- Identify intercepts by setting variables to zero: x-intercept \((3, 0)\) and y-intercept \((0, -3)\).
- Express the equation in function notation by solving for \(y\): \(y = x - 3\).
- State the final function: \(f(x) = x - 3\).
- Final Answer: \(\boxed{f(x) = x - 3}\)
Explanation
- Problem Analysis
We are given the linear equation \(x - y = 3\). To understand the graph of this line, we need to find its intercepts, which are the points where the line crosses the x-axis and the y-axis. Additionally, we need to rewrite this equation in the standard function notation \(f(x) = mx + b\). The x-intercept is found by setting \(y = 0\) and solving for \(x\), and the y-intercept is found by setting \(x = 0\) and solving for \(y\). These two points provide enough information to draw the line on a coordinate plane.
- Finding the Intercepts
To find the x-intercept, we substitute \(y = 0\) into the original equation:
\[x - 0 = 3 \implies x = 3\]
Thus, the x-intercept occurs at the point \((3, 0)\).
To find the y-intercept, we substitute \(x = 0\) into the original equation:
\[0 - y = 3 \implies -y = 3 \implies y = -3\]
Thus, the y-intercept occurs at the point \((0, -3)\). These two points, \((3, 0)\) and \((0, -3)\), define the line on the Cartesian plane.
- Function Notation Representation
To express the equation \(x - y = 3\) in function notation \(f(x) = mx + b\), we must isolate \(y\) on one side. Adding \(y\) to both sides of the equation gives:
\[x = y + 3\]
Subtracting 3 from both sides results in:
\[y = x - 3\]
By definition, if \(y\) is a function of \(x\), we can write this as \(f(x) = y\). Therefore, the function notation for the line is:
\[f(x) = x - 3\]
This confirms that the slope \(m\) is \(1\) and the y-intercept \(b\) is \(-3\), which matches our intercept calculation.
Examples
In real life, linear equations like this are used to model relationships with a constant rate of change. For example, if you are saving money for a trip, you might start with a deficit (the y-intercept) and save a fixed amount each week (the slope). Graphing this helps visualize your progress over time, showing exactly when your balance will reach zero (the x-intercept) and when you will have positive funds. Understanding how to switch between different forms of an equation allows you to analyze these trends effectively.