## Quadratic Functions

 2.3 Quadratic Functions

 Quadratic Function Properties A quadratic function can be expressed in the form $$f(x)=ax^2+bx+c$$, where $$a$$, $$b$$ and $$c$$ are constants and $$a \ne 0$$. If $$a\gt 0$$, graph has the shape $$\LARGE \smile$$ which passes through a minimum point. If $$a\lt 0$$, graph has the shape $$\LARGE \frown$$ which passes through a maximum point.

Relationship between position of the graph on $$x$$-axis and its type of roots
 Two Different Real Roots

$$b^2-4ac>0$$

 Two Equal Real Roots

$$b^2-4ac=0$$

 No Real Roots

$$b^2-4ac<0$$

Other form of Quadratic Function Properties
 Vertex Form
• A quadratic function can be expressed in the form $$f(x)=a(x-h)^2+k$$ where $$a$$$$h$$ and $$k$$ are constants and $$a \neq 0$$.
• In this form, $$x=h$$ is an axis of symmetry and $$(h,k)$$ is the coordinate of minimum or maximum point.
 Factored Form
• A quadratic function can be expressed in the form $$f(x)=a(x-p)(x-q)$$ where $$a$$$$p$$ and $$q$$ are constants and $$a \neq 0$$.
• In this form, the roots of the quadratic function are $$p$$ and $$q$$.
• The axis of symmetry, $$x=\dfrac{p+q}{2}$$.
• The coordinate of minimum or maximum point is $$\left[ \dfrac{p+q}{2}, f \left( \dfrac{p+q}{2} \right) \right]$$.

## Quadratic Functions

 2.3 Quadratic Functions

 Quadratic Function Properties A quadratic function can be expressed in the form $$f(x)=ax^2+bx+c$$, where $$a$$, $$b$$ and $$c$$ are constants and $$a \ne 0$$. If $$a\gt 0$$, graph has the shape $$\LARGE \smile$$ which passes through a minimum point. If $$a\lt 0$$, graph has the shape $$\LARGE \frown$$ which passes through a maximum point.

Relationship between position of the graph on $$x$$-axis and its type of roots
 Two Different Real Roots

$$b^2-4ac>0$$

 Two Equal Real Roots

$$b^2-4ac=0$$

 No Real Roots

$$b^2-4ac<0$$

Other form of Quadratic Function Properties
 Vertex Form
• A quadratic function can be expressed in the form $$f(x)=a(x-h)^2+k$$ where $$a$$$$h$$ and $$k$$ are constants and $$a \neq 0$$.
• In this form, $$x=h$$ is an axis of symmetry and $$(h,k)$$ is the coordinate of minimum or maximum point.
 Factored Form
• A quadratic function can be expressed in the form $$f(x)=a(x-p)(x-q)$$ where $$a$$$$p$$ and $$q$$ are constants and $$a \neq 0$$.
• In this form, the roots of the quadratic function are $$p$$ and $$q$$.
• The axis of symmetry, $$x=\dfrac{p+q}{2}$$.
• The coordinate of minimum or maximum point is $$\left[ \dfrac{p+q}{2}, f \left( \dfrac{p+q}{2} \right) \right]$$.