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?

Answer 1

The given equation is a quadratic function in the form of $y=a{x}^2+bx+c$. Here, $a=3$, $b=-24$, and $c=48$.

To determine the general shape of the graph associated with this quadratic function, we can examine the coefficient $a$:

1. Shape of the Graph:
   • If $a>0$ (which it is, since $a=3$), the parabola opens upwards.
   • If $a<0$, the parabola would open downwards.

Since $a=3$ is greater than zero, the graph of the equation $y=3x^2-24x+48$ is a parabola that opens upwards.

2. Vertex and Solutions:
   • The solutions $x=4$ (which is a repeated root or double root) indicate that the vertex of the parabola touches the x-axis at this point. This means that the vertex is located at $x=4$ and is the minimum point of the parabola.

In conclusion, the general shape of the graph associated with the equation $y=3x^2-24x+48$ is an upward-opening parabola that touches the x-axis at the point $(4,0)$.