7-1 Slope

• Objectives: Find the slope of a line given the coordinates of two points on the line

What is Slope?

- SLOPE -

- SLOPE - +SL

OPE

+

+SLOPE

+

Steepness

Rise

Run

Change in Y

Change in

X

Amount of SlantY=mx + b

The Graph of y = mx +b

• Consider the graph of y = x - 25

4

3

2

1

-2

-3

-4

-5

y

-7 -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 7

x

• Compare to the graph

of y = ½x - 2

• Compare to the graph

of y=2x-2

The Graph of y = mx +b

• Consider the graph of y = x - 25

4

3

2

1

-2

-3

-4

-5

y

-7 -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 7

x

• Compare to the graph

of y = -x - 2

• Compare to the graph

of y = -2x - 2

Determining Slope

Rise=1

Slope= 12Slope=3

Rise=6Run =2

Run =2Rise=12Run =4

Determining Slope

Run = -4

Rise= -2

Slope= -2

Slope= 0 Run = 1

Rise= 0Run = n

Rise= 8= -2

Determining Slope

The Slope is UNDEFINED

Rise= nRun = 0

Determining Slope

(-4, -6)

(2, 5)

•Pick 2 points on the line

•Find the change in the Y-coordinates by subtracting(rise)

•Find the change in the X-coordinates by subtracting(run)

•Write as a ratio (rise/run)

5-(-6)2-(-4)

= 11 6

Determining Slope

(x2, y2)

(x1, y1)

•In general, to find the slope given two points on a line:

•Subtract the Y-coordinates (rise)•Subtract the X-coordinates (run)

•Write as a ratio (rise/run)

Y2-Y1

X2-X1

m = Y1-Y2

X1-X2

=

Slope Summary

Slope = 0Positi

ve Slope

Negative Slope

Negative slope is a downer

Undefined Slope

7-2 Point slope form

• Objectives: Write an linear equation in point slope form given the coordinates of a point on the line and the slope of the line

Point-Slope form

€

y − y1 = m(x − x1)

Slope rise/runX coordinate of known point

Y coordinate of known point

Point-slope form

• Write the equation of a line given the point

(x,y)m

y–y1=m(x–x1)

Y–2= 3(x–5)

3with a slope of(5,2)

7-3 Writing equations in Slope-Intercept form

• Objectives: Write a linear equation in Slope-intercept form given the slope and y intercept

Linear Equations (y = mx + b)

• b = y-intercept• plot (0,b) to get your first point

• m = slope

• written as a fraction slope = rise/run• Lean right if positive• Lean left if negative

7-3• Slope intercept form:

y=mx+b

The y coordinate

Slope rise/run

The x coordinate

The y intercept,Where it crosses the y line

Linear Equations (y = mx + b)

y = 1/3 x - 3m = 1/3 (slope rise/run)

positive leans right

plot up 1, right 3

plot down 1, left 3connect the points

b = -3 (y-intercept)plot (0,-3)

Linear Equations (y = mx + b)

y = -1/2 x + 2m = -1/2 (slope rise/run)

negative leans left

plot up 1, left 2

plot down 1, right 2connect the points

b = 2 (y-intercept)plot (0,2)

7-3

Given the slope m and the y intercept b

write an equation in slope intercept form

m= -3 b= -2

y=mx+b

y= -3x-2

Step1: write the equation

Step 2: substitute the given numbers

7-3• The equation can then be graphed

y= -3x-2

b=-2m= -3

y=-3x-2

Rise 3 run to the left 1

Fall 3 and run to the right 1

7-3• Sometimes you may have to manipulate the equation to

get it in slope-intercept form

4x-2y=8Subtract the 4x from both sides

Divide all by -2 to isolate y-4x -4x

-2y= -4x +8

-2 -2 -2

2 -4

y=2x-4

Given two points, write the equation of a line in y intercept form

Steps:

1. Find slope 2. Place a point and the slope in into point slope form

3. Distributive property

4. Additive property of equality

(2,-3) and (4,-2)

€

−2 − −3

4 − 2=

−2 + 3

2=

1

2

€

y − y1 = m(x − x1)

y − −3 =1

2(x − 2)

y + 3 =1

2(x − 2)

€

y + 3 =1

2(x − 2)

y + 3 =1

2x −1

€

y + 3 =1

2x −1

−3 = −3

y =1

2x − 4