The document discusses formulas for calculating distance, midpoints, and slopes of lines on a coordinate plane. It defines key terms like x-axis, y-axis, origin, and introduces the distance, midpoint, and slope formulas. Examples are provided to demonstrate calculating distances and slopes between points and finding midpoints, and describing lines based on whether their slopes are positive, negative, undefined, or zero.

1. Lesson 1-3 Formulas

2. The Coordinate Plane In the coordinate plane, the horizontal number line (called the x- axis) and the vertical number line (called the y- axis) interest at their zero points called the Origin. Definition: x - axis y - axis Origin

3. The Distance Formula Find the distance between (-3, 2) and (4, 1) x 1 = -3, x 2 = 4, y 1 = 2 , y 2 = 1 The distance d between any two points with coordinates and is given by the formula d = . Example: d = d = d =

4. Midpoint Formula Find the midpoint between (-2, 5) and (6, 4) x 1 = -2, x 2 = 6, y 1 = 5, and y 2 = 4 Example: In the coordinate plane, the coordinates of the midpoint of a segment whose endpoints have coordinates and are . M = M =

5. Slope Formula Find the slope between (-2, -1) and (4, 5). Example: Definition: In a coordinate plane, the slope of a line is the ratio of its vertical rise over its horizontal run. Formula: The slope m of a line containing two points with coordinates and is given by the formula where .

6. Describing Lines Lines that have a positive slope rise from left to right. Lines that have a negative slope fall from left to right. Lines that have no slope (the slope is undefined) are vertical. Lines that have a slope equal to zero are horizontal.

7. Some More Examples Find the slope between (4, -5) and (3, -5) and describe it. Since the slope is zero, the line must be horizontal. Find the slope between (3,4) and (3,-2) and describe the line. Since the slope is undefined, the line must be vertical. m = m =

8. Example 3 : Find the slope of the line through the given points and describe the line. (7, 6) and (– 4, 6) Solution: This line is horizontal. x y (7, 6) up 0 left 11 (-11) (– 4, 6) m

9. Example 4: Find the slope of the line through the given points and describe the line. (– 3, – 2) and (– 3, 8) Solution: This line is vertical. x y (– 3, – 2) up 10 right 0 (– 3, 8) undefined m

10. Practice Find the distance between (3, 2) and (-1, 6). Find the midpoint between (7, -2) and (-4, 8). Find the slope between (-3, -1) and (5, 8) and describe the line. Find the slope between (4, 7) and (-4, 5) and describe the line. Find the slope between (6, 5) and (-3, 5) and describe the line.