Question

below is the graph of a trigonometric function. it intersects its midline at \\((-1.7, -10)\\) and again at \\((5.1, -10)\\).

what is the period of the function? give an exact value.

Explanation:

Identify the midline and intersection points

The midline of the trigonometric function is the horizontal line \(y = -10\).
The graph intersects this midline at the points \((-1.7, -10)\) and \((5.1, -10)\).

Determine the fraction of the period between intersections

Looking at the graph, the point \((-1.7, -10)\) lies on a decreasing portion of the wave as it crosses the midline.
The next intersection with the midline is at a local minimum's right, where the curve is increasing, which occurs between the two highlighted points.
The highlighted point \((5.1, -10)\) is on an increasing portion of the wave as it crosses the midline.
The distance between a point where the function crosses its midline going down and the next point where it crosses going up is exactly half of a full period (\(\frac{T}{2}\)).

Calculate the period

$$ \frac{T}{2} = 5.1 - (-1.7) = 5.1 + 1.7 = 6.8 $$

$$ T = 2 \times 6.8 = 13.6 $$

Answer:

\(13.6\)