Warm-Up How would you describe the roof at the right?

Warm-Up slope Anything that isn’t completely vertical has a slope. This is a value used to describe its incline or decline.

Warmer-Upper pitch The slope or pitch of a roof is quite a useful measurement. How do you think a contractor would measure the slope or pitch of a roof?

Warmer-Upper The slope or pitch of a roof is defined as the number of vertical inches of rise for every 12 inches of horizontal run.

Warmer-Upper The steeper the roof, the better it looks, and the longer it lasts. But the cost is higher because of the increase in the amount of building materials.

3.4 Find and Use Slopes of Lines 3.5 Write and Graph Equations of Lines Objectives: 1.To find the slopes of lines 2.To find the slopes of parallel and perpendicular lines 3.To graph and write equations based on the Slope-Intercept Form, Standard Form, or Point-Slope Form of a Line

Slope Summary Summarize your findings about slope in the table below: m > 0m < 0m = 0m = undef Insert Picture Insert Picture Insert Picture Insert Picture As the absolute value of the slope of a line increases, --?--. GSP the line gets steeper.

Slope of a Line The slope of a line (or segment) through P 1 and P 2 with coordinates (x 1,y 1 ) and (x 2,y 2 ) where x 1  x 2 is ryse

Example 2 Find the slope of the line containing the given points. Then describe the line as rising, falling, horizontal, or vertical. 1.(6, − 9) and ( − 3, − 9) 2.(8, 2) and (8, − 5) 3.(−1, 5) and (3, 3) 4.(−2, −2) and (−1, 5)

Example 3 A line through points (5, -3) and ( − 4, y ) has a slope of − 1. Find the value of y.

Parallel and Perpendicular parallel lines Two lines are parallel lines iff they have the same slope. perpendicular lines Two lines are perpendicular lines iff their slopes are negative reciprocals.

Example 4 Tell whether the pair of lines are parallel, perpendicular, or neither 1.Line 1: through ( − 2, 1) and (0, − 5) Line 2: through (0, 1) and ( − 3, 10) 2.Line 1: through ( − 2, 2) and (0, − 1) Line 2: through ( − 4, − 1) and (2, 3)

Example 5 Line k passes through (0, 3) and (5, 2). Graph the line perpendicular to k that passes through point (1, 2).

Example 6 Find the value of y so that the line passing through the points (3, y ) and ( − 5, − 6) is perpendicular to the line that passes through the points ( − 2, − 7) and (10, 1).

Example 7 Find the value of k so that the line through the points ( k – 3, k + 2) and (2, 1) is parallel to the line through the points ( − 1, 1) and (3, 9).

Intercepts x -intercept The x -intercept of a graph is where it intersects the x -axis. a( a, 0) y -intercept The y -intercept of a graph is where it intersects the y -axis. b(0, b )

Slope-Intercept Slope-Intercept Form of a Line: If the graph of a line has slope m and a y -intercept of (0, b ), then the equation of the line can be written in the form y = mx + b. Equation of a Horizontal Line Equation of a Vertical Line y = b (where b is the y -intercept) x = a (where a is the x -intercept)

Example 9 Find the equation of the line with the set of solutions shown in the table. x 13579… y …

Example 10 Graph the equation:

Slope-Intercept To graph an equation in slope-intercept form: 1.Solve for y to put into slope-intercept form. 2.Plot the y -intercept (0, b ). 3.Use the slope m to plot a second point. 4.Connect the dots.

Example 11 Graph the equation:

Standard Form Standard Form of a Line The standard form of a linear equation is A x + B y = C, where A and B are not both zero. A, B, and C are usually integers.

Standard Form To graph an equation in standard form: 1.Write equation in standard form. 2.Let x = 0 and solve for y. This is your y -intercept. 3.Let y = 0 and solve for x. This is your x -intercept. 4.Connect the dots.

Example 12 Without your graphing calculator, graph each of the following: 1. y = − x y = (2/5) x f ( x ) = 1 – 3 x 4. 8 y = −2 x + 20

Example 13 Graph each of the following: 1. x = 1 2. y = −4

Example 14 A line has a slope of −3 and a y -intercept of (0, 5). Write the equation of the line.

Example 15 A line has a slope of ½ and contains the point (8, − 9). Write the equation of the line.

Point-Slope Form Given the slope and a point on a line, you could easily find the equation using the slope-intercept form. Alternatively, you could use the point-slope form of a line. Point-Slope Form of a Line: A line through ( x 1, y 1 ) with slope m can be written in the form y – y 1 = m ( x – x 1 ).

Example 16 Find the equation of the line that contains the points (−2, 5) and (1, 2).

Example 17 Write the equation of the line shown in the graph.

Example 18 Write an equation of the line that passes through the point (−2, 1) and is: 1.Parallel to the line y = −3 x Perpendicular to the line y = −3 x + 1

Example 19 Find the equation of the perpendicular bisector of the segment with endpoints (-4, 3) and (8, -1).