Problem 7

Question

Yes or No? If No, give a reason. (a) Is the graph of $y=-3$ a horizontal line? (b) Is the graph of $x=-3$ a vertical line? (c) Does a line perpendicular to a horizontal line have slope $0 ?$ (d) Does a line perpendicular to a vertical line have slope $0 ?$

Step-by-Step Solution

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Answer

(a) Yes, (b) Yes, (c) No, perpendicular lines have undefined slopes, (d) Yes.

1Step 1: Understanding Horizontal Line Graph

A horizontal line is defined as a line where all points have the same y-coordinate. The equation of such a line is of the form $ y = c $, where $ c $ is a constant.

2Step 2: Analyzing Graph of y = -3

The equation $ y = -3 $ means that for any value of $ x $, $ y $ will always be $-3$. Since all y-coordinates are the same, this graph is a horizontal line.

3Step 3: Understanding Vertical Line Graph

A vertical line is defined as a line where all points have the same x-coordinate. The equation of such a line is of the form $ x = c $, where $ c $ is a constant.

4Step 4: Analyzing Graph of x = -3

The equation $ x = -3 $ means that for any value of $ y $, $ x $ will always be $-3$. Since all x-coordinates are the same, this graph is a vertical line.

5Step 5: Defining Perpendicular Line Slope

A line is perpendicular to another if the product of their slopes is $-1$. A horizontal line has a slope of $0$. Perpendicular lines to horizontal lines must have undefined slopes as they are vertical.

6Step 6: Checking Perpendicular Line to Horizontal

Since a line perpendicular to a horizontal line has an undefined slope (being vertical), it cannot have a slope of $0$. Thus, the statement is false.

7Step 7: Checking Perpendicular Line to Vertical

A vertical line has an undefined slope, so a line perpendicular to it would have a slope of $0$ since this perpendicular line is horizontal.

Key Concepts

Horizontal LineVertical LinePerpendicular LinesSlope of a Line

Horizontal Line

A horizontal line is a straight line that goes from left to right on a graph. It maintains the same y-coordinate for all its points. This means no matter what x-value you choose, the y-value remains constant.
To understand this better, consider the equation of a horizontal line which is of the form $ y = c $. Here, $ c $ is a constant and represents the y-coordinate where this line lies.

For example, if you have the equation $ y = -3 $, every point on this line has the y-coordinate -3, like (-2, -3), (0, -3), or (4, -3).
This is why the graph of $ y = -3 $ is a horizontal line.

This simple representation makes it easy to identify horizontal lines on a graph, by just observing that they run parallel to the x-axis.

Vertical Line

A vertical line is a straight line that runs up and down the graph. It keeps the same x-coordinate across all its points. Therefore, no matter the y-value, the x-value remains constant.
The general form of a vertical line equation is $ x = c $, with $ c $ acting as the x-coordinate for this constant line.

For instance, the equation $ x = -3 $ indicates that every point along this line has an x-coordinate of -3, like (-3, 1), (-3, 0), or (-3, -4).
Consequently, this is why the graph of $ x = -3 $ is a vertical line.

This characteristic of keeping a consistent x-coordinate makes it straightforward to spot vertical lines on a graph, as they will always run parallel to the y-axis.

Perpendicular Lines

Perpendicular lines are lines that intersect at a right angle (90 degrees). For two lines to be perpendicular, the product of their slopes must be -1. This rule helps us understand the relationship between different types of lines on a graph.
Horizontal lines, which have a slope of zero, have perpendicular counterparts that are vertical lines with undefined slopes.

For example, a line perpendicular to the horizontal line $ y = 2 $ would be a vertical line such as $ x = 3 $.
The vertical line $ x = 3 $ has an undefined slope, making it perfectly perpendicular to any horizontal line like $ y = 2 $.

This principle is consistently applied in geometry to determine perpendicularity among various lines in a coordinate plane.

Slope of a Line

The slope of a line is a measure of its steepness, defined as the ratio of the change in y to the change in x between two points on the line. The formula to calculate slope $ m $ is given by: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \]
Horizontal lines have a slope of zero because there is no change in y as x changes, while vertical lines do not have a defined slope because x doesn't change but y does, leading to division by zero.

If you have the line equation $ y = 5 $, its slope $ m = 0 $ because it doesn't rise in elevation (no change in y).
On the contrary, the equation $ x = -2 $ lacks a slope value, as any vertical movement implies changes in y but no changes in x.

Understanding slope is crucial for analyzing and predicting the behavior of linear relationships in mathematical contexts.

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