1. The problem involves analyzing a parabola centered at the origin with vertex at (0,0) and points (-3,4) and (2,4) on the curve. 2. The general form of a parabola opening upwards with vertex at the origin is given by the formula: $$y = ax^2$$ where $a$ is a constant that determines the width and direction of the parabola. 3. To find $a$, use one of the given points on the parabola. Using point $(-3,4)$: $$4 = a(-3)^2$$ $$4 = 9a$$ 4. Solve for $a$: $$a = \frac{4}{9}$$ 5. Therefore, the equation of the parabola is: $$y = \frac{4}{9}x^2$$ 6. Check with the other point $(2,4)$: $$y = \frac{4}{9}(2)^2 = \frac{4}{9} \times 4 = \frac{16}{9} \neq 4$$ This suggests a discrepancy, so let's verify the points carefully. 7. Since both points $(-3,4)$ and $(2,4)$ lie on the parabola, the parabola is symmetric about the y-axis only if these points are equidistant from the vertex. Here, $|-3|=3$ and $|2|=2$, so the parabola might be shifted or the points are not on the same parabola. 8. Alternatively, consider the parabola in vertex form: $$y = a(x - h)^2 + k$$ Given vertex at $(0,0)$, $h=0$, $k=0$, so: $$y = ax^2$$ 9. Since the points $(-3,4)$ and $(2,4)$ both satisfy $y=4$, set up equations: $$4 = a(-3)^2 = 9a$$ $$4 = a(2)^2 = 4a$$ 10. From these: $$a = \frac{4}{9}$$ $$a = 1$$ 11. Contradiction means the points do not lie on the same parabola with vertex at origin. 12. Possibly, the parabola is not centered at origin or the points are open circles (not on the curve). 13. Since the problem states open circles at $(-3,4)$ and $(2,4)$, these points are not on the parabola but near it. 14. The parabola passes through $(0,0)$ and opens upwards, so the equation is: $$y = ax^2$$ 15. Without additional points on the curve, the exact value of $a$ cannot be determined. Final answer: The parabola equation is $$y = ax^2$$ with vertex at $(0,0)$ and $a > 0$ (opens upwards). The points $(-3,4)$ and $(2,4)$ are not on the parabola but indicated as open circles.