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: ( I don't get the vertical stretch part)
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
Think about the axis (horizontal x, versus vertical y) and then think about compression (getting fatter) and stretch (getting skinnier). You might want to graph a few to see if you can see the results. You might want to also go over the notes your teacher gave you : )
Sure, I'd be happy to explain the concept of vertical stretch versus horizontal stretch versus compression.
In the case of the function g(x) = 2(x - 1)^3 - 1, the factor of 2 in front of the bracketed term (x - 1)^3 indicates a vertical stretch. A vertical stretch increases the height of the graph vertically. In this case, every y-coordinate of the original function is multiplied by 2, which creates a steeper curve.
On the other hand, a horizontal stretch or compression would affect the width of the graph. If the factor in front of the x is greater than 1, it indicates a compression, meaning the graph gets narrower. If the factor is between 0 and 1, it indicates a stretch, meaning the graph gets wider.
In the given function g(x), there is no factor in front of the x term, which means there is no horizontal stretch or compression.
I hope that clarifies the difference between vertical and horizontal transformations for you! Let me know if you have any more questions.
To understand the concepts of vertical stretch, horizontal stretch, and compression, let's first define what they mean.
1. Vertical stretch: A vertical stretch makes the graph of the function "taller" or "shorter" by multiplying the y-coordinates of all points on the graph by a factor greater than 1. In other words, it stretches the function vertically without changing its width.
2. Horizontal stretch: A horizontal stretch makes the graph of the function "wider" or "narrower" by multiplying the x-coordinates of all points on the graph by a factor greater than 1. This stretches the function horizontally without changing its height.
3. Compression: Compression occurs when the graph of a function is vertically or horizontally condensed. It is the opposite of stretching and involves multiplying the coordinates by a factor less than 1.
Now, let's apply these concepts to the given function, g(x) = 2(x - 1)^3 - 1, based on the reference points (-1, -1), (0, 0), and (1, 1).
We start with the original function, ƒ(x) = x^3.
1. Vertical stretch by a factor of 2:
Multiplying the y-coordinates by 2 stretches the graph vertically. For example, the point (1, 1) on the original function will become (1, 2) after the vertical stretch.
2. Translation of 1 unit to the right:
Adding 1 to the x-coordinate of each point shifts the graph 1 unit to the right. For example, the point (0, 0) on the original function will become (1, 0) after the translation.
3. Translation of 1 unit down:
Subtracting 1 from the y-coordinate of each point shifts the graph 1 unit down. For example, the point (1, 2) after the vertical stretch and the translation to the right will become (1, 1) after the additional translation down.
By applying these transformations to the reference points, we obtain the following transformed points:
(-1, -1) -> (0, -2) -> (1, -1)
(0, 0) -> (1, 0) -> (2, -1)
(1, 1) -> (2, 2) -> (3, 1)
Plotting these transformed points on the same coordinate plane as the graph of ƒ(x) = x^3 will give you the graph of g(x) = 2(x - 1)^3 - 1.
To understand the concept of vertical stretch, horizontal stretch, and compression, let's look at the general equation of a function:
ƒ(x) = a * ƒ(b(x - h)) + k
Where:
- 'a' represents the vertical stretch or compression factor.
- 'b' represents the horizontal stretch or compression factor.
- (h, k) represents the translation (or shift) of the graph.
In the given function g(x) = 2(x - 1)^3 - 1, we can identify the transformations:
1. Vertical Stretch or Compression: The factor 'a' determines the vertical stretch or compression. If 'a' is greater than 1, it results in a vertical stretch, and if 'a' is between 0 and 1, it results in a vertical compression.
In g(x) = 2(x - 1)^3 - 1, the factor 'a' is 2, which means there is a vertical stretch by a factor of 2. This means that each y-coordinate of the original graph is multiplied by 2. For example, if the point (0, 0) lies on the original graph, after the vertical stretch, it will become (0, 0 * 2) = (0, 0).
2. Horizontal Stretch or Compression: The factor 'b' determines the horizontal stretch or compression. If 'b' is greater than 1, it results in a horizontal compression, and if 'b' is between 0 and 1, it results in a horizontal stretch.
In the given function g(x) = 2(x - 1)^3 - 1, there is no factor 'b' present. Hence, there is no horizontal stretch or compression.
3. Translation (Shift): The values (h, k) determine the translation or shift of the graph. 'h' represents the horizontal shift, and 'k' represents the vertical shift.
In g(x) = 2(x - 1)^3 - 1, there is a translation of 1 unit to the right (positive h) and 1 unit down (negative k). This means that each x-coordinate of the original graph is shifted 1 unit to the right, and each y-coordinate is shifted 1 unit down.
Using these transformations, we can now graph g(x) on the same coordinate plane as f(x) by applying the transformations to the reference points (-1, -1), (0, 0), and (1, 1).