f(x) = 6x^2 + 7x − 20
Find the maximum or minimum value of f(x)
the coefficient of x^2 is positive, so the parabola opens upward.
as always, the vertex is at x = -b/2a = -7/12
To find the maximum or minimum value of a quadratic function, you can use a concept called vertex form. The vertex form of a quadratic function is given by the equation f(x) = a(x - h)^2 + k, where (h, k) represents the coordinates of the vertex.
Step 1: Convert the given quadratic function to vertex form.
Given function: f(x) = 6x^2 + 7x - 20
To convert it to vertex form:
a. Complete the square for the quadratic term (x^2 + bx) by halving the coefficient of x and squaring it.
Completing the square: (x^2 + 7/6x) = (x^2 + 7/6x + (7/12)^2) - (7/12)^2
b. Rewrite the function with the completed square term.
f(x) = (x^2 + 7/6x + 49/144) - 49/144 - 20
c. Simplify the expression by combining the constant terms.
f(x) = (x + 7/12)^2 - 1489/144
So, the quadratic function f(x) can be written in vertex form as f(x) = (x + 7/12)^2 - 1489/144.
Step 2: Identify the coordinates of the vertex.
Comparing the vertex form equation f(x) = a(x - h)^2 + k, we can see that the vertex is located at the point (-h, k). In our case, the vertex is (-7/12, -1489/144).
Since the coefficient of the square term (a) is positive in the given quadratic function, the parabola opens upward, and the vertex represents the minimum point.
Therefore, the minimum value of f(x) is -1489/144, which occurs at x = -7/12.