Intro U4D9 Warmup Evaluate each equation for x = –1, 0, and y = 3x 2. y = x – 7 3. y = 2x y = 6x – 2 –3, 0, 3 –8, –7, –6 3, 5, 7 –8, –2, 4 Collect Interims

HW Check: Collect U4D8 HW WS Mad Minute B52 ONE MINUTE MAXIMUM

Intro U4 D8 SLOPE Objective: To evaluate the slope and y- intercept of a function.

OBJ: Learn to find the slope of a line and use slope to understand and draw graphs. Slope of a Line 21 MAR 16

Linear equations have constant slope. For a line on the coordinate plane, slope is the following ratio: vertical change horizontal change change in y change in x = This ratio is often referred to as, or “rise over run,” where rise indicates the number of units moved up or down and run indicates the number of units moved to the left or right. Slope can be positive, negative, zero, or undefined. A line with positive slope goes up from left to right. A line with negative slope goes down from left to right. rise run

If you know any two points on a line, or two solutions of a linear equation, you can find the slope of the line without graphing. The slope of a line through the points (x 1, y 1 ) and (x 2, y 2 ) is as follows: y2 – y1y2 – y1x2 – x1x2 – x1y2 – y1y2 – y1x2 – x1x2 – x1

Find the slope of the line that passes through (–2, –3) and (4, 6). Additional Example 1: Finding Slope, Given Two Points Let (x 1, y 1 ) be (–2, –3) and (x 2, y 2 ) be (4, 6). 6 – (–3) 4 – (–2) Substitute 6 for y 2, –3 for y 1, 4 for x 2, and –2 for x = The slope of the line that passes through (–2, –3) and (4, 6) is. 3 2 = y 2 – y 1 x 2 – x =

Find the slope of the line that passes through (–4, –6) and (2, 3). Try This: Example 1 Let (x 1, y 1 ) be (–4, –6) and (x 2, y 2 ) be (2, 3). 3 – (–6) 2 – (–4) Substitute 3 for y 2, –6 for y 1, 2 for x 2, and –4 for x = The slope of the line that passes through (–4, –6) and (2, 3) is. 3 2 = y 2 – y 1 x 2 – x =

Use the graph of the line to determine its slope. Additional Example 2: Finding Slope from a Graph

Additional Example 2 Continued Choose two points on the line: (0, 1) and (3, –4). Guess by looking at the graph: rise run = –5 3 = – 5 3 Use the slope formula. Let (3, –4) be (x 1, y 1 ) and (0, 1) be (x 2, y 2 ). 1 – (–4) 0 – 3 = y 2 – y 1 x 2 – x 1 5 –3 = 5 3 = – –5 3

Notice that if you switch (x 1, y 1 ) and (x 2, y 2 ), you get the same slope: 5 3 The slope of the given line is –. Let (0, 1) be (x 1, y 1 ) and (3, –4) be (x 2, y 2 ). Additional Example 2 Continued –4 – 1 3 – 0 = y 2 – y 1 x 2 – x 1 –5 3 = 5 3 = –

Use the graph of the line to determine its slope. Try This: Example 2

Try This: Example 2 Continued Choose two points on the line: (1, 1) and (0, –1). Guess by looking at the graph: rise run = 2 1 = 2 Use the slope formula. Let (1, 1) be (x 1, y 1 ) and (0, –1) be (x 2, y 2 ). = y 2 – y 1 x 2 – x 1 –2 –1 = –1 – 1 0 – 1 = 2 1 2

Recall that two parallel lines have the same slope. The slopes of two perpendicular lines are negative reciprocals of each other.

Additional Example 3A: Identifying Parallel and Perpendicular Lines by Slope Tell whether the lines passing through the given points are parallel or perpendicular. A. line 1: (–6, 4) and (2, –5); line 2: (–1, –4) and (8, 4) slope of line 1: slope of line 2: Line 1 has a slope equal to – and line 2 has a slope equal to, – and are negative reciprocals of each other, so the lines are perpendicular = y 2 – y 1 x 2 – x 1 –9 8 = –5 – 4 2 – (–6) 4 – (–4) 8 – (–1) = y 2 – y 1 x 2 – x = 9 8 = –

Additional Example 3B: Identifying Parallel and Perpendicular Lines by Slope B. line 1: (0, 5) and (6, –2); line 2: (–1, 3) and (5, –4) Both lines have a slope equal to –, so the lines are parallel. 7 6 slope of line 1: slope of line 2: = y 2 – y 1 x 2 – x 1 –7 6 = –2 – 5 6 – 0 = y 2 – y 1 x 2 – x = – –7 6 = 7 6 = – –4 – 3 5 – (–1)

Try This: Example 3A Tell whether the lines passing through the given points are parallel or perpendicular. A. line 1: (–8, 2) and (0, –7); line 2: (–3, –6) and (6, 2) slope of line 1: slope of line 2: Line 1 has a slope equal to – and line 2 has a slope equal to, – and are negative reciprocals of each other, so the lines are perpendicular = y 2 – y 1 x 2 – x 1 –9 8 = –7 – 2 0 – (–8) 2 – (–6) 6 – (–3) = y 2 – y 1 x 2 – x = 9 8 = –

Try This: Example 3B B. line 1: (1, 1) and (2, 2); line 2: (1, –2) and (2, -1) Line 1 has a slope equal to 1 and line 2 has a slope equal to –1. 1 and –1 are negative reciprocals of each other, so the lines are perpendicular. slope of line 1: slope of line 2: = y 2 – y 1 x 2 – x = 2 – 1 = y 2 – y 1 x 2 – x 1 –1 1 = –1 – (–2) 2 – (1) = 1 = –1

Additional Example 4: Graphing a Line Using a Point and the Slope Graph the line passing through (3, 1) with slope 2. Plot the point (3, 1). Then move 2 units up and right 1 unit and plot the point (4, 3). Use a straightedge to connect the two points. The slope is 2, or. So for every 2 units up, you will move right 1 unit, and for every 2 units down, you will move left 1 unit. 2 1

Additional Example 4 Continued 1 2 (3, 1)

Try This: Example 4 Graph the line passing through (1, 1) with slope 2. Plot the point (1, 1). Then move 2 units up and right 1 unit and plot the point (2, 3). Use a straightedge to connect the two points. The slope is 2, or. So for every 2 units up, you will move right 1 unit, and for every 2 units down, you will move left 1 unit. 2 1

Try This: Example 4 Continued 1 2 (1, 1)

HOMEWORK CW: U4D9 CW WS HW: U4D9 HW WS