- Use the given equation \(y - 3 = \frac{1}{2}(x + 2)\) as the primary definition of the line.
- Set the variable \(x\) to 0 to find the \(y\)-intercept.
- Perform the algebraic calculation: \(y - 3 = \frac{1}{2}(0 + 2)\), which simplifies to \(y - 3 = 1\), resulting in \(y = 4\).
- Identify the point on the coordinate plane as \(\boxed{(0, 4)}\).
Explanation
- Analysis of the Linear Equation
The linear function is defined by the equation \(y - 3 = \frac{1}{2}(x + 2)\). This is given in point-slope form, \(y - y_1 = m(x - x_1)\), where \(m\) is the slope and \((x_1, y_1)\) is a point on the line. In this specific equation, the slope \(m\) is \(\frac{1}{2}\) and a point on the line is \((-2, 3)\). Although a table is provided with the point \((-6, -5)\), we prioritize the defining equation to determine the properties of the line.
- Calculating the y-intercept
The \(y\)-intercept of a linear function occurs at the point where the \(x\)-coordinate is equal to 0. To find the \(y\)-intercept, we substitute \(x = 0\) into the given equation and solve for \(y\):
\[y - 3 = \frac{1}{2}(0 + 2)\]
Now, simplify the right side of the equation:
\[y - 3 = \frac{1}{2}(2)\]
\[y - 3 = 1\]
Finally, solve for \(y\) by adding 3 to both sides:
\[y = 1 + 3 = 4\]
The \(y\)-intercept is the point \((0, 4)\).
- Conclusion
By setting \(x = 0\) in the provided linear equation \(y - 3 = \frac{1}{2}(x + 2)\), we determined that the corresponding \(y\)-value is 4. Thus, the line crosses the \(y\)-axis at the point \((0, 4)\). This corresponds to the last option provided. The final answer is \(\boxed{(0, 4)}\).
Examples
Understanding how to identify the y-intercept of a line is fundamental in many real-world applications, such as analyzing business costs. For example, if a company's total production cost is modeled by a linear equation, the y-intercept represents the "fixed costs" (costs that exist even when zero units are produced, such as rent). By finding the y-intercept of the line \(y = \frac{1}{2}x + 4\), we see that the fixed costs are \(\$4\). This helps business managers plan budgets and understand their baseline expenses before production even begins.