Finding The Equation Of A Linear Function Given Two Points

In the realm of mathematics, linear functions hold a position of fundamental importance. These functions, characterized by their straight-line graphs, serve as versatile tools for modeling a wide array of real-world phenomena. From the simple relationship between distance and time at a constant speed to the more complex dynamics of supply and demand in economics, linear functions provide a clear and concise way to represent direct proportionality. At their core, linear functions embody the concept of a constant rate of change, a cornerstone of mathematical analysis. This constant rate, known as the slope, dictates the steepness and direction of the line, while the y-intercept marks the point where the line intersects the vertical axis. Together, these two parameters define the unique character of a given linear function, making it possible to predict outputs for any given input and to trace the function's path across the coordinate plane. Mastering linear functions is not just an academic exercise; it's an essential step towards unlocking a deeper understanding of the mathematical world and its applications in everyday life. By grasping the principles of slope, intercept, and the equation of a line, students and practitioners alike gain a powerful toolkit for problem-solving and analytical thinking. This foundational knowledge paves the way for exploring more advanced mathematical concepts, such as calculus and differential equations, where linear functions often serve as building blocks for more complex models.

The specific challenge we are addressing here is a classic problem in linear algebra: finding the equation of a linear function when given two points that lie on its graph. This scenario is incredibly common in both theoretical mathematics and practical applications. Imagine, for instance, that you're tracking the growth of a plant over time. You might have two data points: after 10 days, the plant is 2 inches tall, and after -6 days (i.e., 6 days ago), it was 6 inches tall. Assuming the plant's growth is approximately linear, how can you determine a function that models its height at any given time? This is precisely the kind of problem our method addresses. The key insight is that two points uniquely define a line in a two-dimensional plane. Therefore, if we can find the slope and y-intercept of the line passing through these points, we can write the equation of the linear function in slope-intercept form, which is a standard and highly useful representation. This form, y = mx + b, explicitly shows the slope (m) and the y-intercept (b), making it easy to visualize the function's behavior and make predictions. The process of finding this equation involves several steps, including calculating the slope using the two given points, and then using either point along with the slope to solve for the y-intercept. This approach is not only mathematically sound but also provides a clear and systematic way to tackle a wide range of problems involving linear relationships.

To solve this problem, we'll follow a step-by-step approach, making use of the fundamental properties of linear functions and their representation in slope-intercept form. The slope-intercept form of a linear equation is given by ${ f(x) = mx + b }$, where ${ m }$ represents the slope of the line and ${ b }$ represents the y-intercept. Our goal is to determine the values of ${ m }$ and ${ b }$ using the given information:

Step 1: Calculate the Slope (m)

The slope of a line passing through two points ${ (x_1, y_1) }$ and ${ (x_2, y_2) }$ is given by the formula: ${ m = \frac{y_2 - y_1}{x_2 - x_1} }$

In our case, we have two points: (10, 2) and (-6, 6). Let's assign these values: ${ x_1 = 10, y_1 = 2 }$ ${ x_2 = -6, y_2 = 6 }$

Now, substitute these values into the slope formula: ${ m = \frac{6 - 2}{-6 - 10} = \frac{4}{-16} = -\frac{1}{4} }$

So, the slope of the linear function is ${ -\frac{1}{4} }$. This negative slope indicates that the line is decreasing as x increases, meaning the function's value goes down as we move from left to right on the graph.

Step 2: Find the y-intercept (b)

Now that we have the slope, we can use the slope-intercept form ${ f(x) = mx + b }$ and one of the given points to solve for ${ b }$. Let's use the point (10, 2). This means we'll substitute ${ x = 10 }$ and ${ f(x) = 2 }$ into the equation: ${ 2 = -\frac{1}{4}(10) + b }$

Now, solve for ${ b }$: ${ 2 = -\frac{10}{4} + b }$ ${ 2 = -\frac{5}{2} + b }$ ${ b = 2 + \frac{5}{2} }$ ${ b = \frac{4}{2} + \frac{5}{2} }$ ${ b = \frac{9}{2} }$

So, the y-intercept is ${ \frac{9}{2} }$, or 4.5. This is the point where the line crosses the y-axis, and it represents the value of the function when ${ x = 0 }$.

Step 3: Write the Equation in Slope-Intercept Form

Now that we have both the slope ${ m = -\frac{1}{4} }$ and the y-intercept ${ b = \frac{9}{2} }$, we can write the equation of the linear function in slope-intercept form: ${ f(x) = -\frac{1}{4}x + \frac{9}{2} }$

This is the final equation of the linear function that passes through the points (10, 2) and (-6, 6). This equation allows us to calculate the function's value for any given x. For example, if we wanted to find the value of the function at ${ x = 4 }$, we would simply substitute 4 for x in the equation: ${ f(4) = -\frac{1}{4}(4) + \frac{9}{2} = -1 + \frac{9}{2} = \frac{7}{2} }$

So, ${ f(4) = \frac{7}{2} }$. This demonstrates the power of having the equation of the linear function: it allows us to easily find the function's value at any point, and it provides a clear and concise way to represent the relationship between x and f(x).

