Family Characteristics
What is the general parent equation and signature graph shape of the Absolute Value function family?
Equation: f(x)=|x|; Shape: "V"-Shape
In the function f(x) = |x| + 4, what specific shift occurs compared to the parent function?
Shift UP 4 units; vertical shift where k > 0
When finding the intervals of increase, decrease, or constant on a graph, which variable's values (x or y) do you use to write the interval?
The x-values.
Write the equation for an absolute value parent function that has been shifted DOWN 5 units.
f(x) = |x| - 5
What is the zero of a function?
The $x$-intercept (the value of x when y = 0).
What is the standard domain of the parent quadratic function f(x)=x2 written in interval notation?
(-inf, inf)
What transformation takes place when 1/2 is placed directly in front of the entire function, such as f(x) = 1/2(x)3
A vertical compression; the graph compresses toward the x-axis
What is the definition of a y-intercept of a graph?
The point where the graph crosses the y-axis (where x = 0).
Write the equation of a quadratic function that has been shifted RIGHT 2 units.
f(x) = (x-2)^2
Identify the parent function and transformations for f(x) = 2^(2x)
Parent Function: Exponential (y = 2^x); Transformation: Horizontal compression by a factor of 1/2
Which function family's parent graph features both a horizontal asymptote and a vertical asymptote?
The rational function family; parent function: f(x)=1/x
Describe the directional shift and number of units for the function f(x) = (x+3)3 + 4
Shift LEFT 3 units, horizontal shift of -3; UP 4 units, vertical shift of 4 units
Describe the end behavior for a typical parent cubic function f(x)=x^3 as x -> -inf and x -> inf
As x -> -inf, f(x) -> -inf; As x -> inf, f(x) -> inf
Write the equation of a cubic parent function that is vertically stretched by a factor of 4 and shifted UP 1 unit.
f(x) = 4x^3 + 1
If a graph is flat (horizontal) between x = - 2 and x = 3, how do you express this characteristic?
The graph is constant on the interval (-2, 3).
Name the parent family function that has a restricted domain of [0, inf) and a restricted range of [0, inf).
The square root function family; parent function: f(x)= sqrt(x)
Describe the difference between the transformations caused by f(x)=2|x| and f(x)=1/2|x|
2|x| causes a vertical stretch by a factor of 2, while 1/2|x| causes a vertical compression by a factor of 1/2
How do you determine whether a relative extrema point on a graph is a local maximum or a local minimum?
A local maximum is a peak where the graph turns from increasing to decreasing; a local minimum is a valley where it turns from decreasing to increasing.
Write the equation for a quadratic function shifted LEFT 3 units, and shifted UP 4 units.
f(x) = (x+3)^2 + 4
What is the end behavior for the logarithmic parent function?
As x -> 0, f(x) -> -inf
As x -> inf, f(x) -> inf
How does the range of a cubic parent function f(x)= x3 differ from the range of a quadratic parent function f(x)= x2 in interval notation?
The cubic range is (-inf, inf) while the quadratic range is restricted to [0, inf).
List all three transformations in correct order for the equation f(x)=(2x+1)^2 - 1
Horizontal compression by a factor of 1/2, shift LEFT 1 unit, and shift DOWN 1 unit.
If a graph starts ar (-5,0) and extends infinitely up and to the right, what is the domain and range in interval notation.
Domain: [-5, inf); Range: [0, inf)
Write the transformed equation if the parent function f(x) = |x| undergoes a vertical compression of 1/2, a shift RIGHT 6 units, and a shift DOWN 3 units.
f(x) = 1/2|x - 6| - 3
Given the transformed graph f(x) = (x+3)^3 + 4, state its domain, range, and end behavior.
Domain: (-inf, inf); Range: (-inf, inf); End Behavior: As x -> -inf, f(x) -> -inf,
As x -> inf, f(x) -> inf