Question

the height of an arch is modeled by the equation \\(y = -x^2 - 8x + 16\\). which of the following gives the width of the arch at its base?

• the \\(y\\)-coordinate of the vertex
• double the \\(x\\)-coordinate of the vertex
• the \\(y\\)-intercept of the equation
• the difference between the zeroes

Explanation:

Identify the physical meaning of the base of the arch

The base of the arch lies on the ground where the height \(y = 0\). The horizontal positions of the base are the zeros (roots) of the quadratic equation \(y = -x^2 - 8x + 16\).

Determine the width of the arch at its base

The width is the horizontal distance between the two points where the arch meets the ground. This distance is the difference between the two zeros of the quadratic equation.

Answer:

• the \(y\)-coordinate of the vertex
• double the \(x\)-coordinate of the vertex
• the \(y\)-intercept of the equation
• the difference between the zeroes (Correct answer)