Find a point-slope form for the line with slope $\frac{1}{4}$ and passing through the point $(-3,-9)$.
The problem asks us to find the point-slope form of a line given its slope and a point it passes through. The slope is given as \(m = \frac{1}{4}\) and the point is \((x_1, y_1) = (-3, -9)\). The point-slope form of a linear equation is defined by the formula:
\[y - y_1 = m(x - x_1)\]
This formula describes a line with slope \(m\) that passes through the point \((x_1, y_1)\).
Now we substitute the values \(m = \frac{1}{4}\), \(x_1 = -3\), and \(y_1 = -9\) into the formula:
\[y - (-9) = \frac{1}{4}(x - (-3))\]
Simplifying the signs inside the expressions (remembering that subtracting a negative is equivalent to adding a positive), we get:
\[y + 9 = \frac{1}{4}(x + 3)\]
This is the equation of the line in point-slope form.
The equation of the line passing through \((-3, -9)\) with a slope of \(\frac{1}{4}\) in point-slope form is:
\[\boxed{y + 9 = \frac{1}{4}(x + 3)}\]
The point-slope form is a fundamental tool in coordinate geometry for finding the equation of a line when you know a single point and the rate of change (slope). For example, if you are designing a ramp for a building where the incline (slope) is specified as \(\frac{1}{4}\) and the ramp must start at a certain ground coordinate, you can use this formula to map out the exact position of the ramp surface. This ensures that the construction is precise and meets design specifications.