Download Linear Functions

Linear Functions
Compare and Contrast
Yards to Square Yards
Number of Feet
Area of Square (yd2)
Yards to Feet
Number of Yards
Length of a Side of a Square Yard
Tables: Linear or Nonlinear
 Is the rate of change constant (the same)?
+2
+2
+2
x
y
2
50
4
35
6
20
8
5
• linear
-15
-15
-15
+3
+3
+3
x
y
1
1
4
16
7
49
10
100
• nonlinear
+15
+33
+51
Identify: Linear or Nonlinear Table?
x
y
0
20
5
16
10
12
15
8
• linear
x
0
2
4
6
y
0
2
8
18
 nonlinear
Graphs: Linear or Nonlinear
 Is the graph a straight line?
 linear
 nonlinear
 linear
Identify: Linear or Nonlinear Graph?
 linear
 linear
 nonlinear
Equations: Linear or Nonlinear
 REMEMBER: x1 = x and x0 = 1
 In “y = ” form, is x raised to a power of 1 or
0?
 Does x appear in the numerator?
y = x3 + 1
y = 6/x
y=x+4
linear
nonlinear
y=½x
linear
nonlinear
y=4
linear
Identify:Linear or Nonlinear Equation?
y=
.6x1
linear
y=
x2
+8
nonlinear
y=
3x
2
linear
+1
y = 2/x + 5
nonlinear
Pointers to Keep in Mind
 A table is linear if the rate of change is constant.
There is a common difference.
• A graph is linear if it is a straight line.
 An equation is linear if the power of x is either 1
or 0 and it appears in the numerator.
Exit Slip
 Identify if linear or nonlinear.
1. Table A
3. Equation
x
3
6
9
12
y
12
10
8
6
y=
2. Graph
a
b
c
8
x
+5
Review of Formulas
Formula for Slope
y2  y1
m
x 2  x1
Standard Form
Ax  By  C
*where A>0 and A, B, C are integers
Slope-intercept Form
Point-Slope Form
y  mx  b
y  y1  mx  x1 
3
Change y  x  2 into
4
standard form.
3
4 y  4 x 42
4
3x  4y  8
3x  4y  8
x-intercepts and y-intercepts
The intercept is the point(s)
where the graph crosses the axis.
To find an intercept, set the other
variable equal to zero.
3x  5y 15
3x  50 15
3x  15
x 5
 5, 0  is the
x -intercept
Graphing Lines
*You
need at least 2 points to
graph a line.
Using x and y intercepts:
•Find the x and y intercepts
•Plot the points
•Draw your line
Graph using x and y intercepts
2x – 3y = -12
x-intercept
2x = -12
x = -6
(-6, 0)
y-intercept
-3y = -12
y=4
(0, 4)
6
B: ( 0 , 4 )
4
B
2
A: ( -6 , 0 )
-10
A -5
-2
Graph using x and y intercepts
6x + 9y = 18
x-intercept
6x = 18
x=3
(3, 0)
y-intercept
9y = 18
y=2
(0, 2)
4
D: ( 0 , 2 )
2
D
C: ( 3 , 0 )
C
-2
5
Vertical Lines
 Slope is undefined.
 Equation form is x = #.
Write an equation of a line and graph it with
undefined slope and passes through (1, 0).
x=1
Write an equation of a line that passes through
(3, 5) and (3, -2).
x=3
Horizontal Lines
 Slope is zero.
 Equation form is y = #.
Write an equation of a line and graph it with
zero slope and y-intercept of -2.
y = -2
Write an equation of a line and graph it that
passes through (2, 4) and (-3, 4).
y=4
Review of Formulas
Formula for Slope
y2  y1
m
x 2  x1
Standard Form
Ax  By  C
*where A>0 and A, B, C are integers
Slope-intercept Form
Point-Slope Form
y  mx  b
y  y1  mx  x1 
Change 3x  5y 15 into slopeintercept form and identify the
slope and y-intercept.
5y  3x 15
5y 3x 15


5
5 5
3
3
y  x  3 m  5 and b  5
5
Write an equation for the line that
passes through (-2, 5) and (1, 7):
75 2

Find the slope: m 
12 3
Use pointslope form:
2
y  5  x  2 
3
Graphing Lines
Using slope-intercept form y = mx + b:
•Change the equation to y = mx + b.
•Plot the y-intercept.
•Use the numerator of the slope to count the
corresponding number of spaces up/down.
•Use the denominator of the slope to count the
corresponding number of spaces left/right.
•Draw your line.
Graph using slope-intercept form
y = -4x + 1:
Slope
m = -4 = -4
1
4
2
E: ( 0 , 1 )
E
5
y-intercept
(0, 1)
-2
F: ( 1 , -3 )
F
-4
Graph using slope-intercept form
3x - 4y = 8
y = 3x - 2
4
4
2
G: ( 4 , 1 )
G
Slope
m=3
4
5
y-intercept
(0, -2)
-2
H
-4
H: ( 0 , -2 )