Question

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 \( y = ax^2 + bx + c \). 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 \( (4, 0) \).