The vertex of a quadratic function in the standard form, y = ax² + bx + c, can be determined using the vertex formula. For the given function y = 3x² + 2x + 1, we identify:
- a = 3
- b = 2
- c = 1
To find the x-coordinate of the vertex, we use the formula:
x = -b / (2a)
Substituting the values of a and b:
x = -2 / (2 * 3) = -2 / 6 = -1/3
Next, we can find the y-coordinate of the vertex by substituting x = -1/3 back into the equation:
y = 3(-1/3)² + 2(-1/3) + 1
Calculating each term:
- 3(-1/3)² = 3(1/9) = 1/3
- 2(-1/3) = -2/3
- 1 = 1
Now, summing these values gives:
y = 1/3 – 2/3 + 1 = -1/3 + 1 = 2/3
Thus, the coordinates of the vertex are:
(-1/3, 2/3)
In conclusion, the vertex of the graph of the function y = 3x² + 2x + 1 is at the point (-1/3, 2/3).