Finding The Equation Of A Linear Function Passing Through Two Points

In mathematics, a linear function is a function whose graph is a straight line. These functions are fundamental in various fields, from basic algebra to advanced calculus and real-world applications. Determining the formula for a linear function is a common task, especially when given two points that the line passes through. This article will guide you through the process of finding the formula for the linear function whose graph passes through the points $(-21, 15)$ and $(-7, 9)$. We'll explore the underlying concepts, step-by-step calculations, and practical insights to ensure a comprehensive understanding.

Understanding Linear Functions

Before diving into the calculations, it's crucial to understand the basic form of a linear function. A linear function can be represented by the equation:

$$f(x) = mx + b$$

Where:

• $f(x)$ is the value of the function at $x$.
• $m$ is the slope of the line, representing the rate of change of $f(x)$ with respect to $x$.
• $b$ is the y-intercept, the point where the line crosses the y-axis.

To find the formula for a specific linear function, we need to determine the values of $m$ and $b$. When given two points, we can first calculate the slope and then use one of the points to find the y-intercept. This approach provides a systematic way to define the linear function that fits the given data.

Calculating the Slope ($m$)

The slope of a line is a measure of its steepness and direction. It tells us how much the y-value changes for each unit change in the x-value. Given two points $(x_1, y_1)$ and $(x_2, y_2)$, the slope $m$ can be calculated using the formula:

$$m = \frac{y_2 - y_1}{x_2 - x_1}$$

This formula essentially calculates the "rise over run," where the rise is the difference in the y-values and the run is the difference in the x-values. By understanding and applying this formula, we can accurately determine the slope of any line given two points on that line.

Finding the Y-intercept ($b$)

The y-intercept is the point where the line intersects the y-axis, which occurs when $x = 0$. Once we have calculated the slope $m$, we can use one of the given points and the slope-intercept form of the linear equation ($f(x) = mx + b$) to solve for $b$. This involves substituting the x and y values of the point, along with the calculated slope, into the equation and solving for the unknown $b$. This method provides a straightforward way to find the y-intercept, which is essential for defining the complete linear function.

Step-by-Step Solution

Now, let's apply these concepts to find the formula for the linear function that passes through the points $(-21, 15)$ and $(-7, 9)$.

Step 1: Calculate the Slope

Given the points $(-21, 15)$ and $(-7, 9)$, we can label them as follows:

• $(x_1, y_1) = (-21, 15)$
• $(x_2, y_2) = (-7, 9)$

Using the slope formula:

$$m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{9 - 15}{-7 - (-21)}$$

$$m = \frac{-6}{-7 + 21} = \frac{-6}{14}$$

$$m = -\frac{3}{7}$$

So, the slope of the line is $- \frac{3}{7}$. This negative slope indicates that the line is decreasing, meaning that as x increases, y decreases. The magnitude of the slope, $3/7$, tells us the rate at which y decreases for each unit increase in x. Understanding the slope is crucial for visualizing and interpreting the linear function.

Step 2: Find the Y-intercept

Now that we have the slope, we can use one of the points and the slope-intercept form ($f(x) = mx + b$) to find the y-intercept $b$. Let's use the point $(-7, 9)$:

$$9 = -\frac{3}{7}(-7) + b$$

$$9 = 3 + b$$

Subtract 3 from both sides:

$$b = 9 - 3$$

$$b = 6$$

Thus, the y-intercept is 6. This means that the line crosses the y-axis at the point (0, 6). Knowing the y-intercept provides a fixed point on the line, which is essential for accurately graphing and understanding the linear function. The y-intercept, along with the slope, completely defines the linear function.

Step 3: Write the Formula

Now that we have both the slope $m = -\frac{3}{7}$ and the y-intercept $b = 6$, we can write the formula for the linear function:

$$f(x) = -\frac{3}{7}x + 6$$

This is the linear function that passes through the points $(-21, 15)$ and $(-7, 9)$. The equation represents a straight line on a coordinate plane, and any point on this line will satisfy this equation. The formula provides a concise and precise way to describe the relationship between x and y for this specific linear function.

Verification

To ensure our formula is correct, we can verify it by plugging in the original points $(-21, 15)$ and $(-7, 9)$ into the equation:

For $(-21, 15)$:

$$f(-21) = -\frac{3}{7}(-21) + 6$$

$$f(-21) = 9 + 6$$

$$f(-21) = 15$$

For $(-7, 9)$:

$$f(-7) = -\frac{3}{7}(-7) + 6$$

$$f(-7) = 3 + 6$$

$$f(-7) = 9$$

Both points satisfy the equation, so our formula is correct. This verification step is crucial in mathematics to ensure the accuracy of the solution. By plugging the original points back into the derived equation, we can confirm that the equation indeed represents the line passing through those points.

Practical Applications and Implications

Linear functions have numerous practical applications in various fields. They are used to model relationships between two variables that exhibit a constant rate of change. Here are a few examples:

1. Economics: Linear functions can represent cost functions, revenue functions, and supply-demand relationships. For instance, a linear cost function might describe the total cost of production as a function of the number of units produced.
2. Physics: Linear functions are used to describe motion with constant velocity, where the distance traveled is a linear function of time. The equation $d = vt + d_0$, where $d$ is distance, $v$ is velocity, $t$ is time, and $d_0$ is the initial distance, is a linear function.
3. Computer Graphics: Linear interpolation is used to create smooth transitions between colors or positions in animations and 3D graphics. This involves using linear functions to calculate intermediate values between two known values.
4. Data Analysis: Linear regression is a statistical technique used to model the relationship between two variables by fitting a linear equation to observed data. This is commonly used to make predictions based on historical data.
5. Engineering: Linear functions are used in circuit analysis, structural analysis, and control systems. For example, Ohm's Law, $V = IR$, is a linear function relating voltage ($V$), current ($I$), and resistance ($R$).

Understanding linear functions is essential for problem-solving and modeling real-world phenomena. The ability to find the formula for a linear function given two points is a fundamental skill in mathematics and has wide-ranging implications in various disciplines. The process we've outlined provides a systematic approach to solving such problems, ensuring accuracy and clarity.

Conclusion

In this article, we've walked through the process of finding the formula for a linear function whose graph passes through the points $(-21, 15)$ and $(-7, 9)$. We began by understanding the basic form of a linear function, $f(x) = mx + b$, and the significance of the slope $m$ and y-intercept $b$. We then calculated the slope using the slope formula and found the y-intercept by substituting one of the points and the slope into the linear equation. Finally, we combined these values to write the formula for the linear function:

$$f(x) = -\frac{3}{7}x + 6$$

We verified the formula by plugging in the original points and demonstrated its correctness. Additionally, we discussed the practical applications and implications of linear functions in various fields, highlighting their importance in problem-solving and modeling real-world situations.

Understanding and applying these concepts allows you to confidently find the equation of any linear function given two points, a skill that is invaluable in mathematics and beyond. Whether you're a student learning algebra or a professional working with mathematical models, mastering the techniques discussed here will enhance your analytical and problem-solving capabilities. The ability to work with linear functions is a cornerstone of mathematical literacy and is essential for success in many academic and professional pursuits. By following the step-by-step process outlined in this article, you can confidently tackle similar problems and deepen your understanding of linear functions.