Definition of Slope

The slope of the line through the distinct points (x1, y1) and (x2, y2) is

where x2 – x1 = 0.

Definition of Slope

The slope of the line through the distinct points (x1, y1) and (x2, y2) is

where x2 – x1 = 0.y1

y2

yChange in y

Change in x=

Rise

Run=

y2 – y1

x2 – x1

(x1, y1)

x1 x2

(x2, y2)Risey2 – y1

Runx2 – x1

x

Date:Topic: Lines and Slope (1.2)

Find the slope of the line with points (3, 2) and (-5, 5)

slope 5 2 5 3

3

8

The Possibilities for a Line’s Slope

Positive Slope

x

y

m > 0

Line rises from left to right.

Zero Slope

x

y

m = 0

Line is horizontal.m is

undefined

Undefined Slope

x

y

Line is vertical.

Negative Slope

x

y

m < 0

Line falls from left to right.

m = 2 0

Slope-Intercept Form of the Equation of a nonvertical line with slope m and y-intercept b is

y = mx + b

Find the slope and the y-intercept of the line whose equation is 2x – 3y + 6 = 0.

+ 3y +3y 2x – 3y + 6 = 0

To isolate the y-term, add 3y on both sides.2x + 6 = 3y

Reverse the two sides. (This step is optional.)3y = 2x + 6

Divide both sides by 3.y = 2/3x + 2 3 3

The slope is

2/3.

The y-intercept is 2.

Graphing y=mx+b Using the Slope and Y-Intercept

• Plot the y-intercept, b.• Plot a second point using the slope, m, rise over run.• Draw a line through the two points.

Graph the line whose equation is:y = 2/3 x + 2

The slope is 2/3.

The y-intercept is 2.

We plot the second point on the line by starting at (0, 2), the first point. Then move 2 units up (the rise) and 3 units to the right (the run). This gives us a second point at (3, 4).

-5 -4 -3 -2 -1 1 2 3 4 5

5

4

3

2

1

-1-2

-3

-4-5

We need two points in order to graph the line: We can use the y-intercept, 2, to obtain the first point (0, 2).

Point-Slope Form of the equation of a nonvertical line

of slope m that passes through the point (x1, y1) is

y – y1 = m(x – x1)Write the point-slope form of the equation of the line passing through (-1,3) with a slope of 4. Then solve the equation for y.

Solution We use the point-slope equation of a line with m = 4, x1= -1, and

y1 = 3.

This is the point-slope form of the equation.y – y1 = m(x – x1)

Substitute the given values.y – 3 = 4[x – (-1)]

We now have the point-slope form of the equation for the given line.y – 3 = 4(x + 1)

We can solve the equation for y by applying the distributive property.y – 3 = 4x + 4

y = 4x + 7

+3 + 3 Add 3 to both sides.

If given two points find the slope using the points, and use one of the coordinates in the equation

Equations of Lines

Point-slope form: y – y1 = m(x – x1)

Slope-intercept form: y = mx + b

Horizontal line: y = b

Vertical line: x = a

General form: Ax + By + C = 0

Complete Student CheckpointIndicate whether each line has a positive, negative, zero, or undefined slope.

Write the equation of the line in slope-intercept form. (Assume scale is 1)

x-axis

y-axisa

bcd

x-axis

y-axis

positive

undefined

zero

negative

y-intercept3

-3

6slope-1/2

equation:

y 1

2x 3

Use the given conditions to write an equation in point-slope form and slope-intercept form.Passing through (-3,2) and (3,6). Send you answer to me using the calculator

y y1 m(x x1)

y (2) 6 2

3 ( 3)(x ( 3))

y 2 2

3(x 3)

y 2 2

3x 2

y 2

3x 4

Slope and Parallel LinesIf two nonvertical lines are parallel, then they have the same slope.

Write an equation of the line passing through (-3, 2) and parallel to the line whose equation is y = 2x + 1. Express the equation in point-slope form and y-intercept form.

y = 2x + 1

-5 -4 -3 -2 -1 1 2 3 4 5

5

4

3

2

1

-1-2

-3

-4-5

(-3, 2)

Rise = 2

Run = 1y – y1 = m(x – x1)

y1 = 2 m = 2 x1 = -3

Parallel lines have the same slope. Because the slope of the given line is 2, m = 2 for the new equation.

y – 2 = 2[x – (-3)]

y – 2 = 2(x + 3)

y – 2 = 2x + 6 Apply the distributive property.

y = 2x + 8 This is the slope-intercept form of the equation.

DAY 2

Slope and Perpendicular Lines

Slope and Perpendicular Lines• If two nonvertical lines are perpendicular, then the

product of their slopes is –1.(2/3) • (-3/2) = -1

• Slopes are negative reciprocals of each other

Slope and Perpendicular Lines• If two nonvertical lines are perpendicular, then the

product of their slopes is –1.(2/3) • (-3/2) = -1

• Slopes are negative reciprocals of each other

Two lines that intersect at a right angle (90°) are said to be perpendicular. There is a relationship between the slopes of perpendicular lines.

90°

Find the slope of any line that is perpendicular to the line whose equation is 2x + 4y – 4 = 0.

Slope is –1/2.

y = -1/2x + 1

Any line perpendicular to this line has a slope that is the negative reciprocal, 2.

4y = -2x + 4

You TryWrite an equation of the line passing through (-3,6) and perpendicular to the line whose equation is y=1/3 x +4 Express in point-slope form and slope-intercept form.

Write an equation of the line passing through (-2,5) and parallel to the line whose equation is y=3x+1. Express in point-slope form and slope-intercept form.

y y1 m(x x1)

y 5 3(x ( 2))

y 5 3x 6

y 3x 11

perpendicular slope: 3

1 3

y y1 m(x x1)

y 6 3(x ( 3))y 6 3x 9

y 3x 3

Graphs and Viewing Windows• Bob purchased a house 8 years ago for $42,000. This

year is was appraised at $67,500.a) A linear equation V=mt + b, 0 ≤ t ≤ 15, represents

the value V of the house for 15 years after it was purchased. Determine m and b.

b) Graph the equation and trace to estimate in how many years after the purchase of this house it will be worth $72,500

c) Write and solve an equation algebraically to determine how many years after purchase this house will be worth $74,000.