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4 votes
Determine the equation of

the cubic or quartic function f,
represented by the graph below.
AY
y
X
-4-2 O 2 4
(-2,-3)
2
-4

1 Answer

3 votes

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 = ..........................

Determine the equation of the cubic or quartic function f, represented by the graph-example-1
User Nafsaka
by
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