geo 3.6&7 slope

Review of slope from Algebra IGraphing lines of a known slope and intercept

Slope Review from Algebra I

Slope

The ratio that describes the tilt of a line is its slope.

To calculate slope, you use this ratio.

Slope = (Vertical Change) = Rise

(Horizontal Change) Run

Slope Equation

m = y2 – y1

x2 – x1

m is the slopepoints (x1 , y1) & (x2 , y2)

Slope

A positive slope rises to the right

A negative slope falls to the right

Finding Slope on a GraphRemember: Rise over Run.

Rise: 2

Run: 4

Rise: -3

Run: 2

Ratio is 2/4 Ratio is -3/2

We’re reading from left to right. So start at the left most point and then figure out how to get to the next point.

Finding Slope from 2 Points You can find the slope of the line

using the ratio. slope = difference of y – coordinates

difference of x – coordinates.

The y-coordinate you use first in the numerator must correspond to the x-coordinate you use first in the denominator.

Slope Equation

m = y2 – y1

x2 – x1

m is the slopepoints (x1 , y1) & (x2 , y2)

Find the slope of the line through C(-2, 6) and D(4, 3)Slope = difference in y-coordinates difference in x-coordinates

= (3 – 6) y-coordinates (4 – (-2)) x-coordinatesSlope = -3 / 6 = -1/2Down 1, to the Right 2. Cause of Rise

(of –1) over Run (+2).

Find the Slope of the Line through each pair of points:

V(8, -1) and Q(0, -7)

S(-4, 3) and R(-10, 9)

Find the Slope of the Line through each pair of points:

V(8, -1) and Q(0, -7)

S(-4, 3) and R(-10, 9)

m = 3/4

m = -1

= (-1 / 1) if you need a ratio

Special Cases

Horizontal and Vertical lines are special cases

This is a horizontal line.

The points are (-3, 2) and (1, 2).

Therefore, Y = 2.

Find the slope.

Slope = (2 – 2) / (1 – (-3) = 0 /4 = 0

The slope for a horizontal line (or anything Y = ?) is zero.

Special Cases Horizontal and Vertical lines are

special casesThis is a vertical line.

The points are (-4, 1) and (-4, 3).

Therefore, X = -4.

Find the slope.

Slope = (1 – 3) / (-4 – (-4) = -2 /0 = Undefined

Slope is, therefore, UNDEFINED for vertical linesSlope is, therefore, UNDEFINED for vertical lines.

Finding the Equation of a Line

Formats for a Linear Equation

Standard Form: ax + by = c

Slope-Intercept : y = mx + b

Use your properties of algebra to convert between the two

(Addition Property, Division Property, etc)

Finding the Equation of a Line

Use your slope equation with any point on the line and the point (x, y)

For example the points C(-2, 6) and D(4, 3) earlier had a slope of -1/2

m = y2 – y1 -1 = y – 6

x2 – x1 2 x – (-2)

2( y-6 ) = -1 ( x – (-2) )2y - 12 = -x +2y = (-1/2) x + 7

Graphing Lines

Graphing Lines

This is the graph of y=(-1/2)x + 3.

The slope of the line is (-2/4) or (-1/2).

The Y-INTERCEPT of the line is the point where the line crosses the Y-AXIS.

• The CONSTANT in the equation is the same as the y-intercept.

Graphing Lines

This is the graph of y=(-1/2)x + 3.

The slope of the line is (-2/4) or (-1/2).

y = (-1/2)x + 3

Slope always a ratio

For whole numbers divide by 1

Y-Intercept = Constant

Using Slope-Intercept Form Using the Slope-Intercept Form, you

can graph without having to pick points and make a table.

y = mx + b Slope-Intercept Formy = mx + b Slope-Intercept Form m = Slope of the line. (Ratio) b = Y-Intercept. (Constant) Linear Equations can always be put

in this format. It is like solving for y.

To Graph with y = mx + b

1) Start with b. Since b is where the line of the equation hits the y-axis, its your first point. Point = (0, b)

2) Take the slope, or m. Starting at b, move along the RISE and RUN of the ratio.

3) Where you end up is your second point.4) Connect the two dots with a line. (This

is the graph of your linear equation).

Lets Graph Together!

y = (-1/3)x + 2

Lets Graph Together!

y = (-1/3)x + 2 1) b = 2 so, plot

(0, 2)

(0, 2)

Lets Graph Together!

y = (-1/3)x + 2 1) b = 2 so, (0, 2)2) Rise: -1, Run:

+3

(0, 2)

Lets Graph Together!

y = (-1/3)x + 2 1) b = 2 so, (0, 2)2) Rise: -1, Run:

+33) Graph next

dot.(0, 2)

(2, 1)

Lets Graph Together!

y = (-1/3)x + 2 1) b = 2 so, (0, 2)2) Rise: -1, Run:

+33) Graph next dot4) Connect dots

with straight line

(0, 2)(2, 1)

Finding Parallel and Perpendicular Lines

Parallel Lines

Parallel lines have the same slope

Find the equation using the same process we used above with the slope and the new point

Example of Parallel Line

Find a line parallel to y = (-1/2)x + 7 through point ( 10, 3)

-1 = y – 3 2 x – 102(y – 3) = -1 (x – 10)Cross multiplied2y – 6 = -x + 10 Distributive

Property2y = -x +16 Added 6 to both

sides y = (-1/2)x + 8 Divided by 2

Perpendicular Line

The slope of a perpendicular line is the negative inverse of the original slope

For example, if the original slope is -1/2, the perpendicular slope is 2 (Ratio form 2/1)

To find a perpendicular line through a given point, use the perpendicular slope and the given point in the slope equation

Example of Perpendicular Line Find a line perpendicular to y = (-1/2)x

+ 7 through point ( 10, 3) Perpendicular slope = 2 (same as 2 / 1

) 2 = y – 3 Slope equation 1 x – 101 (y – 3) = 2 (x – 10) Cross multipliedy – 3 = 2x - 20 Distributive

Propertyy = 2x - 17Added 3 to both sides

Practice

Even problems in the sets below Textbook p168: even of (12-18, 24,

34-42) Textbook p175: even of (8-16, 24-32,

38-44)