is t 2. 2.Domains must be restricted because after a certain time, the volume cannot become negative. 3.The initial value from the graph is the y-intercept, and you can find this from the initial equation by looking at it in the form y = ax 2 + bx + c, and notice that ‘c’ is the y-intercept. 4.Container A empties out a greater rate because: (a) the coefficient of t 2 is more than double for A than B (b) the graph for Container A is a steeper curve, reaching 0L faster.