Understanding the Slope of f(t) = 2t – 6

Calculus is a challenging subject that requires a deep understanding of mathematical concepts and principles. One concept that every calculus student encounters is the slope of a function. Slope is a measure of how steep a line is in graphical terms, and its value can help us determine the direction and rate of change of a function over a given interval. In this article, we will explore the slope of a simple linear function, f(t) = 2t – 6, and learn how to calculate and interpret its slope.

Before diving into the details of slope calculation, it is essential to understand what a linear function is. Linear functions describe relationships between two variables that are proportional to each other. In our case, the function f(t) relates the independent variable t (time) to the dependent variable f(t), which measures the output or value of the function at any given time. The function has a slope of 2, which means that for every increase of one unit in the input value t, the output value f(t) increases by 2 units.

To visualize the function graphically, we can plot several points on the coordinate axes and connect them with a straight line. For example, if we substitute t = 0 into the function, we obtain f(0) = 2(0) – 6 = -6. This means that when the time is zero, the output value of the function is -6. Similarly, if we plug in t = 1, we get f(1) = 2(1) – 6 = -4. This tells us that when the time is one unit, the output value of the function is -4. If we continue this process by substituting other values of t, we can generate a table of values that represents the function of f(t).

t | f(t)

———

0 | -6

1 | -4

2 | -2

3 | 0

4 | 2

Once we have plotted these points on the coordinate plane, we can connect them with a straight line that represents the function of f(t). The slope of this line is given by the formula:

m = (y2 – y1)/(x2 – x1)

where m stands for the slope, (x1, y1) and (x2, y2) are two points on the line, and the numerator and denominator represent the differences in the y-coordinates and x-coordinates, respectively. In our case, we can choose any two points on the line to calculate the slope, but it is best to pick the endpoints since they are the easiest to identify. Let us take the points (0, -6) and (4, 2) to calculate the slope of the line.

m = (2 – (-6))/(4 – 0)

m = 8/4

m = 2

This confirms that the slope of the function f(t) is indeed 2, as we expected.

So what does a slope of 2 mean for our function? One interpretation is that the function is increasing at a constant rate of 2 units per one unit of time. This means that if we draw a tangent line to the function at any point, the slope of that line will be equal to 2, and it will represent the rate of change of the function at that particular moment. Moreover, a positive slope indicates a positive correlation between the variables, which means that as time increases, so does the output value of the function. Conversely, a negative slope would signify a negative correlation, where an increase in time leads to a decrease in the function’s output value.

In conclusion, the slope of a function is a critical concept in calculus that measures the rate of change and direction of a function over a given interval. The slope of a linear function, such as f(t) = 2t – 6, can be calculated using the formula m = (y2 – y1)/(x2 – x1), where m stands for slope, and (x1, y1) and (x2, y2) are two points on the line. A slope of 2 in our function indicates a constant increase of 2 units per one unit of time and signifies a positive correlation between the variables. By understanding the concept of slope and its interpretation, we can gain a deeper insight into the workings of functions and better understand how they behave over time.