Find The Linear Function Passing Through (-21, 15) And (-7, 9)
Finding the equation of a line that passes through two given points is a fundamental concept in algebra. This article will delve into the step-by-step process of determining the linear function whose graph passes through the points (-21, 15) and (-7, 9). We'll explore the underlying principles, formulas, and calculations involved, ensuring a clear understanding of how to arrive at the solution. This comprehensive guide is designed to help students, educators, and anyone interested in linear functions and their applications.
Understanding Linear Functions
Before we dive into the specifics of our problem, let's establish a solid understanding of linear functions. A linear function is a function whose graph is a straight line. It can be represented in the slope-intercept form, which is written as:
y = mx + b
where:
yrepresents the dependent variable.xrepresents the independent variable.mrepresents the slope of the line, indicating its steepness and direction.brepresents the y-intercept, the point where the line crosses the y-axis.
The slope (m) is a crucial aspect of a linear function. It quantifies the rate at which the y-value changes with respect to the x-value. A positive slope indicates an upward-sloping line, while a negative slope indicates a downward-sloping line. A slope of zero represents a horizontal line. The slope can be calculated using two points on the line, (x1, y1) and (x2, y2), using the following formula:
m = (y2 - y1) / (x2 - x1)
The y-intercept (b) is the point where the line intersects the y-axis. At this point, the x-coordinate is always 0. The y-intercept is essential for defining the vertical position of the line on the coordinate plane.
In our problem, we are given two points, (-21, 15) and (-7, 9), and our goal is to find the linear function that passes through these points. This means we need to determine the values of m (the slope) and b (the y-intercept) that define the line. The slope-intercept form of a linear equation, y = mx + b, provides a clear and concise way to represent the relationship between x and y. By identifying the slope and y-intercept, we can fully describe the line and its behavior. The ability to find the equation of a line given two points is a fundamental skill in algebra and has numerous applications in various fields, such as physics, engineering, and economics.
Step 1: Calculate the Slope (m)
Our first step in finding the linear function is to calculate the slope (m). We have two points: (-21, 15) and (-7, 9). Let's designate these points as follows:
- (x1, y1) = (-21, 15)
- (x2, y2) = (-7, 9)
Now, we can use the slope formula:
m = (y2 - y1) / (x2 - x1)
Substitute the values into the formula:
m = (9 - 15) / (-7 - (-21))
Simplify the equation:
m = -6 / 14
Reduce the fraction to its simplest form:
m = -3 / 7
Therefore, the slope (m) of the line is -3/7. This negative slope indicates that the line slopes downward from left to right. For every 7 units we move to the right along the x-axis, the line descends 3 units along the y-axis. The slope is a crucial characteristic of the line, as it determines its steepness and direction. A steeper line will have a larger absolute value of the slope, while a flatter line will have a smaller absolute value. Understanding the slope is essential for interpreting the relationship between the variables represented by the line.
By accurately calculating the slope, we have taken a significant step toward finding the complete equation of the line. The slope provides valuable information about the line's inclination, and now we can use this information, along with one of the given points, to determine the y-intercept. This process involves substituting the slope and the coordinates of a point into the slope-intercept form (y = mx + b) and solving for b. This will give us the vertical position of the line and allow us to write the complete equation of the linear function.
Step 2: Find the y-intercept (b)
Now that we have the slope (m = -3/7), we can proceed to find the y-intercept (b). To do this, we will use the slope-intercept form of a linear equation, which is:
y = mx + b
We can substitute the slope (-3/7) and the coordinates of one of the given points into this equation. Let's use the point (-7, 9). Substituting these values, we get:
9 = (-3/7) * (-7) + b
Simplify the equation:
9 = 3 + b
Now, solve for b:
b = 9 - 3
b = 6
Therefore, the y-intercept (b) is 6. This means that the line intersects the y-axis at the point (0, 6). The y-intercept is a critical component of the linear equation, as it defines the vertical position of the line. Knowing the y-intercept, along with the slope, allows us to fully describe the line and its behavior.
Alternatively, we could have used the other point, (-21, 15), to find the y-intercept. Substituting these values into the slope-intercept form, we get:
15 = (-3/7) * (-21) + b
Simplify the equation:
15 = 9 + b
Solve for b:
b = 15 - 9
b = 6
As expected, we arrive at the same y-intercept (b = 6). This consistency confirms the accuracy of our calculations. The fact that we can use either point to find the y-intercept highlights the fundamental property of a linear function: all points on the line satisfy the same equation. With the slope and y-intercept now determined, we have all the necessary information to write the complete equation of the linear function.
Step 3: Write the Equation of the Linear Function
With the slope (m = -3/7) and the y-intercept (b = 6), we can now write the equation of the linear function in slope-intercept form:
y = mx + b
Substitute the values of m and b:
y = (-3/7)x + 6
This is the equation of the linear function whose graph passes through the points (-21, 15) and (-7, 9). This equation completely defines the line, allowing us to predict the y-value for any given x-value and vice versa. The slope (-3/7) indicates the rate of change of y with respect to x, while the y-intercept (6) specifies the point where the line crosses the y-axis.
To express the equation in function notation, we can replace y with f(x):
f(x) = (-3/7)x + 6
This notation emphasizes that the y-value is a function of x. The equation f(x) = (-3/7)x + 6 represents the unique linear function that satisfies the given conditions. We can verify this equation by substituting the x-coordinates of the given points into the function and checking if the resulting y-values match the given y-coordinates.
For example, let's substitute x = -21:
f(-21) = (-3/7) * (-21) + 6
f(-21) = 9 + 6
f(-21) = 15
This matches the y-coordinate of the point (-21, 15). Similarly, let's substitute x = -7:
f(-7) = (-3/7) * (-7) + 6
f(-7) = 3 + 6
f(-7) = 9
This matches the y-coordinate of the point (-7, 9). These verifications confirm that the equation f(x) = (-3/7)x + 6 accurately represents the linear function passing through the given points. This process of finding the equation of a line is a fundamental skill in algebra and has numerous applications in various fields.
Conclusion
In this article, we successfully determined the linear function whose graph passes through the points (-21, 15) and (-7, 9). We followed a step-by-step process, which included calculating the slope, finding the y-intercept, and writing the equation in slope-intercept form. The resulting equation is:
f(x) = (-3/7)x + 6
This equation provides a complete description of the line, allowing us to understand its slope, y-intercept, and the relationship between x and y. The ability to find the equation of a line given two points is a fundamental skill in mathematics with widespread applications in various fields. This process reinforces the understanding of linear functions and their graphical representation. By mastering these concepts, you can effectively analyze and model real-world phenomena that exhibit linear relationships.
The process we followed highlights the importance of understanding the key components of a linear function: the slope and the y-intercept. The slope defines the direction and steepness of the line, while the y-intercept determines its vertical position. By carefully calculating these values, we can accurately represent any linear relationship. This skill is not only valuable in mathematics but also in fields such as physics, engineering, economics, and computer science, where linear models are frequently used to analyze data and make predictions. The techniques discussed in this article provide a solid foundation for further exploration of linear functions and their applications.
