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 downward. The graph opens downward. 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 entire graph lies above the x-axis.
Answers
There are no human answers yet.
Answered by
GPT-4o mini
The equation given is a quadratic equation in the form \(y = ax^2 + bx + c\), where \(a = 3\), \(b = -24\), and \(c = 48\).
1. **Determine if the graph opens upward or downward**:
Since the coefficient of \(x^2\) (which is \(a\)) is positive (\(a = 3\)), the graph opens **upward**.
2. **Vertex position**:
The vertex of a quadratic function \(y = ax^2 + bx + c\) can be found using the formula:
\[
x_{vertex} = -\frac{b}{2a}
\]
Substituting the values of \(a\) and \(b\):
\[
x_{vertex} = -\frac{-24}{2 \cdot 3} = \frac{24}{6} = 4
\]
This means the vertex is located at \(x = 4\), which is on the right side of the y-axis.
3. **Position of the entire graph relative to the x-axis**:
To determine if the entire graph lies above the x-axis, we can assess the y-values at the solutions (roots) \(x = 4\):
\[
y = 3(4^2) - 24(4) + 48 = 3(16) - 96 + 48 = 48 - 96 + 48 = 0
\]
Since the vertex represents the minimum point of the graph, and the graph opens upwards, with the minimum value (the vertex) being at \(y = 0\) when \(x = 4\), it means that the graph touches the x-axis at \(y = 0\), but does not lie entirely above the x-axis — it intersects the x-axis.
Based on the above analysis, the correct identification of the general shape of the graph is:
- **The graph opens upward.**
1. **Determine if the graph opens upward or downward**:
Since the coefficient of \(x^2\) (which is \(a\)) is positive (\(a = 3\)), the graph opens **upward**.
2. **Vertex position**:
The vertex of a quadratic function \(y = ax^2 + bx + c\) can be found using the formula:
\[
x_{vertex} = -\frac{b}{2a}
\]
Substituting the values of \(a\) and \(b\):
\[
x_{vertex} = -\frac{-24}{2 \cdot 3} = \frac{24}{6} = 4
\]
This means the vertex is located at \(x = 4\), which is on the right side of the y-axis.
3. **Position of the entire graph relative to the x-axis**:
To determine if the entire graph lies above the x-axis, we can assess the y-values at the solutions (roots) \(x = 4\):
\[
y = 3(4^2) - 24(4) + 48 = 3(16) - 96 + 48 = 48 - 96 + 48 = 0
\]
Since the vertex represents the minimum point of the graph, and the graph opens upwards, with the minimum value (the vertex) being at \(y = 0\) when \(x = 4\), it means that the graph touches the x-axis at \(y = 0\), but does not lie entirely above the x-axis — it intersects the x-axis.
Based on the above analysis, the correct identification of the general shape of the graph is:
- **The graph opens upward.**
Submit Your Answer
We prioritize human answers over AI answers.
If you are human, and you can answer this question, please submit your answer.