Question

The height of an arch is modeled by the equation . Which of the following gives the width of the arch at its base?

1) The y-coordinate of the vertex
2) Double the x-coordinate of the vertex
3) The y-intercept of the equation
4) The difference between the zeroes

Answer 1

The height of an arch is modeled by the equation . 2) Double the x-coordinate of the vertex gives the width of the arch at its base.

Step-by-step explanation:

The width of the arch at its base is typically represented by the distance between the two points where the arch intersects the x-axis. In the context of the given equation modeling the arch's height, this width is precisely determined by 2)double the x-coordinate of the vertex. The vertex form of a quadratic equation
`\(y = a(x - h)^2 + k\)` represents the vertex as the point
`\((h, k)\),` where
`\(h\)` is the x-coordinate of the vertex.

In the equation modeling the arch's height, the x-coordinate of the vertex is a critical factor in determining the base width. Doubling the x-coordinate provides the horizontal distance between the points where the arch meets the x-axis, offering a straightforward and accurate representation of the arch's width at its base.

Understanding the relationship between the vertex form of a quadratic equation and the geometry it models helps elucidate how the x-coordinate of the vertex directly relates to the arch's width. This connection enhances the comprehension of mathematical modeling and its application to real-world scenarios, such as describing the dimensions of an arch.