Ellen thinks that if a line has no slope, then it never touches the [tex]$y$[/tex]-axis. To evaluate this statement and find a counterexample, let's consider the lines provided in the options:
1. [tex]\(x = 0\)[/tex]
2. [tex]\(y = 0\)[/tex]
3. [tex]\(x = 1\)[/tex]
4. [tex]\(y = 1\)[/tex]
We need to determine whether these lines touch the [tex]$y$[/tex]-axis and their slopes. Let's break it down:
1. [tex]\(x = 0\)[/tex]:
- This is a vertical line.
- It intersects the [tex]$y$[/tex]-axis at all points where [tex]$y$[/tex] can take any value (essentially the entire [tex]$y$[/tex]-axis).
- A vertical line is usually described as having an undefined slope.
2. [tex]\(y = 0\)[/tex]:
- This is a horizontal line.
- It intersects the [tex]$y$[/tex]-axis at exactly one point: the origin [tex]\((0,0)\)[/tex].
- A horizontal line has a slope of zero.
3. [tex]\(x = 1\)[/tex]:
- This is a vertical line.
- It runs parallel to the [tex]$y$[/tex]-axis and intersects the [tex]$x$[/tex]-axis at [tex]\(x = 1\)[/tex].
- Vertical lines have undefined slope and do not intersect the [tex]$y$[/tex]-axis at any specific single point other than the entire [tex]$y$[/tex]-axis collectively.
4. [tex]\(y = 1\)[/tex]:
- This is a horizontal line.
- It intersects the [tex]$y$[/tex]-axis at the point [tex]\((0, 1)\)[/tex].
- A horizontal line has a slope of zero.
Now, Ellen believes that if a line has no slope, then it never touches the [tex]$y$[/tex]-axis. But a horizontal line with a zero slope can intersect the [tex]$y$[/tex]-axis. Among the given options:
- Both [tex]\(y = 0\)[/tex] and [tex]\(y = 1\)[/tex] are horizontal lines with a zero slope.
- These lines intersect the [tex]$y$[/tex]-axis at points [tex]\( (0, 0) \)[/tex] and [tex]\( (0, 1) \)[/tex] respectively.
Therefore, Ellen's statement is proven incorrect by the line [tex]\( y = 0 \)[/tex] because it is a horizontal line (no slope), and it intersects the [tex]$y$[/tex]-axis at the origin [tex]\((0,0)\)[/tex].
So, the correct answer is:
[tex]\[ \boxed{2} \][/tex]
