Question

the quadratic regression graphed on the coordinate grid represents the height of a road surface x meters from the center of the road.
road surface height
graph of a quadratic curve on a coordinate grid with x-axis from -7 to 7 and y-axis from -1.25 to 1.25, showing the curve rising to a peak near the center (around x=0) and then falling
what does the graph of the regression model show?
○ the height of the surface decreases from the center out to the sides of the road.
○ the height of the surface increases, then decreases, from the center out to the sides of the road.
○ the height of the surface increases from the center out to the sides of the road.
○ the height of the surface remains the same the entire distance across the road.

Explanation:

A quadratic regression graph is a parabola. If the parabola opens down - ward (as in this case, since the vertex is a maximum point), when moving from the center (vertex) outwards (increasing \(x\) values), the \(y\) - values (height) first increase (but in this case, since the vertex is at the center \(x = 0\), for \(x\) values close to \(0\) and moving away, the function is decreasing). For a quadratic function \(y=ax^{2}+bx + c\) with \(a<0\), the function has a maximum at \(x =-\frac{b}{2a}\). As \(|x|\) increases from \(x =-\frac{b}{2a}\) (the center of the road), the \(y\) - values (height) decrease.

Answer:

The height of the surface decreases from the center out to the sides of the road.