Question

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**.