Question

Find the slope of the line passing through the pairs of points and describe the line as rising, falling, horizontal or vertical. a. (2 , 1) , (4 , 5) b. (-1 , 0) , (3 , -5) c. (2 , 1) , (-3 , 1) d. (-1 , 2) , (-1 ,- 5)

Answer 1

The slope is calculated using the difference in y-coordinates divided by the difference in x-coordinates. If the slope is positive, the line rises; if negative, it falls. A zero slope implies a horizontal line, and an undefined slope indicates a vertical line.

Step-by-step explanation:

To find the slope of the line passing through two points (x1, y1) and (x2, y2), we use the formula:

Slope (m) = (y2 - y1) / (x2 - x1)

1. For points (2, 1) and (4, 5), the slope is (5 - 1) / (4 - 2) = 4 / 2 = 2. The line is rising since the slope is positive.
2. For points (-1, 0) and (3, -5), the slope is (-5 - 0) / (3 - (-1)) = -5 / 4 = -1.25. The line is falling since the slope is negative.
3. For points (2, 1) and (-3, 1), the slope is (1 - 1) / (-3 - 2) = 0. The line is horizontal since the slope is zero.
4. For points (-1, 2) and (-1, -5), the slope is not defined as the denominator in the slope formula would be 0. The line is vertical.

These slopes allow us to describe the lines. If a slope is positive, the line rises from left to right. If the slope is negative, the line falls from left to right. A zero slope indicates a horizontal line, while an undefined slope (when x1 = x2) indicates a vertical line.

Answer 2

`a.\ m=2,\ \text{the line is rising}\\\\b.\ m=-(5)/(4),\ \text{the line is falling}\\\\c.\ m=0,\ \text{the line is horizontal}\\\\d.\ m\ is\ unde fined,\ \text{the line is vertical}`

Step-by-step explanation:

The formula of a slope:

`m=(y_2-y_1)/(x_2-x_1)`

If

m > 0, then a line is rising

m < 0, then a line is falling

m = 0, then a line is horizontal

m is undefined, then a line is vertical

a.

(2, 1) and (4, 5)

`m=(5-1)/(4-2)=(4)/(2)=2>0\to\text{rising}`

b.

(-1, 0) and (3, -5)

`m=(-5-0)/(3-(-1))=(-5)/(4)=-(5)/(4)<0\to\text{falling}`

c.

(2, 1) and (-3, 1)

`m=(1-1)/(-3-2)=(0)/(-5)=0\to\text{horizontal}`

d.

(-1, 2) and (-1, -5)