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

Investigation 1 Click on the button and use the activity, to discover something about the actual value of the slope of a line. Then complete the table on the next slide.

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, --?--. the line gets steeper. Copy and complete in your notebook

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 rise

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) 0horizontal undefined vertical -1/2falling 7rising

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

Investigation 2 Use the Geometer’s Sketchpad activity to complete the two following postulates, and then add them to your Because I Said So… Postulate page.

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) -3 & -3 parallel -3/2 & 2/3 perpendicular

Example 5 Line k passes through (0, 3) and (5, 2). Graph the line perpendicular to k that passes through point (1, 2). Find slope, use it’s negative reciprocal to find slope of new line, then use new slope to plot the 2 nd point of new line.

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). -18

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). K=2

Tangent

Tangent tangent A line is a tangent if and only if it intersects a circle in one point.

Investigation 3 Use the Geometer’s Sketchpad activity to discover the relationship between a radius and a line tangent to a circle.

Tangent Line Theorem In a plane, a line is tangent to a circle if and only if the line is perpendicular to a radius of the circle at its endpoint on the circle.

Example 8 The center of a circle has coordinates (1, 2). The point (3, -1) lies on this circle. Find the slope of the tangent line at (3, -1).

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 )

Investigation 4 Use the Geometer’s Sketchpad Activity “Equations of Lines” to complete the Slope-Intercept Form of a Line.

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. 1.Find slope 2.Plug in x, y and slope into y=mx+b 3.Solve for “b” 4.Write the equation using slope and y-intercept x 13579… y … 3 5=3(1) + b 2 y = 3x + 2

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: In your notebook 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: In your notebook 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. Y = -3x + 5

Example 15 A line has a slope of ½ and contains the point (8, − 9). Write the equation of the line. HINT: Plug all value into slope-intercept form first y=1/2x -13

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). y=-x +3

Example 17 Write the equation of the line shown in the graph. y= -1/3x + 1/2

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 y=-3x - 5 y= 1/3x - 5

Example 19 Find the equation of the perpendicular bisector of the segment with endpoints (-4, 3) and (8, -1). HINT: find midpoint, then use that point to find formula of new line y= -3x + 7

Example 20 The center of a circle has coordinates (1, 2). The point (3, −1) lies on this circle. Find the equation of the tangent line at (3, −1).

Assignment Give me 1.5 hours: P : 1-14 all, 16, 19, 23, 26-28, 33, 42, 43 P : 4-44 multiples of 4, 30, 33, 53-59, 67, 68 Challenge Problems