What is the equation for finding the vertex of a parabola given the quadratic function? Show your work.
To find the vertex of a parabola given a quadratic function, you can use the formula:
x = -b / 2a
The quadratic function generally takes the form of f(x) = ax^2 + bx + c, where a, b, and c are coefficients.
To find the vertex, you need to identify the values of a and b from the quadratic function. Once you have those values, you can substitute them into the formula:
x = -b / 2a
For example, let's say we have the quadratic function: f(x) = 2x^2 + 4x + 3
From this function, we can determine that a = 2 and b = 4.
Substituting those values into the formula, we get:
x = -4 / (2 * 2)
Simplifying further, we obtain:
x = -4 / 4
Which simplifies to:
x = -1
Therefore, the x-coordinate of the vertex is -1.
To find the y-coordinate of the vertex, we substitute the value of x back into the original quadratic function:
f(x) = 2x^2 + 4x + 3
f(-1) = 2(-1)^2 + 4(-1) + 3
Simplifying further, we get:
f(-1) = 2(1) - 4 + 3
f(-1) = 2 - 4 + 3
f(-1) = 1
Therefore, the y-coordinate of the vertex is 1.
Hence, the vertex of the parabola defined by the quadratic function f(x) = 2x^2 + 4x + 3 is (-1, 1).