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) = - 1/2(x-3)^3
answer:
The transformations of the graph of f (x)
that produce the graph of g (x) are:
• a vertical compression by a factor of 1/2
• a reflection across the x-axis
• a translation of 3 units to the right
thank you again
To identify the transformations that produce the graph of g(x) = -1/2(x-3)^3 from the graph of f(x) = x^3, you can follow these steps:
Step 1: Identify the transformations one by one.
1. Vertical Compression by a factor of 1/2:
The coefficient in front of the function's equation determines the vertical scaling. In this case, the -1/2 coefficient indicates a vertical compression by a factor of 1/2. This means that all the y-values will be multiplied by 1/2, causing the graph to become narrower.
2. Reflection across the x-axis:
A negative sign in front of the function's equation reflects the graph across the x-axis. This means that all the y-values will be multiplied by -1, causing the graph to be flipped upside down.
3. Translation 3 units to the right:
The term inside the parentheses, (x-3), represents a horizontal shift. In this case, it indicates a translation of 3 units to the right. This means that all the x-values will be decreased by 3, shifting the graph to the right.
Step 2: Apply the transformations to the reference points (-1, -1), (0, 0), and (1, 1).
For f(x) = x^3:
- (-1, -1) remains the same
- (0, 0) remains the same
- (1, 1) remains the same
Now, let's apply the transformations to these reference points to get the corresponding points on g(x) = -1/2(x-3)^3:
1. Vertical Compression by a factor of 1/2:
Multiply the y-values of the reference points by 1/2:
- (-1, -1) => (-1, -1/2)
- (0, 0) => (0, 0)
- (1, 1) => (1, 1/2)
2. Reflection across the x-axis:
Multiply the y-values obtained from the previous step by -1:
-1/2 => 1/2
0 => 0
1/2 => -1/2
3. Translation 3 units to the right:
Add 3 to the x-values obtained from the previous step:
(-1, -1/2) => (2, -1/2)
(0, 0) => (3, 0)
(1, -1/2) => (4, -1/2)
Now, you have the corresponding points for g(x) = -1/2(x-3)^3:
(-1, -1/2), (0, 0), (1, -1/2) transform into (2, -1/2), (3, 0), (4, -1/2).
Finally, plot these transformed points on the same coordinate plane as the graph of f(x) = x^3, and connect them to create the graph of g(x) = -1/2(x-3)^3.