Choose a linear function for the line represented by the point-slope equation $y-5=3(x-2)$.
We are given the point-slope form of a linear equation, \(y - 5 = 3(x - 2)\), and our objective is to rewrite this equation in the slope-intercept form, \(f(x) = mx + b\), where \(m\) is the slope and \(b\) is the \(y\)-intercept. This form allows us to easily identify the function representing the line.
To convert the equation \(y - 5 = 3(x - 2)\) into slope-intercept form, we first distribute the slope \(3\) across the terms inside the parentheses:
\[y - 5 = 3 \cdot x - 3 \cdot 2\]
\[y - 5 = 3x - 6\]
Next, we isolate \(y\) on the left side by adding \(5\) to both sides of the equation:
\[y = 3x - 6 + 5\]
Combining the constant terms \(-6\) and \(5\), we get:
\[y = 3x - 1\]
Thus, the function in slope-intercept form is \(f(x) = 3x - 1\).
Comparing our derived function \(f(x) = 3x - 1\) with the given choices:
The correct option is \(f(x) = 3x - 1\), which corresponds to the second choice. The final answer is \(\boxed{f(x) = 3x - 1}\).
Linear functions are used in various fields like economics, engineering, and physics to model relationships that have a constant rate of change. For example, if a taxi charges an initial fee of R\$5 and R\$3 for every kilometer traveled, the total cost \(y\) for a distance \(x\) can be expressed as \(y - 5 = 3x\), which is equivalent to \(y = 3x + 5\). Converting between forms of linear equations helps in quickly identifying key properties like the starting value (y-intercept) and the rate of increase (slope).