The turning point of a quadratic function given by the equation f(x) = ax² + bx + c can be found using the vertex formula. The vertex represents either the maximum or minimum point of the parabola, depending on the direction it opens. For the function in question, f(x) = x² + 3x + 1, we have:
- a = 1
- b = 3
- c = 1
To find the x-coordinate of the turning point (vertex), we use the formula:
x = -b / (2a)
Plugging in our values:
x = -3 / (2 * 1) = -3 / 2 = -1.5
Next, we need to find the y-coordinate by substituting x back into the function:
f(-1.5) = (-1.5)² + 3(-1.5) + 1
Calculating this:
- f(-1.5) = 2.25 – 4.5 + 1 = -1.25
Thus, the coordinates of the turning point (vertex) of the function f(x) = x² + 3x + 1 are:
Turning point: (-1.5, -1.25)
This point is where the graph of the function reaches its minimum value, since the parabola opens upwards (a > 0).