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 + 3))^3 + 4
answer:
g (x) are:
• a horizontal compression by a factor of __1/2
• a translation of 3 units to the left and 4 units up
again--I don't get the compression part ( I know it has to do with the first number/ term..the a part)
thank you...i have a test on this and don't get it at all
To understand the concept of compression and how it is related to the coefficient of the function, let's take a closer look at the transformation of a function.
In the given function g(x) = (2(x + 3))^3 + 4, the coefficient 2 represents the horizontal compression of the graph of the function ƒ(x) = x^3.
A horizontal compression occurs when all the x-values of the original function are multiplied by a factor less than 1. In this case, the factor is 2.
To see this, let's compare the reference points of the original function ƒ(x) = x^3 with the transformed function g(x):
For ƒ(x) = x^3:
(-1, (-1)^3) = (-1, -1)
(0, 0^3) = (0, 0)
(1, (1)^3) = (1, 1)
Now let's apply the transformation to each reference point for g(x):
(-1, -1):
First, we shift the point 3 units to the left by subtracting 3 from the x-coordinate: (-1 - 3, -1) = (-4, -1)
Then we compress the x-coordinate by multiplying it by 1/2: (-4 * 1/2, -1) = (-2, -1)
Finally, we shift the point 4 units up by adding 4 to the y-coordinate: (-2, -1 + 4) = (-2, 3)
(0, 0):
We perform the same steps:
(0 - 3, 0) = (-3, 0)
(-3 * 1/2, 0) = (-3/2, 0)
(-3/2, 0 + 4) = (-3/2, 4)
(1, 1):
(1 - 3, 1) = (-2, 1)
(-2 * 1/2, 1) = (-1, 1)
(-1, 1 + 4) = (-1, 5)
Now we have the transformed reference points for g(x):
(-4, 3)
(-3/2, 4)
(-1, 5)
By plotting these points on the coordinate plane and connecting them, you will obtain the graph of the transformed function g(x) = (2(x + 3))^3 + 4.
Remember to label each point and use a smooth curve to connect them.