Question

the quadratic regression graphed on the coordinate grid represents the height of a road surface x meters from the center of the road. 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:

The graph is a quadratic (parabola) opening downward. The center of the road is around \( x = 0 \) (vertex region). As \( x \) moves from 0 (center) out to positive or negative sides (sides of the road), the \( y \)-value (height) first increases a bit near the center then decreases (since it's a downward - opening parabola, the vertex is the maximum point, and as we move away from the vertex (center) towards the sides, the height decreases after reaching the peak near the center). Wait, no, looking at the graph: the vertex is around \( x = 0 \) to \( x = 2 \) area, and as we go from the center (around \( x = 0 \)) out to the sides (larger \( |x| \)), the height first increases a little (from \( x=-5 \) to \( x = 0 - 2 \)) and then decreases (from \( x = 0 - 2 \) to \( x = 5 \)). Wait, no, the key is the shape: a downward - opening parabola has a maximum at the vertex. So from the center (vertex area) out to the sides, the height increases until the vertex and then decreases? Wait, no, the \( x \) - axis: the center of the road is probably at \( x = 0 \) (since \( x \) is distance from center). So when \( x = 0 \), the height is around 0.25 - 0.5? Wait, the graph: at \( x=-5 \), it's 0 (intercept), then it goes up to a peak (around \( x = 0 - 2 \)) and then comes down to 0 at \( x = 5 \). So from the center ( \( x = 0 \)) out to the sides ( \( x=-5 \) or \( x = 5 \)), the height first increases (from \( x=-5 \) to the peak near \( x = 0 - 2 \)) and then decreases (from the peak to \( x = 5 \))? Wait, no, the question is "from the center out to the sides". The center is at \( x = 0 \). So as we move from \( x = 0 \) (center) to \( x = 5 \) (side) or \( x=-5 \) (side), what happens to the height? Looking at the graph, at \( x = 0 \), the height is around 0.25 - 0.5, at \( x = 2 \), it's still around 0.25 - 0.5, then at \( x = 5 \), it's 0. So from center (\( x = 0 \)) out to sides (\( x = 5 \) or \( x=-5 \)), the height decreases. Wait, the first option: "The height of the surface decreases from the center out to the sides of the road." Let's check the options:

Option 1: The height of the surface decreases from the center out to the sides of the road. - If the center is at \( x = 0 \), as \( x \) moves from 0 to 5 (side) or - 5 (side), the height goes from around 0.25 - 0.5 to 0, so it decreases.

Option 2: Increases then decreases - but from center out, does it increase first? At \( x = 0 \), height is, say, 0.3, at \( x = 2 \), still 0.3, then at \( x = 5 \), 0. So it's more like decreases from center out.

Option 3: Increases from center out - no, because at \( x = 5 \), height is 0, which is less than at center.

Option 4: Remains same - no, because it goes from 0 at \( x=-5 \), up to a peak, then down to 0 at \( x = 5 \), so it's not constant.

Wait, maybe I misread. Let's re - analyze: The graph is a quadratic regression, so it's a parabola. The vertex is the maximum point. So the parabola opens downward (since it has a maximum). So the general shape: as we move away from the vertex (center) in either direction (towards the sides), the \( y \) - value (height) decreases. Wait, the vertex is the highest point. So if the center is at the vertex (the peak), then from the center (peak) out to the sides, the height decreases. That matches option 1: "The height of the surface decreases from the center out to the sides of the road."

Answer:

The height of the surface decreases from the center out to the sides of the road. (The correct option is the first one: "The height of the surface decreases from the center out to the sides of the road.")