The equation of the parabola is given in the standard quadratic form as f(x) = 5x² + 30x + 6. To find the vertex of this parabola, we can use the vertex formula derived from the general form of a quadratic equation f(x) = ax² + bx + c, where:
- a = 5
- b = 30
- c = 6
The x-coordinate of the vertex can be calculated using the formula:
x = -b/(2a)
Substituting the values of a and b into the equation, we have:
x = -30/(2 * 5) = -30/10 = -3
Next, we substitute x = -3 back into the original equation to find the y-coordinate of the vertex:
f(-3) = 5(-3)² + 30(-3) + 6
= 5(9) – 90 + 6
= 45 – 90 + 6
= -39
Therefore, the vertex of the parabola is at the point:
(-3, -39)
This point represents the maximum or minimum of the parabola, depending on the orientation. Since the coefficient of x² (which is 5) is positive, the parabola opens upwards, and therefore the vertex at (-3, -39) is a minimum point.