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