The quartic polynomial with the specified characteristics is y = 2(x + 3)^2(x - 3)^2 + 2.
To find the equation of a quartic polynomial with the specified characteristics, we can consider the general form of a quartic polynomial:
y = ax^4 + bx^3 + cx^2 + dx + e.
Since the graph is symmetric about the y-axis, both the cubic and linear terms (bx^3 and dx) must be zero. Additionally, since there are local maxima at (-3, 4) and (3, 4), the quartic polynomial has double roots at these points, leading to factors (x + 3)^2 and (x - 3)^2. The y-intercept of 2 implies that the constant term e is 2.
Now, let's construct the polynomial using these considerations:
y = a(x + 3)^2(x - 3)^2 + 2.
To determine the value of a, we can use the fact that the local maxima occur at (-3, 4) and (3, 4). Plugging in these points, we get:
4 = a(-3 + 3)^2(-3 - 3)^2 + 2 = 2 times a.
Therefore, a = 2, and the equation of the quartic polynomial is:
y = 2(x + 3)^2(x - 3)^2 + 2.
The question probable may be:
1. Find the equation of a quartic polynomial whose graph is symmetric about the y-axis and has local maxima at (-3, 4) and (3, 4) and a y-intercept of 2.
y = ..........................