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Learning intentions:
In this section we will examine:
- Relationships and how they change from a graphical perspective.
- Qualitative and quantitative descriptions of rates of change.
Relationships
Two variables are often linked by a relationship.
- In some instances we have a mathematical rule which describes the relationship.
- In other situations no rule exists, instead we rely on data and observations.
In either case we are often interested in analysing the relationship. In particular we are often interested in the rate of change of one variable with respect to the change in another.
Graphing the relationship between two variables
Graphing the relationship between two variables is often useful in order to analyse the relationship.
- Click on the images below to view dynamic simulations of filling different shaped vases with water.
Constant rate of change
Variable rate of change
10A - VIDEO EXAMPLE 1:
Water is being poured into each of the following containers at a constant rate. Draw a graph to represent how the height of the water changes as time progresses.
Introduction to rates of change
Rate of change can be defined as how one quantity (Q1) changes in relation to another quantity (Q2) changing. Mathematically this can be expressed as:
It is important to remember that the rate of change is a quantity.
Qualitative description of rates of change
We can describe the rate of change without a numerical value. We can describe rates as:
- Positive, negative or zero.
- Constant or variable.
- Large or small.
- Increasing or decreasing.
Positive, negative and zero rates of change
Rates can be positive or negative and even zero.
- A positive rate of change indicates that as one variable increases so to does the other variable in the relationship (see Figure 1a).
- A zero rate of change indicates that as one variable increases there is no change in the other variable (see Figure 1b).
- A negative rate of change indicates that as one variable increases the other variable in the relationship decreases (see Figure 1c).
Figure 1 - (a) positive rate of change (b) zero rate of change (c) negative rate of change.
10A - VIDEO EXAMPLE 2:
Consider the function shown in the graph below. State the intervals where the rate of change is positive and negative for the relationship.
Constant and variable rates of change
Rates can be constant (as seen in Figure 1) or variable (as seen in Figure 2).
- A relationship with a constant rate is one where the the rate of change of one quantity with respect to another does not change.
- A relationship with a variable rate is one where the rate of change of one quantity increases or decreases in the relationship.
Figure 2 - (a) Variable positive rate of change (b) variable negative rate of change.
Large or small rates of change
By inspecting the rate of change we can comment on the magnitude of the rate; that is, how large or small the rate of change is.
Constant rates of change are considered in section 10B.
Variable rates of change are considered in section 10C.
Average rates of change are considered in section 10C.
Instantaneous rates of change are considered in section 10D.
Figure 3 - (a) Large and (c) small rates of change.
Increasing and decreasing rates of change
Rates can also be described as increasing or decreasing:
- A rate of changing is said to be increasing if the rate of change of one quantity with respect to another is getting larger as the independent variable increases (refer to Figure 2a).
- A rate of changing is said to be decreasing if the rate of change of one quantity with respect to another is getting smaller as the independent variable increases (refer to Figure 2b).
Quantitative description of rates of change
We can describe the rate of change with a numerical value. Numerically, we can calculate the:
- Average rate of change.
- Instantaneous rate of change.
Average rate of change
The average rate of change is found by calculating the gradient of a line segment joining two points on a graph.
Figure 4 - The average rate of change between x = 2 and x = 7.
Instantaneous rate of change
The instantaneous rate of change is found by calculating the gradient (m) of a tangent line to a graph at the desired point. The gradient of the tangent describes the rate of change at that point or instant.
Figure 5 - Instantaneous rates of change
Success criteria:
You will be successful if you can:
- Graph relationships in a given context.
- Understand qualitative and quantitative descriptions of rates of change.
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