Slope: Measuring the Steepness and Direction

In the context of linear functions, the slope serves as a crucial indicator of the line's steepness and direction. Numerically, it quantifies the rate at which the function's output changes relative to its input. A positive slope signifies an upward trend, meaning that as the input (x) increases, the output (f(x)) also increases. Conversely, a negative slope indicates a downward trend, where an increase in input leads to a decrease in output. The magnitude of the slope further refines this understanding: a larger absolute value of the slope corresponds to a steeper line, while a smaller absolute value suggests a gentler incline or decline. Mathematically, the slope (m) is defined as the change in y divided by the change in x, often expressed as ${ \frac{\Delta y}{\Delta x} }$ or ${ \frac{rise}{run} }$. This ratio captures the fundamental characteristic of linear functions: their constant rate of change. No matter which two points you choose on the line, the slope calculated between them will always be the same. This consistency is a defining feature of linear relationships and makes the slope a powerful tool for analyzing and predicting the behavior of these functions. In practical terms, the slope can represent various rates of change, such as speed (change in distance over time), cost per unit (change in cost over quantity), or the growth rate of a population (change in population size over time). Understanding the slope allows us to interpret the meaning of a linear function in real-world contexts and to make informed decisions based on the relationship it represents.

Y-intercept: The Starting Point

The y-intercept, denoted as 'b' in the slope-intercept form ${ f(x) = mx + b }$, marks the point where the line intersects the y-axis. This intersection occurs when the input value, x, is equal to zero. Therefore, the y-intercept represents the value of the function when the input is zero, making it a crucial reference point for understanding the function's behavior. In graphical terms, the y-intercept is the vertical coordinate of the point where the line crosses the y-axis. It provides a visual anchor for the line and helps to define its position on the coordinate plane. In practical applications, the y-intercept often has a meaningful interpretation related to the initial condition or starting value of the scenario being modeled. For example, if a linear function represents the cost of a service based on the number of hours used, the y-intercept might represent a fixed fee or initial charge that applies regardless of how many hours are used. Similarly, if a linear function models the height of a plant over time, the y-intercept could represent the plant's initial height before any growth occurred. Understanding the y-intercept allows us to interpret the practical meaning of a linear function in specific contexts and to make sense of the starting conditions or baseline values being represented. It complements the information provided by the slope, offering a complete picture of the linear relationship being modeled.

Slope-Intercept Form: A Powerful Representation

The slope-intercept form, expressed as ${ f(x) = mx + b }$, is a cornerstone representation of linear functions in mathematics. This form is celebrated for its clarity and utility in revealing the key characteristics of a line: its slope (m) and y-intercept (b). By simply examining the equation in slope-intercept form, one can immediately identify the line's steepness, direction, and the point at which it crosses the y-axis. This directness makes the slope-intercept form invaluable for both analysis and graphing linear functions. The slope, as previously discussed, quantifies the rate of change of the function, while the y-intercept pinpoints the function's value when the input is zero. Together, these two parameters provide a comprehensive description of the line's position and orientation on the coordinate plane. Moreover, the slope-intercept form facilitates the prediction of outputs for any given input. By substituting a value for x into the equation, one can readily calculate the corresponding value of f(x). This predictive capability is essential in numerous applications, from forecasting trends to solving real-world problems modeled by linear functions. The slope-intercept form also serves as a bridge to other forms of linear equations, such as the point-slope form and the standard form. It allows for easy conversion between these different representations, providing flexibility in problem-solving and analysis. In essence, the slope-intercept form is a powerful tool for understanding, manipulating, and applying linear functions in a wide range of mathematical and practical contexts.

In summary, we have successfully determined the equation of the linear function that passes through the points (10, 2) and (-6, 6). By employing the fundamental concepts of slope and y-intercept, we systematically calculated the slope as ${ -\frac{1}{4} }$ and the y-intercept as ${ \frac{9}{2} }$. These values were then incorporated into the slope-intercept form, yielding the equation:

${ f(x) = -\frac{1}{4}x + \frac{9}{2} }$

This equation represents the unique linear function that satisfies the given conditions. It encapsulates the relationship between the input (x) and the output (f(x)), allowing us to predict the function's value for any x and to visualize its behavior as a straight line on the coordinate plane. The significance of this solution extends beyond the specific problem at hand. The process of finding the equation of a linear function from two points is a fundamental skill in mathematics and has wide-ranging applications in various fields. From physics and engineering to economics and computer science, linear functions serve as building blocks for more complex models and analyses. The ability to determine the equation of a linear function empowers us to analyze data, make predictions, and solve real-world problems involving linear relationships. Furthermore, this exercise reinforces our understanding of key mathematical concepts, such as slope, y-intercept, and the slope-intercept form. It highlights the importance of systematic problem-solving and the power of mathematical tools in unraveling complex relationships. By mastering this skill, we gain a valuable foundation for further exploration in mathematics and its applications.