Question

which equation represents the parabola shown on the graph? \\(\bigcirc\\ y^2 = 1.5x\\) \\(\bigcirc\\ x^2 = 1.5y\\) \\(\bigcirc\\ y^2 = 6x\\) \\(\bigcirc\\ x^2 = 6y\\)

Explanation:

Step1: Identify the parabola's orientation

The parabola opens upwards, so its standard form is \(x^2 = 4py\) (opens up/down), while \(y^2 = 4px\) opens left/right. Eliminate options with \(y^2\) (first and third).

Step2: Test a point on the parabola

The graph passes near \((0, 1.5)\)? Wait, no, look at the vertex at \((0,0)\) and the shape. Let's test the point \((3, 3)\)? Wait, no, check the options \(x^2 = 1.5y\) and \(x^2 = 6y\). Let's take a point on the parabola, say when \(x = 3\), what's \(y\)? From the graph, when \(x = 3\), \(y\) is around 6? Wait, no, let's plug into the equations.

For \(x^2 = 1.5y\): If \(x = 3\), then \(9 = 1.5y \implies y = 6\)? No, the graph at \(x = 3\) has \(y\) around 1.5? Wait, no, maybe the point \((3, 6)\) is not on the graph. Wait, maybe I misread. Wait the graph: when \(x = 3\), \(y\) is 1.5? Wait no, the pink dot is \((0, 1.5)\)? No, the parabola's vertex is at the origin, and it opens upwards. Let's check the standard form \(x^2 = 4py\). Let's take a point on the parabola. From the graph, when \(x = 3\), \(y = 6\)? Wait no, let's check the options.

Wait, the options are \(x^2 = 1.5y\) and \(x^2 = 6y\). Let's plug \(x = 3\) into \(x^2 = 6y\): \(9 = 6y \implies y = 1.5\). Which matches the graph (the point \((3, 1.5)\) is on the parabola). For \(x^2 = 1.5y\), \(x = 3\) gives \(9 = 1.5y \implies y = 6\), which is not on the graph. So the correct equation is \(x^2 = 6y\).

Answer:

\(x^2 = 6y\) (the fourth option: \(x^2 = 6y\))