- Use a spreadsheet to create a scatterplot
- Use a spreadsheet to generate a trendline for a scatterplot
- Analyze a trendline for appropriateness and fit
Modeling Basics
In many real-world situations a dependent relationship can be identified between two quantities, the independent variable and the dependent variable.
For example, say you purchase a brand-new car from a dealership. The value of your car begins to decline as soon as you sign the sales contract and drive the car off the lot. Notwithstanding other variables such as vehicle damage or economic fluctuations, the value of your new car going forward will be largely dependent upon how long you’ve owned it. This type of decrease in value over time is called depreciation.
Populations are also dependent upon time. It is not difficult to visualize growth in the population of people on the Earth over decades and centuries or of a number of bacteria in a petri dish over minutes and hours. Other types of growth depend on quantities other than time. For example, revenue, the amount of money collected when you sell an item, depends upon the number of items sold. The growth (or decay, as in the value of your new car over time) in each of these situations can be modeled mathematically.
The notation [latex]y=f\left(x\right)[/latex] defines a function named [latex]f[/latex]. This is read as “[latex]y[/latex] is a function of [latex]x[/latex].” The letter [latex]x[/latex] represents the input value, or independent variable. The letter [latex]y[/latex] or [latex]f\left(x\right)[/latex], represents the output value, or dependent variable.
Ordered Pairs as Coordinates for Data Points in the Plane
Recall that ordered pairs in the form [latex]\left(x, y\right)[/latex] give information about a point in the plane.
The graph of an equation contains a set of such points having input, [latex]x[/latex]-values and corresponding output, [latex]y[/latex]-values. Each ordered pair (data point) on the graph of such a mathematical relation satisfies the equation of the relation (makes it a true statement when substituted for the variables).
model basics
The independent variable, representing the input quantity is graphed on the horizontal axis. The dependent variable, representing the output quantity is graphed on the vertical axis.
Each point on the graph represents a correspondence between one piece of information from the set of input values with one piece of information from the set of output values.
This correspondence, called a relation, is written in the form of an ordered pair as (input, output) and it satisfies the mathematical statement of the relation between the two quantities described by the equation. A well-defined relation with no ambiguity regarding the output variable is also called a function.
There are several terms used to denote the coordinates of an ordered pair, (input, output), depending on the discipline applying the math or the particular situation to which it is being applied. You may see them variously referred to as any of the following:
[latex]\left(x, y\right) \quad \left(x, f\left(x\right)\right) \quad \left(t, p\left(t\right)\right) \quad\left(\text{input,} \text{ output}\right) \quad \left(\text{explanatory variable,}\text{ response variable}\right) \quad[/latex].
The letters or terms for the coordinates may change depending on the situation, but conceptually they all represent ordered pairs containing a coordinate from the set of all inputs to the relation along with its corresponding coordinate from the set of all outputs of the relation. A relation in which there is a correspondence between two variables is sometimes called bivariate.
We don’t know what the equation is when we begin to build a model. Instead, we use what we know of the nature of the situation to choose a type of equation or function to apply, and what we observe in the data of the situation to write a particular equation that best fits it. Some situations will be best approximated by linear relationships, that is by equations of lines, while others will require a non-linear relationship such as an exponential function. You will become familiar with linear, exponential, and logistic modeling in this section but you should know that these three types of relationships are just a few of the many types of equations that can be used to model real-world situations.