- Identify the given equation as a point-slope form \(y - y_1 = m(x - x_1)\) with slope \(m = \frac{1}{2}\) and point \((4, 8)\).
- Distribute the slope to the terms inside the parentheses to get \(y - 8 = \frac{1}{2}x - 2\).
- Add 8 to both sides of the equation to isolate \(y\), resulting in the slope-intercept form \(f(x) = \frac{1}{2}x + 6\).
- The correct linear function is \(\boxed{f(x) = \frac{1}{2}x + 6}\).
Explanation
- Analyzing the Point-Slope Equation
The given equation is in point-slope form, which is represented as \(y - y_1 = m(x - x_1)\), where \(m\) is the slope and \((x_1, y_1)\) is a known point on the line. For the equation \(y - 8 = \frac{1}{2}(x - 4)\), we can identify the slope as \(m = \frac{1}{2}\) and the point on the line as \((4, 8)\). Our objective is to rewrite this equation into the slope-intercept form, \(f(x) = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept.
- Converting to Slope-Intercept Form
To express the equation in the form \(y = mx + b\), we start by distributing the slope \(\frac{1}{2}\) on the right side of the equation:
\[y - 8 = \frac{1}{2}x - \frac{1}{2} \times 4\]
Simplifying the constant term on the right side:
\[y - 8 = \frac{1}{2}x - 2\]
Next, to isolate \(y\), we add 8 to both sides of the equation:
\[y = \frac{1}{2}x - 2 + 8\]
Combining the constant terms \(-2\) and \(8\), we get:
\[y = \frac{1}{2}x + 6\]
Therefore, the linear function is \(f(x) = \frac{1}{2}x + 6\).
- Conclusion
Comparing our derived function \(f(x) = \frac{1}{2}x + 6\) with the given choices, it matches the second option. The linear function that represents the line given by the point-slope equation \(y - 8 = \frac{1}{2}(x - 4)\) is \(\boxed{f(x) = \frac{1}{2}x + 6}\).
Examples
In real-life scenarios, linear functions are frequently used to model costs or relationships between two variables. For example, if a taxi charges a flat fee of \(\$6\) (the y-intercept) plus \(\$0.50\) for every mile traveled (the slope), the cost \(f(x)\) for \(x\) miles is modeled by the function \(f(x) = 0.5x + 6\). Converting equations from different forms (like point-slope) to slope-intercept form helps in quickly identifying these starting costs and rates of change, making it easier to predict expenses or analyze trends.