Write an equation in slope-intercept form of a linear function f whose graph satisfies the given conditions. The graph of ƒ passes through (−6, 4) and is perpendicular to the line that has an x intercept of 2 and a y-intercept of -4.

Verified step by step guidance

1. Identify the slope of the given line using its intercepts. The line passes through the points (2, 0) and (0, -4). Use the slope formula: \(m = \frac{y_2 - y_1}{x_2 - x_1}\).
2. Calculate the slope of the given line by substituting the points: \(m = \frac{0 - (-4)}{2 - 0} = \frac{4}{2}\).
3. Determine the slope of the line perpendicular to the given line. Recall that perpendicular slopes are negative reciprocals, so if the original slope is \(m\), the perpendicular slope is \(-\frac{1}{m}\).
4. Use the point-slope form of a line equation with the perpendicular slope and the given point \((-6, 4)\): \(y - y_1 = m(x - x_1)\).
5. Rewrite the equation from the previous step into slope-intercept form \(y = mx + b\) by solving for \(y\) and simplifying.

Key Concepts

Slope-Intercept Form of a Linear Equation

The slope-intercept form is y = mx + b, where m represents the slope and b is the y-intercept. This form makes it easy to identify the slope and where the line crosses the y-axis, which is essential for graphing and writing linear equations.

Finding the Slope from Intercepts

A line's slope can be found using two points, such as the x- and y-intercepts. The slope m is calculated as the change in y divided by the change in x (m = (y2 - y1) / (x2 - x1)). Knowing the slope of the given line is necessary to find the slope of the perpendicular line.

Perpendicular Lines and Their Slopes

Two lines are perpendicular if their slopes are negative reciprocals, meaning m1 * m2 = -1. If one line has slope m, the perpendicular line's slope is -1/m. This relationship helps determine the slope of the required line passing through a given point.