Question

complex numbers online practice
complete this assessment to review what youve learned. it will not count toward your grade.
given the equation $y = -x^2 + 2x + 48$ with solutions of $x = -6$ and $x = 8$, which of the following identifies the general shape of its associated graph? (1 point)

• the vertex is to the left of the $y$-axis.
• the graph opens upward.
• the graph crosses the $x$-axis at $x = -6$ and $x = 8$.
• the graph has the shape of a straight line.

check answer remaining attempts: 3

Explanation:

1. Analyze the first option: The vertex of \( y = -x^2 + 2x + 48 \) has an x - coordinate of \( x=-\frac{b}{2a}=-\frac{2}{2\times(-1)} = 1 \), which is to the right of the y - axis, so this option is wrong.
2. Analyze the second option: For a quadratic function \( y = ax^{2}+bx + c \), if \( a<0 \), the parabola opens downward. Here \( a=- 1<0 \), so the graph opens downward, not upward. This option is wrong.
3. Analyze the third option: The solutions of the equation \( -x^{2}+2x + 48 = 0 \) are \( x=-6 \) and \( x = 8 \). The x - intercepts of the graph of a quadratic function \( y=ax^{2}+bx + c \) are the solutions of the equation \( ax^{2}+bx + c = 0 \). So the graph of \( y=-x^{2}+2x + 48 \) crosses the x - axis at \( x=-6 \) and \( x = 8 \). This option is correct.
4. Analyze the fourth option: The function \( y=-x^{2}+2x + 48 \) is a quadratic function, and its graph is a parabola (a U - shaped or inverted U - shaped curve), not a straight line. A straight line is the graph of a linear function (of the form \( y = mx + b\)). This option is wrong.

Answer:

C. The graph crosses the x - axis at \( x=-6 \) and \( x = 8 \)