Question

which statement justifies a zero between x = 1 and x = 2 given the table of values?

x	f(x)
-1	8
0	5
1	2
2	-7
3	-28

a) f(x) is continuously decreasing.
b) the y-intercept is (0, 5).
c) f(1) is positive and f(2) is negative.
d) there may be multiple zeros.

Explanation:

To determine a zero between \( x = 1 \) and \( x = 2 \), we use the Intermediate Value Theorem (for continuous functions). The key is the sign change of \( f(x) \) between these two points. From the table, \( f(1) = 2 \) (positive) and \( f(2) = -7 \) (negative). A sign change in a continuous function implies a zero in that interval.

• Option A: Just saying it's decreasing doesn't guarantee a zero (e.g., a decreasing function could stay negative or positive).
• Option B: The y - intercept is irrelevant to the zero between \( x = 1 \) and \( x = 2 \).
• Option D: The possibility of multiple zeros doesn't justify a zero between \( x = 1 \) and \( x = 2 \).

Only Option C, which states \( f(1) \) is positive and \( f(2) \) is negative, gives a valid justification for a zero in \( (1, 2) \) (assuming continuity, which is implied by the table's context for such problems).

Answer:

C) \( f(1) \) is positive and \( f(2) \) is negative.