Question

for each line, determine whether the slope is positive, negative, zero, or undefined. line 1: graph of vertical line options: positive, negative, zero, undefined. line 2: graph of line with positive slope options: positive, negative, zero, undefined. line 3: graph of line with negative slope options: positive, negative, zero, undefined. line 4: graph of horizontal line options: positive, negative, zero, undefined.

Explanation:

for Line 1:

Step1: Recall slope definition

The slope of a line is calculated as $m = \frac{\Delta y}{\Delta x}$, where $\Delta y$ is the change in $y$ and $\Delta x$ is the change in $x$. For a vertical line, $\Delta x = 0$, so the slope formula involves division by zero, which is undefined. Line 1 is a vertical line (parallel to the $y$-axis), so its slope is undefined.

for Line 2:

Step1: Analyze the line's direction

A line with a positive slope rises from left to right. Line 2 goes up as we move from left to right (since when $x$ increases, $y$ increases). So, using the slope formula $m=\frac{\Delta y}{\Delta x}$, $\Delta y$ and $\Delta x$ will have the same sign (both positive or both negative here, as it's rising), leading to a positive slope.

for Line 3:

Step1: Analyze the line's direction

A line with a negative slope falls from left to right. Line 3 goes down as we move from left to right (when $x$ increases, $y$ decreases). Using $m = \frac{\Delta y}{\Delta x}$, $\Delta y$ will be negative and $\Delta x$ positive, so their ratio is negative, meaning the slope is negative.

for Line 4:

Step1: Recall slope for horizontal lines

For a horizontal line (parallel to the $x$-axis), $\Delta y = 0$ (since $y$ doesn't change as $x$ changes). Using the slope formula $m=\frac{\Delta y}{\Delta x}$, $\Delta y = 0$, so $m = \frac{0}{\Delta x}=0$ (as long as $\Delta x
eq0$, which it is for a non - vertical line). Line 4 is horizontal, so its slope is zero.

Answer:

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