Question

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

Explanation:

Line 1:

Step1: Recall slope definition

The slope of a line is given by \( m=\frac{\Delta y}{\Delta x}=\frac{y_2 - y_1}{x_2 - x_1} \). For a horizontal line, as \( x \) changes, \( y \) remains constant. So \( \Delta y = 0 \).

Step2: Calculate slope

If \( \Delta y = 0 \), then \( m=\frac{0}{\Delta x}=0 \) (as long as \( \Delta x
eq0 \), which it is for a horizontal line that's not vertical). So Line 1 has a slope of zero.

Line 2:

Step1: Recall slope sign rule

A line with a positive slope rises from left to right (as \( x \) increases, \( y \) increases).

Step2: Analyze Line 2's direction

Looking at Line 2, as we move from left to right (increasing \( x \)), the \( y \)-value also increases. So \( \Delta y>0 \) and \( \Delta x>0 \), so \( m = \frac{\Delta y}{\Delta x}>0 \). Thus, slope is positive.

Line 3:

Step1: Recall vertical line slope

A vertical line has \( \Delta x = 0 \) (since \( x \) doesn't change, only \( y \) does). The slope formula \( m=\frac{\Delta y}{\Delta x} \) has a division by zero, which is undefined.

Step2: Identify Line 3's type

Line 3 is a vertical line (parallel to the \( y \)-axis), so its slope is undefined.

Line 4:

Answer:

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