Question

1. which of the following is the graph of \\(y = -x^4 + 8x^2\\)?

Explanation:

Analyze the polynomial function

Using the Polynomial Factored Form and Algebraic Simplification knowledge points

$$ y = -x^4 + 8x^2 = -x^2(x^2 - 8) $$

Determine the end behavior

Using the Leading Coefficient Test and End Behavior of Polynomials knowledge points

$$
LATEXBLOCK0
$$

Find the zeros and multiplicities

Using the Multiplicity of Roots and Zeros of a Function from Graph knowledge points

$$
LATEXBLOCK1
$$

Identify the local extrema

Using the Polynomial Derivatives and Vertex of a Parabola knowledge points

$$
LATEXBLOCK2
$$

Match with the correct graph

Using the Polynomial Graphing knowledge point

• The first graph has a local minimum at \((0,0)\), local maxima at approximately \((-2, 16)\) and \((2, 16)\), and \(x\)-intercepts at \(0\) and approximately \(\pm 2.8\).
• It opens downwards on both ends, matching the end behavior.
• Therefore, the first graph is the correct representation.

Answer:

• (A) M-shaped symmetric curve opening downwards with local maxima at \((-2, 16)\) and \((2, 16)\), and a local minimum at \((0,0)\) (Correct answer)
• (B) Cubic-like curve passing through the origin with a local maximum in the second quadrant and a local minimum in the fourth quadrant