Question

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)
Responses

The graph opens upward.
The graph opens upward.

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

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

The graph opens downward.

Answer 1

The equation given is \( y = 3x^2 - 24x + 48 \).

1. **Determine the direction the graph opens:**
- The coefficient of \( x^2 \) is positive (3), which means the graph opens upward.

2. **Identify the vertex:**
- The vertex can be calculated using the formula \( x = -\frac{b}{2a} \) where \( a = 3 \) and \( b = -24 \):
\[
x = -\frac{-24}{2 \times 3} = \frac{24}{6} = 4.
\]
The vertex is at (4, y-value), which means it is to the right of the y-axis.

3. **Analyze the y-value at the vertex:**
- Substitute \( x = 4 \) back into the equation to find the y-value at the vertex:
\[
y = 3(4)^2 - 24(4) + 48 = 3(16) - 96 + 48 = 48 - 96 + 48 = 0.
\]
The vertex is at (4, 0), which means the graph touches the x-axis.

4. **Conclude the overall shape of the graph:**
- Since the graph opens upward and the vertex is on the x-axis, the graph lies above the x-axis except at the vertex where it touches it.

Given this information, the most accurate statement from the provided options is:

**The graph opens upward.**