To find the x and y intercepts of the graph represented by the equation y = x² + 3x + 10, we need to calculate each intercept separately.
Y-Intercept
The y-intercept occurs when the value of x is 0. We can find it by substituting x = 0 into the equation:
y = (0)² + 3(0) + 10 = 10
Thus, the y-intercept is at the point (0, 10).
X-Intercepts
The x-intercepts occur when y = 0. We set the equation to zero and solve for x:
0 = x² + 3x + 10
To solve this quadratic equation, we can use the quadratic formula, which is given by:
x = (-b ± √(b² – 4ac)) / 2a
In our equation, a = 1, b = 3, and c = 10. We first calculate the discriminant (b² – 4ac):
Discriminant = 3² – 4(1)(10) = 9 – 40 = -31
Since the discriminant is negative (-31), this means there are no real solutions for x. Thus, there are no x-intercepts.
Conclusion
In summary, the graph of the equation y = x² + 3x + 10 has a y-intercept at the point (0, 10) and no x-intercepts.