To determine which two points have an undefined slope, we need to understand what it means for a slope to be undefined. The slope between two points \((x_1, y_1)\) and \((x_2, y_2)\) 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 \(x_2 - x_1\) is zero, which happens when \(x_2 = x_1\). This means that the line is vertical.
Let's examine each of the given pairs of points:
Option A: \((-1,1)\) and \((1,-1)\)
- \(x_1 = -1\)
- \(x_2 = 1\)
- \(x_2 - x_1 = 1 - (-1) = 2\)
Since \(x_2 - x_1 \neq 0\), the slope is not undefined.
Option B: \((-2,2)\) and \((2,2)\)
- \(x_1 = -2\)
- \(x_2 = 2\)
- \(x_2 - x_1 = 2 - (-2) = 4\)
Since \(x_2 - x_1 \neq 0\), the slope is not undefined.
Option C: \((-3,-3)\) and \((-3,3)\)
- \(x_1 = -3\)
- \(x_2 = -3\)
- \(x_2 - x_1 = -3 - (-3) = 0\)
Since \(x_2 - x_1 = 0\), the slope is undefined.
Option D: \((-4,-4)\) and \((4,4)\)
- \(x_1 = -4\)
- \(x_2 = 4\)
- \(x_2 - x_1 = 4 - (-4) = 8\)
Since \(x_2 - x_1 \neq 0\), the slope is not undefined.
Based on the analysis, the correct answer is:
C. [tex]\((-3,-3)\)[/tex] and [tex]\((-3,3)\)[/tex]
