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 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.
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.
Answers
There are no human answers yet.
Answered by
GPT-4o mini
To determine the general shape of the graph for the equation \( y = 3x^2 - 24x + 48 \), we can analyze the quadratic equation.
1. **Determine the vertex and direction of opening**: The standard form of a quadratic equation is \( y = ax^2 + bx + c \). In this case, \( a = 3 \). Since \( a > 0 \), the graph opens upward.
2. **Find the vertex**: The x-coordinate of the vertex can be found using the formula \( x = -\frac{b}{2a} \).
- Here, \( b = -24 \) and \( a = 3 \).
- \( x = -\frac{-24}{2 \times 3} = \frac{24}{6} = 4 \).
3. **Evaluate the vertex position**: The vertex occurs at \( x = 4 \), which is to the right of the y-axis.
4. **Roots of the equation**: The given solutions (or roots) are \( x = 4 \) and \( x = 4 \), meaning the graph touches the x-axis at one point (the vertex) but does not go below it.
With this analysis, the graph:
- Opens upward.
- The entire graph is above the x-axis except for the vertex point.
Given the options:
- The graph opens upward.
Thus, the correct response is: **The graph opens upward.**
1. **Determine the vertex and direction of opening**: The standard form of a quadratic equation is \( y = ax^2 + bx + c \). In this case, \( a = 3 \). Since \( a > 0 \), the graph opens upward.
2. **Find the vertex**: The x-coordinate of the vertex can be found using the formula \( x = -\frac{b}{2a} \).
- Here, \( b = -24 \) and \( a = 3 \).
- \( x = -\frac{-24}{2 \times 3} = \frac{24}{6} = 4 \).
3. **Evaluate the vertex position**: The vertex occurs at \( x = 4 \), which is to the right of the y-axis.
4. **Roots of the equation**: The given solutions (or roots) are \( x = 4 \) and \( x = 4 \), meaning the graph touches the x-axis at one point (the vertex) but does not go below it.
With this analysis, the graph:
- Opens upward.
- The entire graph is above the x-axis except for the vertex point.
Given the options:
- The graph opens upward.
Thus, the correct response is: **The graph opens upward.**
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