2.3 Families of Functions, Transformations, and Symmetry Flashcards
a general term for four specific ways to manipulate the shape and/or position of a point, line, or geometric figure
geometric transformation
the use of a standard form of a function where a, h, and k are real numbers and “a ≠ 0” that uses the general form of a parent function
algebraic transformation
In the equation below, what do a, h, and k stand for with regards to transforming a graph?
horizontally (h)
vertically (k)
stretch (|a| >1)
compress (0 < |a| < 1)
reflection (a < 1)
all the transformations of a function with similar graphs
family of functions
the graph of any function in the square or quadratic family
parabola
moving the graph left or right without changing its shape so that it coincides with another graph
horizontal translation
Name this transformation
translation
Name this transformation
reflection
Name this transformation
rotation
Name this tranformation
dilation
In the equation below, name the parent function. Then, describe what transformation has occurred?
Square Root Function
Horizontal Shift to the Right 3 units
In the equation below, name the parent function. Then, describe what transformation has occurred?
Square Root Function
Horizontal Shift to the Left 5 units
In the equation below, name the parent function. Then, describe what transformation has occurred?
Absolute Value Function
Horizontal Shift to the Right 1 unit
In the equation below, name the parent function. Then describe what transformation has occurred?
Quadratic
Horizontal Shift to the left 3 units
In the equation below, name the parent function. Then describe what transformation has occurred?
Quadratic
Horizontal Shift to the right 2 units
a graph that has a mirror image of one another
reflection
How do you determine if there is a reflection over the x-axis?
There is a negative in front of the grouping symbol or parent function.
How do you determine if there is a reflection over the y-axis?
There is a negative in front of the variable inside a grouping symbol.
What kind if transformation is this?
Reflection over the x-axis
What kind of transformation is this?
Reflection over the y-axis
Name the transformation for the following graph
Reflection over the x-axis
Name the transformation for the following graph
Reflection over the y-axis
Name the parent function and the transformation that has occurred in the following equation.
Square Root
Reflection over the y-axis
Name the parent function and the transformation that has occurred in the following equation.
Absolute Value
Reflection over the x-axis
What kind of transformation occurs when
Vertical Stretch by “a”
Horizontal Compression by “1/a”
What kind of transformation occurs when
Vertical compression by “a”
Horizontal Stretch by “1/a”
_______________________ are defined for POSITIVE value so “a”
Stretching and Shrinking
If “a” is ___________________, then a reflection occurs along with stretching and shrinking as long as “a” does not equal 1.
Negative
What kind of transformation occurs when
Vertical Translation up k units
What kind of transformation occurs when
Vertical Translation down k units
What kind of transformation occurs when
Vertical translation down 3 given that the parent function is the square root
a transformation that does not change the shape of a graph
rigid transformation
a transformation that changed the shape of the graph
nonrigid transformation
what are the rigid transformations
translating horizontally
translating vertically
reflections
what are the nonrigid transformations
stretching
shrinking
What is the order in which transformations should be performed?
Horizontal Translations (h)
Reflecting/stretching/shrinking (a)
Vertical translations (k)
What pneumonic device to help remember the order of transformations?
H-A-K
What does H-A-K stand for?
Horizontal Translations (h)
Reflecting/stretching/shrinking (a)
Vertical translations (k)
Identity Function
Slope Intercept
Linear Family
a transformation of the identity function where “a” cannot be 0 which can be simplified to a function in slope intercept form
Constant Function
a linear function with slope =0 in the form of:
What is meant to be symmetric about the y-axis?
What is meant to be symmetric about the origin?
Another name for symmetric about the y-axis
Even Function
Another name for symmetric about the origin
Odd Function
How do you test algebraically if a function is even or odd?
Plug in a -x for every x and simplify. If the original function is the result, the function is even. If the opposite of the original function is the result, the function is odd.
How do you solve an inequality by graphing?
Set one side to 0.
Graph the function with a x/y table or transformation rules.
If there is a > symbol, find the intervals above the x-axis.
If there is a < symbol, find the intervals below the x-axis.
The answer should be represented in interval notation.
