Question

for each line, determine whether the slope is positive, negative, zero, or undefined.
line 1
positive
negative
zero
undefined
line 2
positive
negative
zero
undefined
line 3
positive
negative
zero
undefined
line 4
positive
negative
zero
undefined

Explanation:

Step1: Recall slope - definition

The slope $m$ of a line is given by $m=\frac{\Delta y}{\Delta x}=\frac{y_2 - y_1}{x_2 - x_1}$ for two points $(x_1,y_1)$ and $(x_2,y_2)$ on the line. A vertical line has an undefined slope because $\Delta x = 0$, a horizontal line has a slope of 0 because $\Delta y=0$, a line rising from left - to - right has a positive slope ($\Delta y>0$ and $\Delta x>0$), and a line falling from left - to - right has a negative slope ($\Delta y < 0$ and $\Delta x>0$).

Step2: Analyze Line 1

Line 1 is a vertical line. For a vertical line, the $x$ - values of all points on the line are the same. So, if we take two points $(x_1,y_1)$ and $(x_2,y_2)$ on the line, $x_1=x_2$, and the formula for slope $m = \frac{y_2 - y_1}{x_2 - x_1}$ has a denominator of 0. Thus, the slope of Line 1 is undefined.

Step3: Analyze Line 2

Line 2 is a horizontal line. For a horizontal line, the $y$ - values of all points on the line are the same. So, if we take two points $(x_1,y_1)$ and $(x_2,y_2)$ on the line, $y_1 = y_2$, and $m=\frac{y_2 - y_1}{x_2 - x_1}=0$ since $y_2 - y_1 = 0$. Thus, the slope of Line 2 is zero.

Step4: Analyze Line 3

Line 3 is rising from left - to - right. If we take two points $(x_1,y_1)$ and $(x_2,y_2)$ on the line with $x_2>x_1$, then $y_2>y_1$. So, $m=\frac{y_2 - y_1}{x_2 - x_1}>0$. Thus, the slope of Line 3 is positive.

Step5: Analyze Line 4

Line 4 is falling from left - to - right. If we take two points $(x_1,y_1)$ and $(x_2,y_2)$ on the line with $x_2>x_1$, then $y_2

Answer:

Line 1: Undefined
Line 2: Zero
Line 3: Positive
Line 4: Negative