Given the equation y=3x^2-24x+48 with solutions of x=4 and x=4 , which of the following identifies the general shape of its associated graph

The equation you provided, \( y = 3x^2 - 24x + 48 \), is a quadratic equation in standard form \( y = ax^2 + bx + c \), where \( a = 3 \), \( b = -24 \), and \( c = 48 \).

The key characteristics to determine the shape of the graph are:

1. **Leading Coefficient \( (a) \)**: Since \( a = 3 \) (which is positive), the parabola opens upward.

2. **Vertex and Axis of Symmetry**: The axis of symmetry for a quadratic function in standard form can be found using the formula \( x = -\frac{b}{2a} \).

\[
x = -\frac{-24}{2 \cdot 3} = \frac{24}{6} = 4
\]

This confirms that the vertex is located at \( x = 4 \).

3. **Roots (Solutions)**: The equation has solutions at \( x = 4 \) and \( x = 4 \), indicating that it has a double root (also known as a repeated root). This means the vertex touches the x-axis at that point.

Combining all these features, the graph of the equation \( y = 3x^2 - 24x + 48 \) is a parabola that opens upwards, has its vertex at the point (4, y), and touches the x-axis at \( x = 4 \).

In summary, the general shape of the associated graph is an **upward-opening parabola** with a double root at \( x = 4 \).