Lines and Quadratics
3.3 Quadratic Functions
Shape and Structure
Quadratic functions create parabolas, and their form tells us where key features are.
Definition: Forms of a quadratic function
- General form: $f(x)=ax^2+bx+c$, $a\ne0$
- Vertex form: $f(x)=a(x-h)^2+k$
Theorem: Vertex and axis
For $f(x)=a(x-h)^2+k$:
- Vertex is $(h,k)$.
- Axis of symmetry is $x=h$.
Example: Convert to vertex form
Rewrite
$$f(x)=x^2-6x+5$$
in vertex form.
Show Solution
Complete the square:
$$x^2-6x+5=(x^2-6x+9)-9+5=(x-3)^2-4.$$
So vertex form is
$$f(x)=(x-3)^2-4.$$
MyOpenMath: Practice: vertex form conversion (placeholder)
Modeling with Quadratics
Quadratics often model path and area problems.
Example: Find maximum value of a quadratic
Find the maximum value of
$$f(x)=-2x^2+8x+3.$$
Show Solution
For $ax^2+bx+c$, vertex x-coordinate is
$$x=-\frac{b}{2a}=-\frac{8}{2(-2)}=2.$$
Compute output:
$$f(2)=-2(4)+8(2)+3=-8+16+3=11.$$
Since $a<0$, this is a maximum. Maximum value is $11$.