Problem 86 - Applied Calculus

Question

True or False: \(x=3\) is a vertical line.

Step-by-Step Solution

Answer

True, \(x = 3\) is a vertical line because the \(x\) value is constant while \(y\) can vary.

Step 1: Understand what a vertical line is

A vertical line is one that goes straight up and down on the coordinate plane. It is parallel to the y-axis and it intersects the x-axis at exactly one point. In the equation of a vertical line, all the x-coordinates are the same, while the y-coordinates can be any value.

Step 2: Analyze the equation \(x = 3\)

The equation \(x = 3\) indicates that for any value of \(y\), the \(x\) coordinate is always 3. This means no matter where you are up or down the line (changing \(y\)), the \(x\) value is a constant 3.

Step 3: Compare \(x = 3\) to the definition of a vertical line

Based on the definition of a vertical line, the equation \(x = 3\) matches perfectly because it specifies a constant \(x\) value with varying \(y\) values. Therefore, \(x = 3\) is indeed a vertical line.

Step 4: Draw conclusion

Given that \(x = 3\) is a constant for all points on the line and the \(y\) values can vary, this satisfies the condition of a vertical line. Thus, the statement is true.

Key Concepts

Coordinate Plane

A coordinate plane is a two-dimensional surface where we can graphically represent equations and geometric figures. It consists of two perpendicular lines that intersect at a point called the origin. These lines divide the plane into four quadrants. The horizontal line is called the x-axis, and the vertical line is the y-axis.

When working with the coordinate plane:

• Your x-values will be measured along the horizontal line, moving left or right from the origin.
• Your y-values are measured vertically, going up or down from the origin.
• Every point on the plane can be represented as a pair (x, y), where 'x' is the value on the x-axis, and 'y' measures the position on the y-axis.

This grid-like system helps us plot and analyze various types of lines and shapes clearly.

Equation of a Line

Equations are mathematical sentences that depict relationships. When we talk about the equation of a line, we're referring to the rule that describes all the points on that line.

The format of a line equation can vary:

• The most common is the slope-intercept form, which is \( y = mx + b \), where 'm' is the slope, and 'b' is the y-intercept.
• For horizontal lines, the equation looks like \( y = c \), which means 'c' is constant for every x.
• Vertical lines have equations like \( x = k \), meaning x is fixed at k, and 'y' can be anything.

The simplicity of the equation often makes it easier to understand the constant and variable components of a line.

X-Axis

The x-axis is the horizontal number line on the coordinate plane. It helps in determining the position of any point in the horizontal direction. The x-axis passes through the origin and can contain both positive and negative numbers.

Key facts about the x-axis:

• Points on this axis have a y-coordinate of 0.
• When a line is vertical, it intersects the x-axis at just one point since 'x' is fixed, like in the equation \( x = 3 \).
• To find where a line crosses the x-axis, set the y-value to zero in the line's equation.

This line gives a foundation for locating and plotting points across the coordinate plane.

Y-Axis

The y-axis is the vertical line in the coordinate plane. Like the x-axis, it passes through the origin. It helps to establish the position of points vertically. The y-axis, much like a thermometer, can be thought of as indicating whether you move above or below the origin.

Interesting aspects of the y-axis:

• Any points located on this axis have a fixed x-coordinate of 0.
• Vertical lines do not run parallel to the y-axis; they intersect it perpendicularly.
• A line crosses the y-axis where the x-value is zero, common in slope-intercept form equations as the y-intercept.

Understanding the y-axis helps in analyzing vertical movements and the behavior of lines interacting with the y-axis.