Question

1. which of the following is the graph of (y = x^3 + 8x^2 - x - 8)?

Explanation:

Factor the cubic polynomial

Using the Factoring Polynomials knowledge point

$$
LATEXBLOCK0
$$

Determine the zeros of the function

Using the Zeros of a Function from Graph knowledge point

$$
LATEXBLOCK1
$$

Analyze the end behavior

Using the End Behavior of Cubics knowledge point

$$
LATEXBLOCK2
$$

Identify the correct graph

We look for a cubic curve with \(x\)-intercepts at \(x = -8\), \(x = -1\), and \(x = 1\).

• The first graph shown is a parabola (quadratic), which is incorrect.
• The second graph shown is a cubic curve that crosses the \(x\)-axis near \(x = -8\) on the left, and has a local maximum/loop crossing near \(x = -1\) and \(x = 1\). This matches our calculated zeros and end behavior.

Answer:

• (A) The first graph (parabolic curve with vertex near the origin)
• (B) The second graph (cubic curve crossing the x-axis at -8, -1, and 1) (Correct answer)