Given the equation y=−x2+2x+48

Question

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.

Answer 1

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 \( x = 8 \).