The percentage P of companies that are still in business t...

The percentage $P$ of companies that are still in business $t$ years after the fifth year in operation can be represented by the equation $$P=-3 t+50$$ a. Create a table of points that satisfy this equation. b. Create a graph for the equation using your points. Remember to label the graph with units.

The percentage $P$ of companies that are still in business $t$ years after the fifth year in operation can be represented by the equation
$$P=-3 t+50$$
a. Create a table of points that satisfy this equation.
b. Create a graph for the equation using your points. Remember to label the graph with units.

Key Concepts

Graphing Linear Equations

Graphing a linear equation requires plotting the ordered pairs derived from the table of values on a coordinate system and drawing a straight line through them. It's important to label the axes with the correct units (e.g., years for t and percent for P) to accurately illustrate the relationship between time and the percentage of companies still in business. This visual representation aids in understanding the trend and making predictions.

Table of Values

Creating a table of values involves selecting different values for the independent variable (t), substituting them into the equation, and computing the corresponding output values (P). This systematic approach allows one to generate a set of ordered pairs that represent specific points on the graph, making it easier to visualize the linear relationship.

Linear Functions

A linear function describes a constant rate of change between two variables, forming a straight line when graphed. In the equation P = -3t + 50, 'P' is the dependent variable representing the percentage of companies, and 't' is the independent variable representing time. This concept is crucial as it ties the relationship between time and sustained business operation to a predictable, linear pattern.

Slope and Intercept

The slope of a linear function, given by the coefficient of the variable (in this case, -3), indicates the rate of decline or increase; here, it shows that the percentage of companies decreases by 3 units for each additional unit of time. The y-intercept, 50 in the equation, represents the starting value of the percentage when time t is zero. Understanding these helps predict trends and interpret real-world data.

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