Multiple Response Select the transformations of the graph of the parent cubic
function that result in the graph of g(x) = (3(x - 2))^3 +1
.
A. Horizontal stretch by a factor of 3
B. Horizontal compression by a factor of 1/3
C. Vertical stretch by a factor of 3
D. Vertical compression by a factor of 1/3
E. Translation 1 unit up
F. Translation 1 unit down
G. Translation 2 units left
H. Translation 2 units right
the answer is B, E and H
I don't understand why B though
thank you
since x has been replaced by 3x, it only has to be 1/3 as big to produce the new graph.
Play around with a couple of easy functions to see how this works.
Well, B stands for "Horizontal compression by a factor of 1/3." In the equation g(x) = (3(x - 2))^3 + 1, the part (x - 2) represents a horizontal shift of the graph, shifting it 2 units to the right.
Normally, if we had just f(x) = x^3, this would be a standard cubic function. However, in this case, we have g(x) = (3(x - 2))^3 + 1. Let's break it down step by step:
1. First, we have the factor of 3 inside the parentheses. This factor of 3 indicates a horizontal stretch by a factor of 3. It stretches the graph horizontally, making it narrower.
2. Next, we have (x - 2) inside the parentheses. This represents a horizontal shift to the right by 2 units. The graph is moved to the right by 2 units.
3. Finally, the exponent of 3 outside the parentheses represents a vertical stretch. It stretches the graph vertically, making it taller.
So, the correct options are B, E, and H. The horizontal compression, vertical translation, and horizontal translation can be seen in the equation g(x) = (3(x - 2))^3 + 1. I hope that helps!
To understand why option B (horizontal compression by a factor of 1/3) is included in the transformations of the graph of g(x), let's analyze the given function.
The parent cubic function is y = x^3. It can be thought of as a graph that starts at the origin (0,0) and expands outwards in all four quadrants.
In the given function g(x) = (3(x - 2))^3 + 1, we have three main transformations:
1. Horizontal transformation: (x - 2) is the x-value inside the parenthesis. This means the graph is shifted 2 units to the right. So, we have option H (translation 2 units right).
2. Vertical transformation: The entire cubic function is raised to the power of 3, which means that it is vertically stretched. So, we have option C (vertical stretch by a factor of 3).
3. Vertical translation: The entire function is then shifted upwards by 1 unit. This results in option E (translation 1 unit up).
There is no horizontal stretch or compression happening in this function. The reason option B (horizontal compression) is not included is because the (3(x - 2))^3 term does not affect the horizontal shape of the graph. It only shifts it horizontally by 2 units.
Therefore, the correct transformations for the graph of g(x) = (3(x - 2))^3 + 1 are options B (translation 1 unit up), E (horizontal compression by a factor of 1/3), and H (translation 2 units right).
To understand why the correct answer is B, let's break down the given function g(x) = (3(x - 2))^3 + 1 and compare it to the parent cubic function f(x) = x^3.
The parent cubic function f(x) = x^3 is a basic cubic function with no transformations applied to it. The given function g(x) = (3(x - 2))^3 + 1, on the other hand, has three transformations applied to it.
Transformation B states a horizontal compression by a factor of 1/3. This means that the graph of g(x) is horizontally squeezed compared to the graph of f(x). To understand this, we need to look at the term (3(x - 2))^3.
In the parent cubic function f(x) = x^3, the x-values directly correspond to the y-values (i.e., y = x^3). However, in the given function g(x), the x-values inside the parentheses are multiplied by 3, resulting in a compression. Let's see how it works:
If we take a point on the graph of f(x), let's say (2, 8), the x-value is 2, and the corresponding y-value (output) is 8. When we apply the horizontal compression in g(x), we multiply the x-value by 3.
For the point (2, 8):
g(x) = (3(2 - 2))^3 + 1
= (3(0))^3 + 1
= 0^3 + 1
= 1
So, the corresponding point on the graph of g(x) is (1, 1). We can see that the x-value of 2 is compressed to 1, resulting in a squeezed graph horizontally.
Therefore, the correct answer is B because the graph of g(x) = (3(x - 2))^3 + 1 has a horizontal compression by a factor of 1/3.
Now, let's briefly explain the remaining transformations:
Transformation E states a translation 1 unit up. This means the entire graph of g(x) is shifted upwards by 1 unit. We can see this in the "+1" term added to the function g(x).
Transformation H states a translation 2 units right. This means the entire graph of g(x) is shifted rightwards by 2 units. This can be observed in the term (x - 2), where x is shifted 2 units to the right compared to the parent function f(x) = x^3.
To summarize, the answer includes transformations B (horizontal compression by a factor of 1/3), E (translation 1 unit up), and H (translation 2 units right).