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Here are some thoughts; I can't know at this point whether you will understand them.
You need to know that the basic quartic function is y = x^4. Start your transformations with x^4.
It's obvious that the graph shape opens downward, yes? That fact gives us the first transformation: multiply by -1.
y = -x^4
You should now ponder the three given points.
You should realize that the axis of symmetry is x = 6 because the points where x = 0 and x = 12 both lie on the same horizontal line (y = 5) AND x = 6 is halfway between x = 0 and x = 12.
Sometimes, it helps to figure out transformations by working in reverse. That is, we could shift the given graph six units to the left, to get the vertex on the y-axis, and then shift the resulting graph down 13 units to get the vertex at the origin.
Why do this? Because now the original point (12, 5) has shifted to (6, -8) AND we realize that the equation for the shifted graph passing through (6,-8) is:
y = ax^4
where a must be some negative number that causes a horizontal stretch sufficient to get the graph to pass through (6, -8).
Can you determine the value of a, from this information?
Once you know the leading coefficient a, simply reverse the horizontal and vertical shifts.
You will arrive at a function of this form:
f(x) = a(x - h)^4 + k
where h is the horizontal shift and k is the vertical shift.
You're done!
Do you have any specific questions now? :cool:
