Problem 43 - Algebra A Combined Function

Question

Find an equation of each line described. Write each equation in slope- intercept form when possible. With undefined slope, through \(\left(-\frac{3}{4}, 1\right)\)

Step-by-Step Solution

Answer

The equation of the line is \(x = -\frac{3}{4}\); it cannot be written in slope-intercept form.

Step 1: Understand Undefined Slope

A line with an undefined slope is a vertical line, which means that it runs parallel to the y-axis. For any vertical line, the x-coordinates of all points are constant.

Step 2: Identify the Vertical Line Characteristic

Since the line is vertical and passes through the point \(\left(-\frac{3}{4}, 1\right)\), the x-coordinate of this point must be the same for all points on this line.

Step 3: Formulate the Equation

For vertical lines, the equation can be directly derived from the x-coordinate of the given point. The equation for the line is \(x = -\frac{3}{4}\).

Step 4: Verify the Form of the Equation

Since it's a vertical line, it cannot be expressed in slope-intercept form \(y = mx + b\), because it doesn't have a constant slope \(m\). Hence, the equation remains \(x = -\frac{3}{4}\).

Key Concepts

Undefined Slope

An undefined slope is a fundamental concept in understanding vertical lines in coordinate geometry. When we talk about the slope of a line, we refer to the measure of its steepness. Typically, we calculate slope as the "rise over run," which is the ratio of the change in the y-coordinates to the change in the x-coordinates (\[ m = \frac{{y_2 - y_1}}{{x_2 - x_1}} \]). However, when a line is vertical, this presents a unique situation.For vertical lines, the x-coordinates of points on the line are identical, resulting in a zero denominator when calculating the slope. This division by zero means the slope is undefined.
In simpler terms, undefined slope arises because a vertical line does not run left or right, only up and down, so there's no horizontal change.

• Vertical lines have a constant undefined slope.
• They run parallel to the y-axis and cross the x-axis at one consistent point only.

Slope-Intercept Form

The slope-intercept form is a common method for expressing the equation of a line in algebra. It is especially helpful for quickly understanding the direction and position of a line on a graph. The general notation is:\[ y = mx + b \]where:

• m is the slope of the line.
• b is the y-intercept, the point where the line crosses the y-axis.

This form clearly shows both the rate of change of the line (slope) and the line’s starting position on the vertical axis (y-intercept).Unfortunately, not all lines can be described in this form. Vertical lines, for instance, do not have a y-intercept unless they coincide entirely with the y-axis, and their slope is undefined. Therefore, vertical lines cannot be represented in slope-intercept form.

Vertical Line Equation

A vertical line in the Cartesian plane is unique due to its vertical orientation, meaning it never tilts; it goes straight up and down.The defining characteristic of a vertical line is that all points on the line share the same x-coordinate. Thus, the equation of a vertical line can be expressed simply as:\[ x = a \]where \(a\) is the constant x-coordinate for every point along the line. For the problem at hand, since the line passes through \((-\frac{3}{4}, 1)\), every point on this line has an x-coordinate of \(-\frac{3}{4}\). Therefore, the equation for the line is \[ x = -\frac{3}{4} \].It's crucial to remember that this form of equation signifies the line's parallel nature to the y-axis, manifesting an undefined slope.

• Vertical lines are represented in the form \(x = a\).
• They do not have a slope or a y-intercept.