A quadratic function has a series of varying positive gradients on one part and a series of varying negative gradients on the other part. The positive and negative gradients meet at a point where the gradient is zero hence forming a maximum point or a minimum point. The model in part C has a negative gradient in the range . That is,
Additionally, the model has a series of varying positive gradients in the range .
The rates of change, can be represented using a schematic diagram resembling that of a quadratic function as shown in figure 1.1.
Figure 1.1: The schematic model
Therefore, table C can be modelled as a quadratic function.
Using the quadratic regression function in Desmos graphing calculator we obtain the result shown in figure 1
describes the direction the quadratic model opens (up or down). That is, the trend of the number of worms as the number of days’ increases. In this case, the negative value of a denotes a decrease in the number of worms with the increase in the number of days. describes the coefficient of the linear term of the model and shows how the number of worms is linearly to the number of days. On the other hand, describes the initial number of worms.
Solving for x using the quadratic formula,
We ignore since the number of days can’t be negative.
Hence, it takes for all the worms to die.
Given that, . The graph of is shown in figure 3.1.
Figure 3.1: A graph of
At the vertex,
Hence, the vertex is
The vertex form of the equation is in the form where is the vertex.
But we know that,
Therefore, the vertex form is,
The years when
We substitute 10 into the quadratic model and solve for t using the quadratic formula.
The number of crimes committed was equal to 10million in the years and . That is, 1983.5316 and 1998.8774.
That is, approximately during the years 1983 and 1998.
An example of a quadratic model in the form of is . It has a vertical intercept of +12, decreases initially, and then increases. The sketch of the model is shown in figure 4.1.
Figure 4.1: A graph of the model,
An example of a quadratic model in the form of that has maximum value and a vertical intercept at is as shown in the sketch, figure 4.2.
Figure 4.2: A graph of
The quadratic model, is concave up. To ascertain that, we determine the rate of change of As we can see,
, the rate of change of with respect to x is always positive and increases with the increase in the value of x Hence, it is concave up.