Linear Functions
Compare and Contrast
Yards to Feet
Num
ber o
f Fee
t
Number of Yards
Yards to Square Yards
Length of a Side of a Square Yard
Area
of S
quar
e (y
d2 )
Tables: Linear or Nonlinear
• linear • nonlinear
x y
2 50
4 35
6 20
8 5
x y
1 1
4 16
7 49
10 100
Is the rate of change constant (the same)?
+2
+2
+2
-15
-15
-15
+3
+3
+3
+15
+33
+51
Identify: Linear or Nonlinear Table?
• linear
x y
0 20
5 16
10 12
15 8
x 0 2 4 6
y 0 2 8 18
nonlinear
Graphs: Linear or Nonlinear
Is the graph a straight line?
linear linear nonlinear
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 = x + 4y = 6/x
y = 4y = x3 + 1
y = ½ xlinear
nonlinear
linear
nonlinear
linear
Identify:Linear or Nonlinear Equation?
y = 2/x + 5y = x2 + 8y = .6x1 y = + 13x2
linear linearnonlinear nonlinear
Pointers to Keep in Mind
• A graph is linear if it is a straight line.
A table is linear if the rate of change is constant. There is a common difference.
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
2. Graph
3. Equation x 3 6 9 12
y 12 10 8 6
a b c
y = + 58x
Review of Formulas
Formula for Slope m y2 y1x2 x1
Ax By C
ymx b
y y1 m x x1
Standard Form
Slope-intercept Form
Point-Slope Form
*where A>0 and A, B, C are integers
Change into standard form.
y3
4x 2
4 y4 3
4x 4 2
3x 4y 8
3x 4y8
x-intercepts and y-interceptsThe 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 153x 15
x 5 5,0 is the
-interceptx
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-intercept2x = -12
x = -6(-6, 0)
y-intercept-3y = -12
y = 4(0, 4)
6
4
2
-2
-10 -5
B: (0, 4)
A: (-6, 0)
B
A
Graph using x and y intercepts 6x + 9y = 18
x-intercept6x = 18
x = 3(3, 0)
y-intercept9y = 18
y = 2(0, 2)
4
2
-2
5
D: (0, 2)
C: (3, 0)
D
C
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 = 1Write 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 = -2Write an equation of a line and graph it that passes through (2, 4) and (-3, 4).
y = 4
Review of Formulas
Formula for Slope m y2 y1x2 x1
Ax By C
ymx b
y y1 m x x1
Standard Form
Slope-intercept Form
Point-Slope Form
*where A>0 and A, B, C are integers
Change into slope-intercept form and identify the slope and y-intercept.
3x 5y 15
5y 3x 15 5y
5
3x
5
15
53
35
y x 35
5m and b
Write an equation for the line that passes through (-2, 5) and (1, 7):
Find the slope: m 7 5
1 2
2
3
Use point-slope form: y 5
2
3x 2
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:
Slopem = -4 = -4
1
y-intercept(0, 1)
4
2
-2
-4
5
F: (1, -3)
E: (0, 1)
F
E
Graph using slope-intercept form 3x - 4y = 8
Slopem = 3
4 y-intercept(0, -2)
y = 3x - 2 4
4
2
-2
-4
5
H: (0, -2)
G: (4, 1)G
H