The Family of Linear Functions
Parent Function: f(x) = x or y = x
Standard Form: f(x) = ax + b or y = ax + b
Vertical Shift:
Examples:
(Similar to Translation in Geometry)
f(x) = x + b
b>0
b<0
This results in a Vertical Shift b units up.
This results in a Vertical Shift b units down.
f(x) = x + 5
Compared to the parent functions, y = x +5 has a
vertical shift 5 units up. Instead of a y-intercept of (0,0),
it has a y-intercept of (0,5).
f(x) = x – 5
Compared to the parent functions, y = x +5 has a
vertical shift 5 units up. Instead of a y-intercept of (0,0),
it has a y-intercept of (0,5).
Vertical Shrink or Stretch:
Examples:
(Similar to Dilation in Geometry)
f(x) = ax
|a| > 1
This results in a Vertical Stretch with a Scale
Factor of a.
f(x) = 5x
Compared to y=x, y=5x is a vertical stretch, with a
scale factor of 5. It has a steeper slope of 5 than the
parent function’s slope of 1.
f(x) =
0 < |a| < 1
Factor of a.
This results in a Vertical Shrink with a Scale
x
Compared to y=x, y=
x is a vertical shrink, with a
scale factor of
. It has a less steep slope of versus
the parent functions slope of 1.
Reflection in the x-axis:
Examples:
(Similar to Reflection in Geometry)
f(x) = ax
a<0
This results in a reflection in the x-axis, which appears
as a mirror image of the parent function.
f(x) = −5x
This transformation causes a Reflection in the
x-axis combined with a Vertical Stretch with a scale
factor of 5. The slope is negative, compared to the
parent function’s positive slope.
Combination of Transformations:
Examples:
f(x) = −ax + b
Compared to the parent function, this function has a
vertical shift of b units, a vertical stretch or shrink with
a scale factor of |a| and a reflection in the x-axis.
f(x) = − x + 6
Compared to the parent function, this function has a
vertical shift of 6 units up, a vertical shrink with a
scale factor of and a reflection in the x-axis.
Other Features:
Examples
Checklist for Items to Include for Each Family of Functions:
1.
2.
3.
4.
5.
6.
7.
Parent Function:
Equation
Graph
Description
Standard Form of the Function:
Equation
Description
Example with Graph
Vertical Shift:
Standard Form(s) of Equation
Comparison to Parent Function, Including Example Equation, Graph, Description
Vertical Stretch and Shrink:
Standard Form(s) of Equation
Comparison to Parent Function, Including Example Equation, Graph, Description
Reflection:
Standard Form(s) of Equation
Comparison to Parent Function, Including Example Equation, Graph, Description
Combination of 2 or More Transformations Compared to the Parent Function
Example Equation, Graph, Description
Other Features, As Appropriate to the Family
Examples for Different Function Variants Illustrated with a Labeled Graph
Include Any Required Formulas
Features Include:
Axis of Symmetry
Vertex
Maximum and Minimum Value
Odd vs. Even Function
End Behavior
Asymptote