For the following exercises, use the graph in Figure 2.45,...

For the following exercises, use the graph in Figure $2.45,$ which shows the profit, $y,$ in thousands of dollars, of a company in a given year, $t,$ where $t$ represents the number of years since $1980 .$ Find the linear function $y,$ where $y$ depends on $t,$ the number of years since 1980 .

For the following exercises, use the graph in Figure $2.45,$ which shows the profit, $y,$ in thousands of dollars, of a company in a given year, $t,$ where $t$ represents the number of years since $1980 .$

Find the linear function $y,$ where $y$ depends on $t,$ the number of years since 1980 .

Key Concepts

Linear Functions

A linear function is a mathematical relationship between two variables where the graph of the function is a straight line. This function is typically written in the slope-intercept form, y = mx + b, where m represents the slope (or rate of change) and b represents the y-intercept (the value of y when x = 0). Linear functions are used to model situations with constant rates of change.

Slope

The slope of a linear function measures the rate at which the dependent variable changes with respect to a change in the independent variable. It is calculated as the ratio of the vertical change to the horizontal change between any two points on the line. In real-world contexts, the slope can represent rates such as speed, growth, or profit accumulation over time.

Y-intercept

The y-intercept is the point on the graph where the line crosses the y-axis. It represents the value of the dependent variable when the independent variable is zero. In time-dependent models, this typically corresponds to the starting value at the reference time point.

Graph Interpretation

Interpreting a graph involves identifying key features such as the slope and y-intercept by analyzing the plotted line. This process allows one to construct the corresponding linear equation that models the relationship between the variables, facilitating predictions and understanding of the underlying trend.

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