To determine which pair of points has an undefined slope, we need to understand when a line's slope becomes undefined. In general, the slope of a line through two points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] is calculated using the formula:
[tex]\[ \text{slope} = \frac{y_2 - y_1}{x_2 - x_1} \][/tex]
A slope becomes undefined if the denominator [tex]\((x_2 - x_1)\)[/tex] is zero. This occurs when the x-coordinates of the two points are the same, meaning the line is vertical.
Let's analyze each option:
### Option A: [tex]\((-1, 1)\)[/tex] and [tex]\((1, -1)\)[/tex]
Calculate the x difference:
[tex]\[ x_2 - x_1 = 1 - (-1) = 2 \][/tex]
The x difference is not zero, so this slope is defined.
### Option B: [tex]\((-2, 2)\)[/tex] and [tex]\((2, 2)\)[/tex]
Calculate the x difference:
[tex]\[ x_2 - x_1 = 2 - (-2) = 4 \][/tex]
The x difference is not zero, so this slope is defined.
### Option C: [tex]\((-3, -3)\)[/tex] and [tex]\((-3, 3)\)[/tex]
Calculate the x difference:
[tex]\[ x_2 - x_1 = -3 - (-3) = 0 \][/tex]
The x difference is zero, so the slope is undefined.
### Option D: [tex]\((-4, -4)\)[/tex] and [tex]\((4, 4)\)[/tex]
Calculate the x difference:
[tex]\[ x_2 - x_1 = 4 - (-4) = 8 \][/tex]
The x difference is not zero, so this slope is defined.
Based on this analysis, the pair of points that has an undefined slope is:
[tex]\[ \boxed{C} \][/tex]
Therefore, the correct answer is [tex]\( (-3, -3) \)[/tex] and [tex]\( (-3, 3) \)[/tex], which corresponds to option [tex]\( C \)[/tex].
