- Identify the slope-intercept form as \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept.
- Substitute the given slope \(m = \frac{1}{4}\) and the point \((0,0)\) into the equation to find that the y-intercept \(b = 0\).
- Simplify the equation to \(y = \frac{1}{4}x\).
- Conclude that the correct option is \(\boxed{y = \frac{1}{4}x}\).
Explanation
- Understanding the Problem and Data Analysis
The problem asks for an equation of a line in slope-intercept form. This form is given by the formula \(y = mx + b\), where \(m\) represents the slope of the line and \(b\) represents the y-intercept. We are provided with the following information: the slope \(m = \frac{1}{4}\) and the requirement that the line must pass through the origin, which is the point \((0,0)\).
- Determining the Equation Parameters
We start with the general slope-intercept form equation: \(y = mx + b\). Substituting the given slope \(m = \frac{1}{4}\) into this equation, we obtain:
\[y = \frac{1}{4}x + b\]
To find the y-intercept \(b\), we use the information that the line passes through the origin \((0,0)\). By substituting \(x = 0\) and \(y = 0\) into the equation, we get:
\[0 = \frac{1}{4}(0) + b\]
\[0 = 0 + b\]
\[b = 0\]
- Finalizing the Equation and Selecting the Answer
Now that we have found the slope \(m = \frac{1}{4}\) and the y-intercept \(b = 0\), we substitute these back into the slope-intercept form:
\[y = \frac{1}{4}x + 0\]
This simplifies to:
\[y = \frac{1}{4}x\]
Comparing this result to the given options, we identify that the last equation is the correct one.
Examples
In many fields, such as engineering and physics, the slope-intercept form is a fundamental tool for modeling linear relationships. For example, if you are calculating the cost of a service where there is no base fee but a constant rate per unit, like charging \(\$0.25\) for every mile traveled, the equation for the total cost \(y\) after \(x\) miles would be \(y = 0.25x\), which is equivalent to \(y = \frac{1}{4}x\). This helps in predicting expenses and planning budgets efficiently.