Finding The Slope Of The Line F(t) = 2t - 6 A Comprehensive Guide
The equation f(t) = 2t - 6 represents a linear function, and understanding its components is crucial in mathematics. This article will explore the slope of the line represented by this equation, and the concept of the y-intercept, by providing a detailed analysis, making it easy to understand how to identify these key features from a linear equation. We'll break down the equation f(t) = 2t - 6 step-by-step, highlighting the significance of the slope and y-intercept in the context of linear functions. Understanding these concepts is fundamental not only in algebra but also in various real-world applications, such as physics, engineering, and economics, where linear models are frequently used to describe relationships between variables.
Decoding the Linear Equation: Slope-Intercept Form
To effectively determine the slope, let's first understand the slope-intercept form of a linear equation. The slope-intercept form is generally expressed as y = mx + b, where m represents the slope and b represents the y-intercept. This form provides a clear and concise way to identify these essential characteristics of a line. In our case, the equation f(t) = 2t - 6 is already in slope-intercept form. We can easily see the correspondence between the general form y = mx + b and our specific equation. Here, f(t) is equivalent to y, t is equivalent to x, the coefficient of t (which is 2) is equivalent to m, and the constant term -6 is equivalent to b. Therefore, by aligning the given equation with the slope-intercept form, we can directly extract the values for the slope and the y-intercept. The slope (m) indicates the steepness and direction of the line, while the y-intercept (b) indicates the point where the line crosses the y-axis. Identifying these values is essential for graphing the line and understanding its behavior. The slope tells us how much the line rises or falls for each unit increase in the horizontal direction, while the y-intercept gives us a fixed point to start with when plotting the line. For instance, a positive slope indicates that the line rises as we move from left to right, while a negative slope indicates that the line falls. A larger absolute value of the slope means the line is steeper. The y-intercept, on the other hand, provides a starting point on the y-axis, which is crucial for accurately positioning the line on the coordinate plane. Understanding these aspects of the slope-intercept form allows us to quickly analyze and interpret linear equations, making it a fundamental tool in algebra and its applications.
Identifying the Slope: The Coefficient of t
In the equation f(t) = 2t - 6, the slope is represented by the coefficient of the variable t. In this case, the coefficient of t is 2. Therefore, the slope of the line is 2. The slope, often denoted by m, is a crucial characteristic of a linear equation, as it describes the rate at which the dependent variable (f(t)) changes with respect to the independent variable (t). A slope of 2 indicates that for every one-unit increase in t, the value of f(t) increases by 2 units. This means the line is ascending or going upwards as we move from left to right on the graph. The steepness of the line is directly proportional to the magnitude of the slope; a larger slope indicates a steeper line, while a smaller slope indicates a flatter line. For example, a line with a slope of 5 would be steeper than a line with a slope of 2. The sign of the slope also tells us about the direction of the line. A positive slope, as we have here, means the line rises from left to right, while a negative slope would mean the line falls from left to right. A zero slope would indicate a horizontal line. Understanding how to identify and interpret the slope is fundamental for analyzing linear relationships and making predictions based on linear models. In real-world contexts, the slope can represent various rates of change, such as the speed of an object, the rate of growth of a population, or the change in cost per unit increase in production. Therefore, being able to quickly determine the slope from a linear equation is an essential skill in both mathematics and its practical applications.
Unveiling the y-intercept: The Constant Term
The y-intercept is the point where the line intersects the y-axis. In the equation f(t) = 2t - 6, the y-intercept is the constant term, which is -6. This means that the line crosses the y-axis at the point (0, -6). The y-intercept, often denoted by b, is an essential feature of a linear equation as it provides a fixed reference point on the graph. It represents the value of the dependent variable (f(t)) when the independent variable (t) is zero. In our equation, the y-intercept of -6 indicates that when t is 0, f(t) is -6. This point is crucial for graphing the line because it gives us a specific location on the y-axis to start from. The y-intercept also has practical significance in various real-world contexts. For example, if f(t) represents the cost of a service and t represents the number of hours, the y-intercept would represent the fixed cost or the initial fee, regardless of the number of hours. Similarly, if f(t) represents the amount of water in a tank and t represents time, the y-intercept would represent the initial amount of water in the tank. Understanding the y-intercept allows us to interpret the initial state or value in a linear relationship, making it a valuable component in analyzing and applying linear models. Knowing both the slope and the y-intercept allows us to fully describe and graph a linear equation, making it a cornerstone of linear algebra.
Putting It All Together: Slope and y-intercept of f(t) = 2t - 6
Therefore, in the equation f(t) = 2t - 6, the slope is 2, and the y-intercept is -6. This means the line rises 2 units for every 1 unit increase in t, and it intersects the y-axis at the point (0, -6). Understanding these values allows us to visualize the line and its behavior. The slope of 2 tells us about the steepness and direction of the line. A positive slope indicates that the line is increasing as we move from left to right, and the value 2 tells us that for every unit increase in t, the value of f(t) increases by 2. This gives us a clear sense of how the line is inclined on the graph. The y-intercept of -6 provides a specific point where the line crosses the y-axis. This point (0, -6) is crucial for graphing the line because it gives us a fixed reference point. We can start plotting the line from this point and use the slope to find other points on the line. Together, the slope and the y-intercept completely define the line. We can use them to accurately graph the line, write its equation, and make predictions about the relationship between t and f(t). In practical applications, knowing the slope and y-intercept allows us to interpret the linear relationship in real-world terms. For example, if this equation represented the cost of a taxi ride, the slope would be the cost per mile, and the y-intercept would be the initial fee. Therefore, understanding how to identify and interpret the slope and y-intercept is fundamental for working with linear equations and their applications.
In conclusion, by analyzing the equation f(t) = 2t - 6 using the slope-intercept form, we've successfully identified that the slope of the line is 2 and the y-intercept is -6. This exercise highlights the importance of understanding the components of a linear equation for interpreting its graphical representation and real-world applications.
