🧑Zayn Alazawi 📅August 18, 2020 📊Grade 11 Math

Quadratic functions, characterized by a degree of 2, manifest as parabolic graphs without domain restrictions. Take, for instance, the quadratic function $$f(x)=2x^2-3x+1$$. The graph of this function forms a parabola.

A critical feature of quadratic functions is the constant nonzero second differences when presented in a table of values. Let's illustrate this with the corrected table for the function $$f(x)=x^2+2x+3$$:

$$x$$ $$f(x)$$
-2 3
-1 2
0 3
1 6
2 11

Here, we observe that the second differences (i.e., the differences between consecutive function values) are consistent (1, 1, 1), affirming a quadratic relationship.

The parabolic nature of the graph is determined by the sign of the second differences. Consider the function $$g(x)=-x^2+4x-2$$. As the second differences in the table are negative (-2, -2, -2), the parabola opens downward.

Quadratic functions can be expressed in various forms. The standard form is $$f(x)=ax^2+bx+c$$, where $$a$$, $$b$$, and $$c$$ are coefficients. For instance, if $$a$$=2, $$b$$=−3, and $$c$$=1, the standard form of $$f(x)$$ is $$f(x)=2x^2−3x+1$$.

Conversion between different forms involves algebraic techniques. To move from standard form to vertex form, completing the square can be employed. For factored form, the function can be expressed as a product of factors. For example, using $$f(x)=x^2+2x+3$$, the factored form might be $$f(x)=(x+1)(x+3)$$, where the roots of the quadratic equation provide the factors.

Vertex form is expressed as $$g(x)=a(x−h)^2+k$$, with $$(h,k)$$ as the vertex of the parabola. If $$h=−1$$ and $$k=4$$, then the vertex form of $$g(x)$$ is $$g(x)=(x+1)^2+4$$.

Understanding these different forms equips us with a versatile toolkit for analyzing and solving problems involving quadratic functions, allowing us to represent them in ways most suitable for specific situations.