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 positive - slope line rises from left to right, a negative - slope line falls from left to right, a zero - slope line is horizontal, and an undefined - slope line is vertical.

Step2: Analyze Line 1

Line 1 rises from left to right. Using the slope formula, as $x$ increases, $y$ increases. So, $\Delta y>0$ and $\Delta x>0$, and $m = \frac{\Delta y}{\Delta x}>0$. The slope of Line 1 is positive.

Step3: Analyze Line 2

Line 2 is a vertical line. For a vertical line, $x_1=x_2$, so $\Delta x=x_2 - x_1 = 0$. Since the slope formula $m=\frac{\Delta y}{\Delta x}$ has a denominator of 0 for a vertical line, the slope of Line 2 is undefined.

Step4: Analyze Line 3

Line 3 falls from left to right. As $x$ increases, $y$ decreases. So, $\Delta y<0$ and $\Delta x>0$, and $m=\frac{\Delta y}{\Delta x}<0$. The slope of Line 3 is negative.

Step5: Analyze Line 4

Line 4 is a horizontal line. For a horizontal line, $y_1 = y_2$, so $\Delta y=y_2 - y_1=0$. Then, $m=\frac{\Delta y}{\Delta x}=0$ (since $\Delta x
eq0$). The slope of Line 4 is zero.

Answer:

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