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

The graph is a U-shape (both ends up), so parent is quartic: $f(x)=x^4$.

Step2: Find vertex (transformations)

Vertex is at $(5, 16)$. This means horizontal shift 5 right, vertical shift 16 up.

Step3: Test vertical stretch

Take a point on the graph: $(6,17)$. Substitute $x=6$, $f(x)=17$ into $f(x)=a(x-5)^4+16$:
$17=a(6-5)^4+16$
$17=a(1)+16$
$a=1$

Step4: Write final equation

No stretch, so use the shift values.

Answer:

$f(x)=(x-5)^4+16$