Question

fill in the equation for this function. y = ?(x - )^4+

Explanation:

Step1: Identify the vertex form of a quartic function

The general form is $y = a(x - h)^4 + k$, where $(h,k)$ is the vertex of the function.

Step2: Find the vertex of the graph

The vertex of the given graph is at the point $(4,2)$. So $h = 4$ and $k=2$.

Step3: Determine the value of $a$

We know the graph also passes through the point $(5, - 1)$. Substitute $x = 5$, $y=-1$, $h = 4$ and $k = 2$ into $y=a(x - h)^4 + k$.
We get $-1=a(5 - 4)^4+2$.
Simplify the equation: $-1=a\times1^4 + 2$, which is $-1=a + 2$.
Solve for $a$: $a=-3$.

Answer:

$y=-3(x - 4)^4+2$