Question

1. which of the following is the graph of \\(y = -(x + 3)(x - 1)(x + 2)(x - 4)\\)?

Explanation:

Identify the zeros of the function

Using the Zeros of a Function from Graph knowledge point

$$
LATEXBLOCK0
$$

Determine the end behavior of the polynomial

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

$$
LATEXBLOCK1
$$

Evaluate the y-intercept

Using the Algebraic Evaluation knowledge point

$$
LATEXBLOCK2
$$

Compare with the given graphical options

We analyze the two visible graphs based on our findings:

• First Graph:
• The graph crosses the x-axis at \(x = -3\), \(x = -2\), \(x = 1\), and \(x = 4\).
• As \(x \to \infty\) and \(x \to -\infty\), the graph goes downwards towards \(-\infty\).
• The y-intercept is negative (at \(y = -24\)).
• This matches all properties of our function.
• Second Graph:
• As \(x \to \pm\infty\), the graph goes upwards towards \(+\infty\), which corresponds to a positive leading coefficient.
• The y-intercept is positive.
• This does not match our function.

Answer:

• (A) The first graph, which opens downwards with x-intercepts at -3, -2, 1, and 4, and a negative y-intercept. (Correct answer)
• (B) The second graph, which opens upwards with a positive y-intercept.