Given the equation y=3x2−24x+48 with solutions of x=4 and x=4 , which of the following identifies the general shape of its associated graph?(1 point)

Question

Responses

The entire graph lies above the x-axis.
The entire graph lies above the x -axis.

The vertex is to the left of the y-axis.
The vertex is to the left of the y -axis.

The graph opens upward.
The graph opens upward.

The graph opens downward.

Answer 1

To analyze the quadratic equation given by \( y = 3x^2 - 24x + 48 \), we can look at its coefficients to determine the shape and orientation of the graph.

1. The coefficient of \( x^2 \) is positive (3), which indicates that the graph opens upward.

2. The solutions of the equation are \( x = 4 \) and \( x = 4 \) (a repeated root), which means that the vertex of the parabola is at \( x = 4 \). To find the y-coordinate of the vertex, we can substitute \( x = 4 \) back into the equation: \[ y = 3(4^2) - 24(4) + 48 = 3(16) - 96 + 48 = 48 - 96 + 48 = 0. \] Thus, the vertex is at \( (4, 0) \), which means it is on the x-axis.

3. Since the graph opens upward and the vertex is at the x-axis, it means that the entire graph does not lie above the x-axis, but rather touches it at the vertex \( (4, 0) \).

Conclusion:

• The graph opens upward.

So, the correct response from the options given is:

• The graph opens upward.