Identify the transformations of the graph of ƒ (x) = x 3
that produce the
graph of the given function g (x). Then graph g (x) on the same coordinate
plane as the graph of ƒ (x) by applying the transformations to the
reference points (-1, -1), (0, 0) , and (1, 1).
g (x) = 2 (x - 1)^3-1
why is the answer:
The transformations of the graph of ƒ (x) that produce the graph of
g (x) are:
• a vertical stretch by a factor of 2
• a translation of 1 unit to the right and 1 unit down
thank you sooooo much
if you can explain the vertical stretch versus horizontal stretch versus compression..I don't get that part
To understand the concept of vertical stretch, horizontal stretch, and compression, let's start by looking at the parent function ƒ(x) = x^3. The graph of this function is a cubic curve.
Vertical Stretch:
A vertical stretch involves increasing or decreasing the vertical size of the graph. In this case, g(x) = 2(x - 1)^3 - 1, there is a vertical stretch by a factor of 2. This means that the graph of g(x) will be twice as tall (stretched vertically) compared to the graph of ƒ(x).
Horizontal Stretch/Compression:
A horizontal stretch or compression refers to changing the horizontal size of the graph. It affects how quickly or slowly the graph changes along the x-axis.
To understand this better, let's focus on the transformation (x - 1) in g(x) = 2(x - 1)^3 - 1. When x is subtracted by 1 within the parentheses, it affects the x-values of the graph. This shift to the right by 1 unit means that each x-value on the graph will be increased by 1, causing the graph to move horizontally.
If the factor inside the parentheses (in this case, x - 1) is greater than 1, it results in a horizontal compression, meaning the graph will be compressed horizontally, or appear narrower. On the other hand, if the factor is less than 1, it will cause a horizontal stretch, making the graph wider.
In our case, since we are subtracting 1 (x - 1), the graph will be compressed horizontally, making it narrower.
Translation:
The "-1" at the end of g(x) = 2(x - 1)^3 - 1 represents a downward shift of the graph by 1 unit. This means that every point on the graph will be moved down by 1 unit.
Now, let's apply these transformations to the reference points (-1, -1), (0, 0), and (1, 1) from the parent function ƒ(x) = x^3:
(-1, -1) shifted 1 unit to the right becomes (0, -1) and then vertically stretched by a factor of 2 becomes (0, -2). Finally, it is shifted 1 unit down, resulting in (0, -3).
(0, 0) shifted 1 unit to the right becomes (1, 0) and then vertically stretched by a factor of 2 becomes (1, 0). Finally, it is shifted 1 unit down, resulting in (1, -1).
(1, 1) shifted 1 unit to the right becomes (2, 1) and then vertically stretched by a factor of 2 becomes (2, 2). Finally, it is shifted 1 unit down, resulting in (2, 1).
Plotting these transformed points on the coordinate plane will give you the graph of g(x).