Calculating Slope Of A Linear Function From A Table

Linear functions are a cornerstone of mathematics, representing relationships where the change between two variables is constant. This concept is fundamental in various fields, from physics and engineering to economics and computer science. Grasping the essence of linear functions is crucial for anyone seeking a solid foundation in mathematical principles.

A linear function can be visualized as a straight line on a graph, and its behavior is fully characterized by its slope and y-intercept. The slope, often denoted by 'm', quantifies the steepness and direction of the line. It represents the rate at which the dependent variable (y) changes with respect to the independent variable (x). A positive slope indicates an increasing function, where y increases as x increases, while a negative slope indicates a decreasing function. A slope of zero signifies a horizontal line, where y remains constant regardless of x.

The y-intercept, denoted by 'b', is the point where the line crosses the y-axis. It represents the value of y when x is zero. Together, the slope and y-intercept uniquely define a linear function, allowing us to predict the output (y) for any given input (x).

The most common way to express a linear function is using the slope-intercept form: y = mx + b, where 'm' is the slope and 'b' is the y-intercept. This form provides a clear and concise way to understand the function's behavior. Another useful form is the point-slope form: y - y1 = m(x - x1), where (x1, y1) is a known point on the line and 'm' is the slope. This form is particularly helpful when you know a point on the line and its slope, but not the y-intercept.

Understanding the concept of slope is paramount. It is the heart of understanding the behavior of a linear function. The slope tells us how much the value of 'y' changes for every unit change in 'x'. It's the constant rate of change that defines a linear relationship. Think of it as the 'rise over run' – the vertical change (rise) divided by the horizontal change (run) between any two points on the line. A steep slope (large absolute value) means that y changes rapidly with x, while a shallow slope (small absolute value) means that y changes gradually with x.

In real-world scenarios, linear functions can model a wide range of phenomena. For example, the relationship between the number of hours worked and the amount earned at an hourly wage can be modeled as a linear function. Similarly, the distance traveled by a car at a constant speed can be represented by a linear function. Recognizing these linear relationships in everyday situations allows us to make predictions and solve practical problems. For instance, we can use a linear function to estimate the cost of a taxi ride based on the distance traveled, or to calculate the amount of time it will take to reach a destination driving at a constant speed.

Calculating Slope from a Table of Values

When a linear function is presented in a table format, determining the slope is straightforward. The key is to remember that the slope represents the constant rate of change. This means that for every unit change in 'x', the change in 'y' is consistent. To calculate the slope from a table, we can choose any two distinct points (x1, y1) and (x2, y2) from the table and apply the slope formula:

m = (y2 - y1) / (x2 - x1)

This formula calculates the change in 'y' (the rise) divided by the change in 'x' (the run) between the two chosen points. Since the function is linear, the slope calculated using any two points will be the same. This consistency is a defining characteristic of linear functions.

Let's illustrate this with a concrete example. Suppose we have the following table representing a linear function:

To find the slope, we can select any two points. Let's choose (-2, -2) as our (x1, y1) and (-1, 1) as our (x2, y2). Plugging these values into the slope formula, we get:

m = (1 - (-2)) / (-1 - (-2)) m = (1 + 2) / (-1 + 2) m = 3 / 1 m = 3

Therefore, the slope of the linear function represented by this table is 3. We can verify this by choosing other pairs of points from the table. For instance, let's choose (0, 4) and (1, 7):

m = (7 - 4) / (1 - 0) m = 3 / 1 m = 3

As expected, we obtain the same slope of 3. This reinforces the fact that the slope of a linear function is constant throughout the line.

Applying the Slope Formula to the Given Problem

Now, let's apply our knowledge of calculating slope from a table to solve the problem presented. We are given the following table representing a linear function:

The question asks us to determine the slope of this function. To do this, we will use the slope formula, m = (y2 - y1) / (x2 - x1), and select two points from the table.

Let's choose the points (-2, -2) and (-1, 1) as our (x1, y1) and (x2, y2), respectively. Plugging these values into the formula, we have:

m = (1 - (-2)) / (-1 - (-2)) m = (1 + 2) / (-1 + 2) m = 3 / 1 m = 3

Therefore, the slope of the function is 3.

We can verify this result by selecting another pair of points from the table. Let's choose (0, 4) and (1, 7):

m = (7 - 4) / (1 - 0) m = 3 / 1 m = 3

Again, we obtain a slope of 3, confirming our previous calculation. This reinforces the understanding that the slope of a linear function is constant, regardless of which two points are chosen for the calculation.

Conclusion and Answer

In conclusion, the slope of the linear function represented by the given table is 3. This signifies that for every unit increase in 'x', the value of 'y' increases by 3. Understanding the concept of slope is crucial for interpreting linear functions and their applications in various mathematical and real-world contexts.

The correct answer to the question "What is the slope of the function?" is:

C. 3

This comprehensive explanation provides a thorough understanding of linear functions, slope calculation, and the application of these concepts to solve the given problem. By understanding the fundamental principles behind linear functions and slope, individuals can confidently tackle similar problems and gain a deeper appreciation for the role of linear relationships in mathematics and beyond.