Question

On a coordinate plane, a curved line labeled f of x with a minimum value of (1.9, negative 5.7) and a maximum value of (0, 2), crosses the x-axis at (negative 0.7, 0), (0.76, 0), and (2.5, 0), and crosses the y-axis at (0, 2).

Which statement is true about the graphed function?

F(x) < 0 over the intervals (-∞, -0.7) and (0.76, 2.5).
F(x) > 0 over the intervals (-∞, -0.7) and (0.76, 2.5).
F(x) < 0 over the intervals (-0.7, 0.76) and (2.5, ∞).
F(x) > 0 over the intervals (-0.7, 0.76) and (0.76, ∞).

Answer 1

(a) F(x) < 0 over the intervals (-∞, -0.7) and (0.76, 2.5)

Explanation:

Wherever the function crosses the x-axis, its sign changes. This lets us make a map of the signs of the function, and identify intervals where it is positive and negative.

Sign changes

The x-intercepts are said to be at x ∈ {-0.7, 0.76, 2.5}. These three crossing points divide the graph into four (4) intervals. The sign of the function is given as positive at x=0 and negative at x = 1.9. So, our sign map is ...

< -0.7 . . . . . negative

-0.7 to 0.76 . . . . . positive (2 at x=0, for example)

0.76 to 2.5 . . . . . negative (-5.7 at 1.9, for example)

> 2.5 . . . . . positive

Choosing from the descriptions, the one that matches is ...

F(x) < 0 over the intervals (-∞, -0.7) and (0.76, 2.5)