Objective: Find the slope of a line given two point Objective: Find the slope of a line given two point. Classify lines by their slope. Compare steepness of lines.

( 𝑥 1 ,𝑦 1 ) ( 𝑥 2 ,𝑦 2 ) 𝑚= 𝑦 2 − 𝑦 1 𝑥 2 − 𝑥 1 𝑚= 𝑟𝑦𝑠𝑒 𝑟𝑢𝑛 Slope The slope is the ratio of vertical change (the rise) to horizontal change (the run). ( 𝑥 1 ,𝑦 1 ) ( 𝑥 2 ,𝑦 2 ) 𝑚= 𝑦 2 − 𝑦 1 𝑥 2 − 𝑥 1 𝑚= 𝑟𝑦𝑠𝑒 𝑟𝑢𝑛 The slope of the line is the same regardless of which two points are used. When calculating the slope of a line, be careful to subtract the coordinates in the correct order.

Classifying Lines by Slope Have student sketch the direction of the slope and all the written information on the picture One of the important uses of slope is to decide whether the dependent variable y decreases, increases, or is constant as the independent variable x increases.

Classifying Lines by Slope Remember ! ! ! All linear equations are functions except the equation of a vertical line.

Example 1 Find the slope. Confirm your answer using the slope formula: (−6,4) and (−3,−2) Then tell whether the line rises, falls, is horizontal or is vertical. Notice that the line falls from left to right and that the slope of the line is negative.

Example 2 Use the formula to find the slope of the line that passes through: (4,7) and (2,3). Confirm your answer using the given points on the graph. Then tell whether the line rises, falls, is horizontal or is vertical. Notice that the line rises from left to right and that the slope of the line is positive.

Example 3 Use the formula to find the slope of the line that passes through: (−1,1) and (3,1). Confirm your answer using the given points on the graph. Then tell whether the line rises, falls, is horizontal or is vertical.

Example 4 Use the formula to find the slope of the line that passes through: (3, 2) and (3, − 3). Confirm your answer using the given points on the graph. Then tell whether the line rises, falls, is horizontal or is vertical. Under fined

Example 5 Use the formula to find the slope of the line that passes through: Then tell whether the line rises, falls, is horizontal or is vertical.

The Adventures of Slope Dude

Comparing Steepness of Lines The larger the absolute value, the steeper the slope. Examples: Tell which line is steeper. 𝑚=−5 or 𝑚=3 𝑚=−2 or 𝑚=2 Line 1: m = 3/2 Line 2: m = 1

Comparing Steepness of Lines Examples: Tell which line is steeper. Line 1: through 1,−4 and 5,2 Line 2: through −2,−5 and 1,−2 Line 1 Line 2 Line 1: m = 3/2 Line 2: m = 1

Essential Questions When using the definition of slope, how do I know which point to call 𝑃 1 (𝑥 1 , 𝑦 1 ) and which to call 𝑃 2 (𝑥 2 , 𝑦 2 )? It makes no difference which point you call 1 and which point you call 2. The slope is still the same. Is it OK to say that a vertical line has no slope? It’s not a good idea since “no slope” could mean slope is zero or that the slope is undefined. Does the slope of a line remain the same regardless of which two points are used?

Objective: Classify the slopes of parallel and perpendicular lines. Sec 1.4g

Parallel & Perpendicular Lines Two lines are parallel if they do not intersect. Two lines in a plane are perpendicular if they intersect to form a right angle. Slopes can be used to determine whether two different lines are parallel or perpendicular.

Slopes of Parallel & Perpendicular Lines Two lines are parallel if and only if they have the same slope. Two lines are perpendicular if and only if their slopes are opposite reciprocals of each other. The term opposite reciprocals refers to two numbers that have opposite signs and are flipped fractions of each other. For example, 3 and − 1 3 2 3 and − 3 2 When you multiply opposite reciprocals you always get a −1.

Example 1 Tell whether the lines are parallel, perpendicular, or neither. Line 1: through (− 1, 2) and (1, 8) Line 2: through (−2, − 9) and (3,6) parallel

Example 2 Tell whether the lines are parallel, perpendicular, or neither. Line 1: through (−2,−2) and (4, 1) Line 2: through (−3, −3) and (1,5) Neither just reciprocals

Example 3 Tell whether the lines are parallel, perpendicular, or neither. Line 1: through (4,−3) and (1, 4) Line 2: through (−3, −3) and (0,4) Neither just opposites

Example 4 Tell whether the lines are parallel, perpendicular, or neither. Line 1: through (4,−3) and (1, 4) Line 2: through (3, −1) and (4, 2) perpendicular

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