To determine which pair of points has an undefined slope, let's review the definition of slope and what conditions result in an undefined slope.
The slope [tex]\( m \)[/tex] between two points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] is given by:
[tex]\[ m = \frac{y_2 - y_1}{x_2 - x_1} \][/tex]
The slope is undefined when the denominator [tex]\( (x_2 - x_1) \)[/tex] is zero. This occurs if the x-coordinates of the two points are the same, indicating a vertical line.
Now, let's analyze each set of points:
A. Points [tex]\( (-1,1) \)[/tex] and [tex]\( (1,-1) \)[/tex]:
- [tex]\( x_1 = -1 \)[/tex] and [tex]\( x_2 = 1 \)[/tex]
- The x-coordinates are different ([tex]\(-1 \neq 1\)[/tex]), so the slope is defined.
B. Points [tex]\( (-2,2) \)[/tex] and [tex]\( (2,2) \)[/tex]:
- [tex]\( x_1 = -2 \)[/tex] and [tex]\( x_2 = 2 \)[/tex]
- The x-coordinates are different ([tex]\(-2 \neq 2\)[/tex]), so the slope is defined.
C. Points [tex]\( (-3,-3) \)[/tex] and [tex]\( (-3,3) \)[/tex]:
- [tex]\( x_1 = -3 \)[/tex] and [tex]\( x_2 = -3 \)[/tex]
- The x-coordinates are the same ([tex]\(-3 = -3\)[/tex]), so the slope is undefined.
D. Points [tex]\( (-4,-4) \)[/tex] and [tex]\( (4,4) \)[/tex]:
- [tex]\( x_1 = -4 \)[/tex] and [tex]\( x_2 = 4 \)[/tex]
- The x-coordinates are different ([tex]\(-4 \neq 4\)[/tex]), so the slope is defined.
By examining the x-coordinates, we can see that the points listed in option C [tex]\( (-3,-3) \)[/tex] and [tex]\( (-3,3) \)[/tex] have the same x-coordinate, causing the slope to be undefined.
Therefore, the correct answer is:
C. [tex]\( (-3,-3) \)[/tex] and [tex]\( (-3,3) \)[/tex]
