1. The problem asks to find the equation of a parabola that is a transformation of the function $f(x) = x^2$. 2. The parent function is $f(x) = x^2$, which is a parabola opening upwards with vertex at $(0,0)$. 3. The graph described is a downward-opening parabola shifted to the left with vertex at approximately $(-1, 3)$. 4. The vertex form of a parabola is given by: $$g(x) = a(x - h)^2 + k$$ where $(h,k)$ is the vertex and $a$ determines the direction and width. 5. Since the parabola opens downward, $a$ is negative. 6. Substitute the vertex $(-1, 3)$ into the vertex form: $$g(x) = a(x - (-1))^2 + 3 = a(x + 1)^2 + 3$$ 7. To find $a$, use a point on the parabola. The parabola intersects the x-axis roughly between $-3.5$ and $1.5$. Let's use the approximate root at $x = 1.5$ where $g(1.5) = 0$. 8. Substitute $x=1.5$ and $g(1.5)=0$: $$0 = a(1.5 + 1)^2 + 3$$ $$0 = a(2.5)^2 + 3$$ $$0 = 6.25a + 3$$ 9. Solve for $a$: $$6.25a = -3$$ $$a = \frac{-3}{6.25} = -0.48$$ 10. The equation of the parabola is: $$g(x) = -0.48(x + 1)^2 + 3$$ This matches the description: a downward-opening parabola shifted left by 1 and up by 3.