To determine which pair of points has an undefined slope, we need to understand what causes a slope to be undefined. The slope between two points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] is given by the formula:
[tex]\[
\text{slope} = \frac{y_2 - y_1}{x_2 - x_1}
\][/tex]
The slope is undefined when the denominator of this fraction is zero, which happens when [tex]\(x_1 = x_2\)[/tex]. This means the line is vertical.
Now, let's evaluate each pair of points:
### Choice A: [tex]\((-1, 1)\)[/tex] and [tex]\((1, -1)\)[/tex]
- [tex]\(x_1 = -1\)[/tex] and [tex]\(x_2 = 1\)[/tex]
- Since [tex]\(-1 \neq 1\)[/tex], the denominator [tex]\((1 - (-1))\)[/tex] is not zero, so the slope is defined.
### Choice B: [tex]\((-2, 2)\)[/tex] and [tex]\((2, 2)\)[/tex]
- [tex]\(x_1 = -2\)[/tex] and [tex]\(x_2 = 2\)[/tex]
- Since [tex]\(-2 \neq 2\)[/tex], the denominator [tex]\((2 - (-2))\)[/tex] is not zero, so the slope is defined.
### Choice C: [tex]\((-3, -3)\)[/tex] and [tex]\((-3, 3)\)[/tex]
- [tex]\(x_1 = -3\)[/tex] and [tex]\(x_2 = -3\)[/tex]
- Since [tex]\(-3 = -3\)[/tex], the denominator [tex]\((-3 - (-3))\)[/tex] is zero, so the slope is undefined.
### Choice D: [tex]\((-4, -4)\)[/tex] and [tex]\((4, 4)\)[/tex]
- [tex]\(x_1 = -4\)[/tex] and [tex]\(x_2 = 4\)[/tex]
- Since [tex]\(-4 \neq 4\)[/tex], the denominator [tex]\((4 - (-4))\)[/tex] is not zero, so the slope is defined.
From this analysis, we see that only Choice C, [tex]\((-3, -3)\)[/tex] and [tex]\((-3, 3)\)[/tex], has an undefined slope because the x-coordinates are the same, making the denominator zero.
Thus, the correct answer is:
[tex]\[
\boxed{3}
\][/tex]
