Let's analyze the problem shown in the image.

The general equation for the polynomial function is given as: $g(x) = a(x - h)^4 + k$ where $h$ and $k$ are the coordinates of the vertex of the function.

Step 1: Identify the vertex $(h, k)$

Looking at the graph, the vertex appears to be at the point $(0, 0)$, meaning $h = 0$ and $k = 0$.

Step 2: Determine the value of $a$

To find $a$, we can substitute another point on the graph into the equation and solve for $a$. From the graph, we can see that the point $(-1, 5)$ is on the curve. Substituting $(-1, 5)$ into the equation: $g(x) = a(x - h)^4 + k$ Substituting $x = -1$, $g(x) = 5$, $h = 0$, and $k = 0$: $5 = a(-1 - 0)^4 + 0$ $5 = a(1)^4$ $5 = a$ So, $a = 5$.

Final Equation:

Now we have all the parameters: $a = 5$, $h = 0$, and $k = 0$. Thus, the equation of the function is: $g(x) = 5(x - 0)^4 + 0$ Simplifying, we get: $g(x) = 5x^4$

This is the equation of the graphed function.

Would you like more details or have any further questions?

Here are 5 related questions you could explore:

1. How do different values of $a$ affect the shape of the graph for a quartic function?
2. What happens to the graph if $h$ and $k$ are non-zero?
3. Can you find the roots of $g(x) = 5x^4$?
4. How does the degree of a polynomial affect the number of turning points in its graph?
5. What is the general behavior of even-powered polynomials compared to odd-powered ones?

Tip: Always use points clearly visible on the graph to calculate constants like $a$, $h$, and $k$ when matching an equation to a graph.