Think of the definition for slope. What can you conclude from the possibilities for different slopes?
See solution.
Practice makes perfect
To determine the characteristics given in the exercise for the slope of a line without using points, we should first recall how the slope is defined.
Slope = change in $y$/change in $x$
Since the definition is a quotient, we can tell that if the changes have same sign the slope is positive.
Notice that the function is slanted upwards, so it is rising from left to right. When this happens we can know that the line has a positive slope. On the other hand, if the changes have opposite signs, their ratio and then the slope would be negative.
Notice that the function is slanted downwards, so it is falling from left to right. When this happens we know that the line has a negative slope. Now we can analyze the case when the line is horizontal. For this, let's analyze the graph of a generic horizontal line.
Notice that as the y values are always the same, no mater what pair of x values we use to find the change, our slope will be of the form slope = 0difference inx-values = 0. Therefore, a horizontal line has a zero slope. The last possibility is having a vertical line.
We can see that there is no change in the x values. Then, no matter what pair of y values we use, our slope will be of the form slope = difference iny-values0. Since division by 0 is undefined, a vertical line has an undefined slope as well.
