Question

How does the graph of g(x) = (x - 3)^3 + 4 compare to the parent function f(x)=x^3 ?​

Answer 1

f(x) has moved:

4 units in the positive y direction i.e upwards

3 units in the positive x direction

Explanation:

to get g(x), f(x) has undergone the following transformations

f(x) = x³

f1(x) = x³ + 4 (translation of 4 units in the positive y direction i.e upwards)

f2(x) = g(x) = (x-3)³ + 4 (translation of 3 units in the positive x direction i.e towards the right)

Answer 2

The modified function has a translation of 3 units to the right and 4 units up compared to the parent function.

Explanation:

Let's change both functions to y.

How does y = (x - 3)^3 + 4 compare with the parent function y = x^3?

Start with y = (x - 3)^3 + 4 and subtract 4 from both sides.

y - 4= (x - 3)^3

Now compare the function above with the parent function below.

y = x^3

We notice two differences. y of the parent function becomes y - 4 in the modified function. x of the parent function becomes x - 3 in the modified function.

When you replace x by x - h, the function is translated h units horizontally. The translation is to the right if h is positive and to the left if h is negative.

When you replace y by y - k, the function is translated k units vertically. The translation is up if k is positive and down if k is negative.

In the modified function, x became x - 3.

Compare x - 3 with x - h.

h = 3

3 is positive, so the modified function was translated 3 units to the right.

In the modified function, y becomes y - 4.

Compare y - 4 with y - k.

k = 4

4 is positive, so the modified function is translated 4 units up.

Answer:

The modified function has a translation of 3 units to the right and 4 units up compared to the parent function.