Fill in the blanks to describe the graph of the equation. y = −x2+x +6 (3 points) The graph of the equation opens and crosses the x-axis at . This means that the equation has .

Answer 1

To analyze the graph of the equation \( y = -x^2 + x + 6 \):

1. Determining the direction of opening: The coefficient of \( x^2 \) is negative (\(-1\)), which means the graph opens downward.

2. Finding the x-axis intersections (x-intercepts): To find where the graph crosses the x-axis, we set \( y = 0 \): \[ 0 = -x^2 + x + 6 \] Rearranging gives: \[ x^2 - x - 6 = 0 \] We can factor this quadratic equation: \[ (x - 3)(x + 2) = 0 \] Thus, the x-intercepts are \( x = 3 \) and \( x = -2 \).

3. Determining the number of x-intercepts: Since the graph crosses the x-axis at two points, this means the equation has two real roots.

Putting it all together:

The graph of the equation opens downward and crosses the x-axis at (3, 0) and (-2, 0). This means that the equation has two real roots.