The x-intercept of a graph is the point where the graph crosses the x-axis. This happens when the value of y is zero. To find the x-intercepts, we set the equation y = x² + x + 6 to zero:
x² + x + 6 = 0
To solve this quadratic equation, we can use the quadratic formula: x = (-b ± √(b² – 4ac)) / 2a, where a = 1, b = 1, and c = 6.
Calculating the discriminant (b² – 4ac): 1² – 4(1)(6) = 1 – 24 = -23.
Since the discriminant is negative, there are no real solutions for x, which means the graph does not intersect the x-axis. Therefore, there are no x-intercepts.
On the other hand, the y-intercept of a graph is the point where the graph crosses the y-axis. This occurs when x equals zero. To find the y-intercept, we substitute x = 0 into the equation:
y = (0)² + (0) + 6 = 6.
Thus, the y-intercept is at the point (0, 6).
In summary, the graph of y = x² + x + 6 has no x-intercepts and a y-intercept at the point (0, 6).