Quadratic functions are functions that can be written in the standard form f(x) =
ax^{2} + bx + c, where a ≠ 0 and a, b, and c are all constants.
The standard form has 3 different types of terms:
Notice that the condition that a ≠ 0 ensure that every function has a quadratic term, but not necessarily a linear term or a constant term. If a = 0, the function has no quadratic term. In that case, the function is not a quadratic function.
Here is how the graph of quadratic functions will look like. Notice the axis of symmetry and the vertex that can either be a maximum or a minimum.
f(x) = 2x^{2} + 3x + -4
f(x) = 2x^{2} - 3x + 5
f(x) = -x^{2} + x + 100
f(x) = 3x^{2} + 6x
f(x) = 5x^{2} + -4
f(x) = 6x^{2}
f(x) = (x + 2)(x+3)
f(x) = 2(x - 3)^{2} + 4
Notice that f(x) = (x + 2)(x+3) = x^{2} + 3x + 2x + 6 = x^{2} + 5x + 6
Notice also that f(x) = 2(x - 3)^{2} + 4 = 2(x^{2} - 6x + 9) + 4 = 2x^{2} - 12x + 18 + 4 =
2x^{2} - 12x + 22
A function may appear to be a quadratic function when in fact it is not quadratic.
For example, f(x) = 4(x^{2} + x) - 4(x^{2} + 8) is not a quadratic function.
f(x) = 4(x^{2} + x) - 4(x^{2} + 8)
f(x) = 4x^{2} + 4x - 4x^{2} + -32
f(x) = 4x + -32
As you can see, this function is linear.
General form
f(x) = ax^{2} + bx + c, where a, b, and c are real numbers, is called the general form of a quadratic function.
Factored form
f(x) = (ax + b)(cx + d), where a, b, c, and d are real numbers, is called the factored form of a quadratic function.
Vertex form
f(x) = a(x - h)^{2} + k, where a, h, and k are real numbers, is called the vertex form of a quadratic function.
Sep 17, 23 09:46 AM
Jun 09, 23 12:04 PM