a linear function whose graph passes through the origin Sara Ahrenholz from ahrenholzsarasays.blogspot.com Linear Function Whose Graph Passes Through the Origin: A Comprehensive Guide Linear functions are one of the most basic and fundamental mathematical concepts that we learn in school. They are used in various disciplines, including physics, economics, engineering, and finance. In this article, we will explore the concept of a linear function whose graph passes through the origin.
What is a Linear Function?
A linear function is a mathematical function that has a constant rate of change. This means that the function increases or decreases at the same rate as the input variable. The general form of a linear function is y = mx + b, where m is the slope of the line, and b is the y-intercept.
What does it mean for a Graph to Pass Through the Origin?
When a graph passes through the origin, it means that the input variable (x) is equal to zero when the output variable (y) is also equal to zero. In other words, the point (0,0) is on the graph of the function.
How to Find the Equation of a Linear Function Whose Graph Passes Through the Origin
To find the equation of a linear function whose graph passes through the origin, we need to know the slope of the line. The slope can be found by taking any two points on the line and using the formula: m = (y2 - y1) / (x2 - x1) Since the graph passes through the origin, one of the points is (0,0). Let's say the other point is (x1, y1). Using the formula, we get: m = (y1 - 0) / (x1 - 0) = y1 / x1 Therefore, the equation of the linear function is y = mx, where m is the slope of the line.
Example
Let's say we have a linear function whose graph passes through the origin and has a slope of 2. The equation of the function is y = 2x. We can verify that the graph passes through the origin by plugging in x = 0 and y = 0: y = 2x 0 = 2(0) 0 = 0 Therefore, the point (0,0) is on the graph of the function.
Properties of a Linear Function Whose Graph Passes Through the Origin
There are several properties of a linear function whose graph passes through the origin, including:
The function is always increasing or decreasing at a constant rate.
The slope of the function is equal to the rate of change.
The y-intercept of the function is always zero.
Applications of a Linear Function Whose Graph Passes Through the Origin
Linear functions whose graphs pass through the origin have various applications in different fields. Some examples include:
In physics, the distance traveled by an object is proportional to its speed.
In economics, the production cost of a product is proportional to the number of units produced.
In engineering, the force required to move an object is proportional to its weight.
In finance, the amount of interest earned on an investment is proportional to the principal.
Conclusion
A linear function whose graph passes through the origin is a fundamental concept in mathematics that has various applications in different fields. It is a simple yet powerful tool that is used to model real-world phenomena. By understanding the properties and applications of linear functions, we can gain a deeper appreciation of the role of mathematics in our daily lives.
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