Line Perpendicular To Y-Axis: Which Equation?
Hey guys! Let's dive into a common math question that might seem tricky at first, but is super easy once you get the hang of it. We're going to break down what it means for a line to be perpendicular to the y-axis and figure out which equation represents that. So, buckle up and let's get started!
Understanding Perpendicular Lines
Okay, so before we jump into the equations, let's make sure we're all on the same page about what perpendicular means. In geometry, perpendicular lines are lines that intersect at a right angle (90 degrees). Think of the corner of a square or a rectangle – that's a perfect example of perpendicular lines in action. Now, when we're talking about the y-axis, which is that vertical line running straight up and down on our graph, what kind of line would cross it at a perfect right angle?
The key here is to visualize it. Imagine the y-axis standing tall. A line perpendicular to it would have to be a horizontal line, stretching from left to right. Got that picture in your head? Good! This understanding is crucial because it directly relates to the type of equation that will represent our line.
Now, let's think about the equations of lines. You've probably heard of the slope-intercept form, which is y = mx + b. This is super helpful because 'm' tells us the slope of the line (how steep it is), and 'b' tells us the y-intercept (where the line crosses the y-axis). But what about horizontal lines? What's their deal in terms of slope and equation?
A horizontal line doesn't have any steepness, right? It's perfectly flat. That means its slope is zero. If we plug that into our slope-intercept form (y = mx + b), we get y = 0x + b, which simplifies to y = b. So, the equation of any horizontal line is simply y equals a constant. This constant 'b' is the y-value that the line passes through. For example, the line y = 3 is a horizontal line that crosses the y-axis at the point (0, 3).
On the flip side, vertical lines are a different beast. They have an undefined slope because they're infinitely steep. Their equations look like x = a, where 'a' is a constant. This means that the x-value is always the same, no matter what the y-value is. For example, the line x = 5 is a vertical line that crosses the x-axis at the point (5, 0).
So, to recap, lines perpendicular to the y-axis are horizontal lines, and their equations take the form y = a constant. Keep this in mind as we analyze the answer choices.
Analyzing the Options
Alright, now that we've got a solid understanding of perpendicular lines and their equations, let's look at the options and see which one fits the bill. Remember, we're searching for an equation that represents a horizontal line, which means it should be in the form y = a constant.
Here are the options we're working with:
A. y = 6x B. y = x C. y = -6 D. x = 6
Let's break down each one and see if it matches our criteria:
Option A: y = 6x
This equation looks like our slope-intercept form (y = mx + b), but it's missing the 'b' term. That's okay, it just means the y-intercept is zero. The important part here is the '6x'. This indicates that the line has a slope of 6, meaning it's neither horizontal nor vertical. It's a diagonal line that slopes upwards as you move from left to right. So, option A is not the answer we're looking for.
Option B: y = x
This is another diagonal line, but this time the slope is 1 (think of it as y = 1x). It's a straight line that passes through the origin (0, 0) and makes a 45-degree angle with both the x and y axes. Again, this is not a horizontal line, so option B is out.
Option C: y = -6
Ding ding ding! This is our winner! This equation is in the form y = a constant, which we know represents a horizontal line. Specifically, this line passes through the point (0, -6) on the y-axis and extends horizontally in both directions. It's perpendicular to the y-axis, just like we want. So, option C is the correct answer.
Option D: x = 6
This equation represents a vertical line. It passes through the point (6, 0) on the x-axis and extends vertically upwards and downwards. While this line is perpendicular to the x-axis, it's parallel to the y-axis, not perpendicular to it. So, option D is incorrect.
The Correct Answer
After analyzing each option, it's clear that the correct answer is C. y = -6. This equation represents a horizontal line, which is perpendicular to the y-axis. We nailed it!
Key Takeaways
Let's quickly recap the key things we learned in this question:
- Perpendicular lines intersect at a right angle (90 degrees).
- A line perpendicular to the y-axis is a horizontal line.
- The equation of a horizontal line is in the form y = a constant.
- The equation of a vertical line is in the form x = a constant.
Understanding these concepts will help you tackle similar questions with confidence. Remember, visualizing the lines and their orientations can make a big difference in understanding the equations.
Practice Makes Perfect
Now that you've aced this question, the best way to solidify your understanding is to practice more problems. Try graphing different equations of lines and identifying which ones are perpendicular to the y-axis or x-axis. You can also explore questions that involve finding the equation of a line given a point and a perpendicular line. The more you practice, the easier it will become!
So, keep up the great work, guys! You're on your way to mastering these concepts. Remember to visualize, understand the key principles, and practice regularly. You've got this!
