1.12 Transformations of functions
Consider the function f(x) and its transformations:
What transformation of f(x) does g(x) = f(x) + 3 represent and how does the graph change?
What transformation of f(x) does g(x) = f(x - 2) represent and how does the graph change?
The function g(x) = 2f(x) is a transformation of the function f(x). What type of transformation is this and how does it affect the graph of f(x)?
The function g(x) = f(3x) is a transformation of the function f(x). What type of transformation is this and how does it affect the graph of f(x)?
What happens to the graph of a function f(x) when it undergoes both an additive and a multiplicative transformation?
How can the domain and range of a function change when it undergoes a transformation?
Given the function f(x) = x^2, find the new function g(x) after it has been translated vertically by the following values of k:
k = 3
k = -2
k = 5
k = -7
The function f(x) = x^3 - 2x^2 + x is translated vertically so that its new equation is \\g(x) = x^3 - 2x^2 + x + 8. Determine the vertical translation.
Given the function f(x) = x^3, find the equation for the function g(x) that represents f(x) translated horizontally by 5 units to the right.
The graph of the function f(x) = |x| is horizontally translated to produce the graph of g(x). If the vertex of g(x) is at the point (2,\, 0), find the equation of g(x).
Given the function f \left( x \right) = \dfrac{1}{x}, find the equation of the function g \left( x \right) such that g \left( x \right) is a vertical dilation of f \left( x \right) by a factor of 3 and a reflection over the x-axis.
For each of the following functions:
Identify the value of b.
Describe the horizontal dilation and reflection, if any, of the graph of the function.
A function g(x) is created by applying a vertical dilation of 2, a horizontal reflection, a horizontal shift of 4 units to the left, and a vertical shift of 3 units down to the parent function f(x) = x^2. Find the equation of g(x).
Given the parent function f(x) = |x|, determine the domain and range of each transformed function:
g(x) = |x - 2|
g(x) = 3|x| + 1
g(x) = -|x + 4|
g(x) = \dfrac{1}{2}|x|
Consider the function f(x) = x^2. Perform the following transformations:
Vertically translate the graph by 3 units up. Write the equation for the transformed function.
Horizontally translate the graph by 2 units to the right. Write the equation for the transformed function.
Vertically dilate the graph by a factor of 2. Write the equation for the transformed function.
Horizontally dilate the graph by a factor of 0.5. Write the equation for the transformed function.
Consider the function f(x) = x^2 and its transformation into g(x) = (x - 1)^2 + 3. If you didn't know the equation of f(x), but you had the graph of g(x), how could you determine the equation of f(x)?
Consider a function f(x) and its transformations g(x) and h(x) defined as follows:
g(x) = f(2x - 3) + 4, which is a combination of horizontal dilation, horizontal translation, and vertical translation.
h(x) = -3f(x + 1) - 2,which is a combination of vertical dilation, reflection over the x-axis, horizontal translation, and vertical translation.
Assume that the graph of f(x) passes through the point (2,\, 5). Determine the coordinates of the corresponding points on the graphs of g(x) and h(x).
Explain the effects of each transformation on the domain and range of f(x) when forming g(x) and h(x).