Question

the graph is a transformation of the parent cubic function or quartic function. determine the equation of the graph.
the equation of the graph is $f(x)=\square$.

Explanation:

Step1: Identify parent function type

The graph is a U-shape (even-degree, minimum point), so parent is quartic: $f(x)=x^4$.

Step2: Locate vertex (transform point)

Vertex is at $(4, 4)$. For parent $x^4$, vertex is $(0,0)$. This is a horizontal shift right 4, vertical shift up 4.

Step3: Check vertical stretch/compression

Test point: when $x=6$, $y=8$. Plug into shifted form $y=a(x-4)^4 + 4$:
$8 = a(6-4)^4 + 4$
$8 = 16a + 4$
$16a = 4$
$a = \frac{4}{16} = \frac{1}{4}$

Step4: Write final equation

Combine transformations: $y=\frac{1}{4}(x-4)^4 + 4$

Answer:

$\frac{1}{4}(x-4)^4 + 4$