Question

below is the graph of a trigonometric function. it intersects its midline at \\((1.25, -2.75)\\) and it has a minimum point at \\((4.75, -6.25)\\).

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

Explanation:

Identify given points

We are given a midline intersection point and a minimum point.

• Midline intersection: \((1.25, -2.75)\)
• Minimum point: \((4.75, -6.25)\)

Relate points to period

Using the Trigonometric Midline concept, the horizontal distance between a midline intersection and an adjacent minimum is a quarter of the period.
Let \(T\) be the period of the function.

$$ \frac{T}{4} = x_{\text{min}} - x_{\text{midline}} $$

Calculate the distance

Using the Trigonometric Period concept, we find the horizontal distance:

$$ 4.75 - 1.25 = 3.5 $$

Solve for the period

Multiply the quarter-period distance by 4 to find the full period:

$$ T = 3.5 \times 4 = 14 $$

Answer:

14