Asked by yeslian c
Given the equation y=−x2+2x+48
with solutions of x=−6
and x=8
, which of the following identifies the general shape of its associated graph?(1 point)
Responses
The graph has the shape of a straight line.
The graph has the shape of a straight line.
The vertex is to the left of the y-axis.
The vertex is to the left of the y -axis.
The graph crosses the x-axis at x=−6
and x=8
.
The graph crosses the x -axis at x equals negative 6 and x equals 8 .
The graph opens upward.
with solutions of x=−6
and x=8
, which of the following identifies the general shape of its associated graph?(1 point)
Responses
The graph has the shape of a straight line.
The graph has the shape of a straight line.
The vertex is to the left of the y-axis.
The vertex is to the left of the y -axis.
The graph crosses the x-axis at x=−6
and x=8
.
The graph crosses the x -axis at x equals negative 6 and x equals 8 .
The graph opens upward.
Answers
There are no human answers yet.
Answered by
GPT-4o mini
Given the equation \( y = -x^2 + 2x + 48 \), we can determine the general shape of the graph.
### Analyzing the equation:
1. **Quadratic nature**: This is a quadratic equation in the form \( y = ax^2 + bx + c \), where \( a = -1 \), \( b = 2 \), and \( c = 48 \).
2. **Sign of \( a \)**: Since \( a < 0 \) (specifically \( a = -1 \)), the parabola opens **downward**.
### Given solutions and implications:
- The solutions \( x = -6 \) and \( x = 8 \) indicate that these are the **x-intercepts** of the graph, meaning the graph crosses the x-axis at these points.
### Assessing the options:
1. **The graph has the shape of a straight line.** - **Incorrect**. Quadratic graphs have a parabolic shape, not a straight line.
2. **The vertex is to the left of the y-axis.** - To find the vertex, we can use the formula for the x-coordinate of the vertex: \( x = -\frac{b}{2a} = -\frac{2}{2(-1)} = 1 \). Since \( 1 \) is to the right of the y-axis, this statement is **incorrect**.
3. **The graph crosses the x-axis at \( x = -6 \) and \( x = 8 \).** - **Correct**. This is consistent with the solutions provided.
4. **The graph opens upward.** - **Incorrect**. As established earlier, the graph opens downward.
### Conclusion:
The correct response is:
**The graph crosses the x-axis at \( x = -6 \) and \( x = 8 \).**
### Analyzing the equation:
1. **Quadratic nature**: This is a quadratic equation in the form \( y = ax^2 + bx + c \), where \( a = -1 \), \( b = 2 \), and \( c = 48 \).
2. **Sign of \( a \)**: Since \( a < 0 \) (specifically \( a = -1 \)), the parabola opens **downward**.
### Given solutions and implications:
- The solutions \( x = -6 \) and \( x = 8 \) indicate that these are the **x-intercepts** of the graph, meaning the graph crosses the x-axis at these points.
### Assessing the options:
1. **The graph has the shape of a straight line.** - **Incorrect**. Quadratic graphs have a parabolic shape, not a straight line.
2. **The vertex is to the left of the y-axis.** - To find the vertex, we can use the formula for the x-coordinate of the vertex: \( x = -\frac{b}{2a} = -\frac{2}{2(-1)} = 1 \). Since \( 1 \) is to the right of the y-axis, this statement is **incorrect**.
3. **The graph crosses the x-axis at \( x = -6 \) and \( x = 8 \).** - **Correct**. This is consistent with the solutions provided.
4. **The graph opens upward.** - **Incorrect**. As established earlier, the graph opens downward.
### Conclusion:
The correct response is:
**The graph crosses the x-axis at \( x = -6 \) and \( x = 8 \).**
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