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 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 upward. The graph opens upward. The graph opens downward.

Answer 1

To analyze the quadratic equation \( y = 3x^2 - 24x + 48 \), we can identify several key characteristics that determine the shape of the graph:

1. **Direction of the Opening**: The coefficient of \( x^2 \) (which is 3) is positive. This means that the graph opens upward.

2. **Vertex Position**: We can find the vertex of a quadratic equation given by \( ax^2 + bx + c \) using the formula for the x-coordinate of the vertex, which is \( x = -\frac{b}{2a} \). Here, \( a = 3 \) and \( b = -24 \):

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

The x-coordinate of the vertex is 4, which is to the right of the y-axis.

3. **Position Relative to the x-axis**: To determine if the entire graph lies above the x-axis, we can check the y-coordinate of the vertex. We can substitute \( x = 4 \) back into the equation to find \( y \):

\[
y = 3(4)^2 - 24(4) + 48
\]
\[
= 3(16) - 96 + 48
\]
\[
= 48 - 96 + 48
\]
\[
= 0
\]

The vertex is at the point (4, 0), meaning that the graph touches the x-axis at this point, but does not lie entirely above it.

Given these points, we can summarize:

- The graph **opens upward**,
- The vertex is at \( x = 4 \) (to the right of the y-axis),
- The graph does not lie entirely above the x-axis; it touches the x-axis at the vertex.

Thus the most relevant answer choices are:
- **The graph opens upward.**

This is the correct option regarding the general shape of the associated graph.