Problem 44 - Understanding Elementary Algebra with Geometry

Question

Sketch the graph of the line satisfying the given conditions. Passing through \((4,3)\) and whose slope is undefined

Step-by-Step Solution

Answer

The graph is a vertical line at x=4.

Step 1: Identify the Slope of the Line

The slope of the line is given as undefined. An undefined slope indicates a vertical line.

Step 2: Understand What a Vertical Line Means

A vertical line has a constant x-coordinate for all points on the line. Therefore, the x-coordinate is the same for every point on the line.

Step 3: Use Given Point to Find Equation

The line passes through the point (4,3). For a vertical line, the x-coordinate remains constant. So, the equation of the line is simply x=4.

Step 4: Sketch the Graph

To draw the graph, plot the point (4,3) on the coordinate plane. Then, draw a straight, vertical line through this point which covers all y-values.

Key Concepts

Vertical Line

When we talk about an undefined slope, it refers to a vertical line. A vertical line goes straight up and down and does not tilt in any direction. This type of line has a unique property where the x-coordinate is constant for all points on the line. This constancy means that if you have a vertical line at x=4, every point on that line will have an x-coordinate of 4 no matter what the y-coordinate is. This uniformity is what makes the line vertical.

Coordinate Plane

The coordinate plane is a two-dimensional surface where we can plot points, lines, and curves. It is formed by two perpendicular lines called axes. The horizontal axis is the x-axis, and the vertical axis is the y-axis. Together, these axes create a grid that allows us to locate any point using a pair of numbers called coordinates. These coordinates follow the format (x, y). The point where the axes meet is called the origin, denoted as (0,0).

Equation of a Line

The equation of a line is a mathematical way of describing that line. For vertical lines, the equation is quite simple. Since every point on a vertical line has the same x-coordinate, the line's equation will be in the form x = constant. For example, if a vertical line passes through (4,3), then its equation is x=4. This tells us that no matter the y-value, the x-value will always be 4 on this line.

Plotting Points

Plotting points on the coordinate plane is an essential skill in graphing lines. To plot the point (4,3), start at the origin (0,0). Move 4 units to the right along the x-axis. From there, move 3 units up along the y-axis. Mark the point where you end up. Once you have your point, you can draw your line. In this case, since it's a vertical line through (4,3), draw a straight line going up and down that passes through this point and extends infinitely in both directions.