Lesson 3-3: Slopes of Lines TARGETS Find slopes of lines. Use slope to identify parallel and perpendicular lines. Targets

Slope of Line where x1 ≠ x2 LESSON 3-3: Slopes of Lines Slope of a Line

A. Find the slope of the line. LESSON 3-3: Slopes of Lines EXAMPLE 1 Find the Slope of a Line A. Find the slope of the line. S(–3, 7) = (x1, y1) O(–1, –1) = (x2, y2). Slope formula Substitution Simplify. Answer: –4 Example 1

B. Find the slope of the line. LESSON 3-3: Slopes of Lines EXAMPLE 1 Find the Slope of a Line B. Find the slope of the line. P(0, 4) = (x1, y1) O(0, –3) = (x2, y2) Slope formula Substitution Simplify. Answer: The slope is undefined. Example 1

C. Find the slope of the line. LESSON 3-3: Slopes of Lines EXAMPLE 1 Find the Slope of a Line C. Find the slope of the line. G(–2, –5) = (x1, y1) (6, 2) = (x2, y2). Slope formula Substitution Simplify. Answer: Example 1

D. Find the slope of the line. LESSON 3-3: Slopes of Lines EXAMPLE 1 Find the Slope of a Line D. Find the slope of the line. X(–2, –1) = (x1, y1) (6, –1) = (x2, y2). Slope formula Substitution Simplify. Answer: 0 Example 1

Classifying Slope Zero Slope Positive Slope Undefined Slope LESSON 3-3: Slopes of Lines Classifying Slope Zero Slope Positive Slope Undefined Slope Negative Slope Classifying Slope

The sales increased at an average of $7.4 million per year. LESSON 3-3: Slopes of Lines EXAMPLE 2 Use Slope as Rate of Change RECREATION In 2000, the annual sales for one manufacturer of camping equipment was $48.9 million. In 2005, the annual sales were $85.9 million. If sales increase at the same rate, what will be the total sales in 2015? Steps 1) Use given data [(0, 48.9) & (5, 85.9)] to graph the line that models the annual sales y as a function of the years x since 2000. The sales increase is constant. 2) Find the slope of the line. Use this rate of change to find the amount of sales in 2015. Use the slope Formula to find the slope. The sales increased at an average of $7.4 million per year. Example 2

Slope formula Use Slope as Rate of Change LESSON 3-3: Slopes of Lines EXAMPLE 2 Use Slope as Rate of Change 3) Use the slope of the line and one known point on the line to calculate the sales y when the years x since 2000 is 15. Slope formula m = 7.4, (x1, y1) = (0, 48.9), (x2, y2) = (15, y2) Solve for y2 Answer: Thus, the sales in 2015 will be about $159.9 million. Example 2

Parallel & Perpendicular LESSON 3-3: Slopes of Lines Parallel & Perpendicular Lines Slopes of Parallel Lines Postulate If line l and line m are parallel and nonvertical, then the lines have the same slope. Slopes of Perpendicular Lines Postulate Line p and line m are perpendicular if and only if the product of their slopes is -1. Parallel & Perpendicular

Step 1 Find the slopes of and . LESSON 3-3: Slopes of Lines EXAMPLE 3 Determine Line Relationships Determine whether and are parallel, perpendicular, or neither for F(1, –3), G(–2, –1), H(5, 0), and J(6, 3). Graph each line to verify your answer. Step 1 Find the slopes of and . Step 2 Determine the relationship, if any, between the lines. Slopes are not the same = not parallel The product of the slopes, , so they are not perpendicular Example 3

Answer: The lines are neither parallel nor perpendicular. LESSON 3-3: Slopes of Lines EXAMPLE 3, cont. Determine Line Relationships Answer: The lines are neither parallel nor perpendicular. Check When graphed, you can see that the lines are not parallel and do not intersect in right angles. Example 3

The slopes of two parallel lines are the same. LESSON 3-3: Slopes of Lines EXAMPLE 4 Use Slope to Graph a Line Graph the line that contains Q(5, 1) and is parallel to MN with M(–2, 4) and N(2, 1). First find the slope of . The slopes of two parallel lines are the same. The slope of the line parallel to through Q(5, 1) is . Graph the line. Start at (5, 1). Move up 3 units and then move left 4 units. Label the point R. Answer: Draw . Example 4