Question

the table shows some values of a polynomial function. deshawn says there are turning points between the x - values - 3 and - 2 and between 0 and 1. he also says there is a relative minimum between the x - values - 3 and - 2 and a relative maximum between 0 and 1. sketch a graph that shows how deshawn could be correct. explain how deshawn could draw a graph using the given points that still be incorrect.

x	-5	-4	-3	-2	-1	0	1	2
f(x)	-1504	120	220	211	308	440	386	-270

which graph shows how deshawn could be correct?

options: a, b, c, d (each with a graph image)

Explanation:

To solve this, we analyze the function's values and Deshawn's claims:

Step 1: Analyze \( x \)-values and \( f(x) \) trends

• Between \( x = -3 \) (\( f(-3) = -1504 \)) and \( x = -2 \) (\( f(-2) = 211 \)): \( f(x) \) increases (from negative to positive), so a turning point (change in slope) could occur here.
• Between \( x = 0 \) (\( f(0) = 446 \)) and \( x = 1 \) (\( f(1) = 395 \)): \( f(x) \) decreases (from 446 to 395), so a relative maximum (peak) could occur here (since it rises then falls).

Step 2: Evaluate the graphs

We need a graph where:

• There is a turning point (slope change) between \( x = -3 \) and \( x = -2 \) (matching the \( f(x) \) increase from \( -3 \) to \( -2 \)).
• There is a relative maximum (peak) between \( x = 0 \) and \( x = 1 \) (matching the \( f(x) \) decrease from \( 0 \) to \( 1 \)).

After analyzing the options (A, B, C, D) against these trends, the correct graph will show a turning point between \( -3 \) and \( -2 \), a relative maximum between \( 0 \) and \( 1 \), and align with the \( f(x) \) values (e.g., \( f(-3) \) is very negative, \( f(0) \) is positive, etc.).

Answer:

(Assuming the correct graph is, for example, D; the exact option depends on the visual graphs, but the key is the graph with a turning point between \( -3 \)–\( -2 \) and a relative maximum between \( 0 \)–\( 1 \), matching the \( f(x) \) trends.)

(Note: If the visual graphs show D as having a peak between 0–1 and a turning point between -3–-2, then the answer is D. Adjust based on the actual graph options.)