Deriving the Formula of the Vertex of Quadratic Functions

In getting the vertex of the quadratic function in general form f(x) = ax^2 + bx + c, we usually need to convert it to the vertex form f(x) = a(x - h)^2 + k. In the latter form, the vertex of the parabola is at (h,k). For example, the function in the general form

f(x) = 2x^2 - 12x + 22

can be rewritten in the vertex form as

f(x) = 2(x - 3)^2 + 4.

In the vertex form, it is easy to see that the vertex is at (h,k) = (3,4).

Aside from this method, we can also use the ordered pair

\left ( \dfrac{-b}{2a}, \dfrac{4ac - b^2}{4a} \right )

in place of (h,k)» Read more

Understanding the Meaning of Discriminant

In the Derivation of the Quadratic Formula, we have learned that the solutions to the quadratic equation ax^2 + bx + c = 0, where a \neq 0, is described by the equation

x = \displaystyle\frac{-b \pm \sqrt{b^2-4ac}}{2a}.

Graphically, getting the solutions of ax^2 + bx + c = 0 is equivalent to getting the value of x when y=0 of the function f(x) = ax^2 + bx + c, a \neq 0. This means that the quadratic formula above describes the root of the quadratic function f.

From the equation, the \pm indicates that the quadratic polynomial can have at most two roots, depending on the value of the expression under the radical sign. That is, the roots of the quadratic polyonomials are

x = \displaystyle\frac{-b + \sqrt{b^2-4ac}}{2a}


 x = \displaystyle\frac{-b - \sqrt{b^2-4ac}}{2a}.

The number of roots, however, will depend on the expression b^2 - 4ac. Notice that if it is negative, there is no root (real root to be exact), since we cannot extract the square root of a negative number. Therefore, if b^2 - 4ac < 0, then, there is no real root. » Read more

The Multiplication Parabola

In the graph below, move points A and B such that they are on opposite sides of the y-axis.  What do you observe?

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Theorem: Given two points on the function f(x) = x^2, whose coordinates are (a,a^2) and (b, b^2) with a < 0 and b > 0, the y-intercept of the line passing through these points is -ab. » Read more

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