Problem 48 - A Graphical Approach to Precalculus with Limits

Question

Find the equation of the line described, giving it in slope-intercept form if possible. Through \((1,-4),\) perpendicular to \(x=4\)

Step-by-Step Solution

Answer

The equation is \(y = -4\).

Step 1: Understand the Given Line

The line described by the equation \(x = 4\) is a vertical line. Vertical lines have an undefined slope.

Step 2: Determine the Slope of the Perpendicular Line

Lines perpendicular to vertical lines are horizontal lines. Horizontal lines have a slope of 0.

Step 3: Use the Point-Slope Form

Since we know the line must pass through the point \((1, -4)\) and has a slope of 0, we can use the point-slope form equation, \(y - y_1 = m(x - x_1)\). Substitute \(m = 0\), \(x_1 = 1\), and \(y_1 = -4\).

Step 4: Simplify into Slope-Intercept Form

Substitute the values into the point-slope form equation to get: \(y - (-4) = 0(x - 1)\). Simplify this to find \(y = -4\), which is the equation in slope-intercept form.

Key Concepts

Perpendicular Lines

When we talk about perpendicular lines in geometry, we're referring to two lines that meet at a 90-degree angle. This specific property leads to some interesting rules about their slopes.

• A vertical line, characterized by an equation like \(x = 4\), doesn't actually have a defined slope. It's a straight up-and-down line.
• Horizontal lines, on the other hand, are always perpendicular to vertical lines. These have a slope of 0.

Because vertical lines are so unique, any line that is perpendicular to one is a horizontal line. The slope relationship is simple: if one line's slope is undefined (like vertical lines), the perpendicular line's slope is 0 (like horizontal lines). This relationship helps us quickly figure out what the slope of the perpendicular line will be.

Point-Slope Form

The point-slope form of the equation of a line is a powerful tool in algebra that particularly shines when you're given a point on the line and the line's slope. It's structured as follows: \[y - y_1 = m(x - x_1)\]

• \(m\) represents the slope of the line.
• \((x_1, y_1)\) is a point on the line.

This form is immensely useful because it straightforwardly incorporates the slope and a point on the line. In our problem, since the perpendicular line has a slope of 0 and passes through the point \((1, -4)\), using point-slope form directly leads us to the equation of the line. To simplify the equation into slope-intercept form, just let the math unfold: start by substituting \(m = 0\) and the point coordinates \((x_1, y_1) = (1, -4)\). This sets up the expression \(y + 4 = 0(x - 1)\), which simplifies down to \(y = -4\), the line's finished equation in slope-intercept form.

Horizontal Lines

Horizontal lines are a fundamental concept in coordinate geometry. Imagine drawing a line perfectly parallel to the x-axis. That's what a horizontal line is.

• Slope: Always 0. No matter where you place the horizontal line on the graph, it never rises or falls. It's perfectly flat.
• Equation: Typically written as \(y = b\). Here, \(b\) is the y-coordinate of every point on the line. For example, \(y = -4\) tells you that the line crosses the y-axis at -4.

In our exercise, the horizontal line derived is described by the equation \(y = -4\). This means no matter what x-value you choose, the corresponding y-value is always -4. It's consistent everywhere on this line across the graph.