The graph of a quadratic function $y=ax^2+bx+c$ (with $a\neq0$) is a curve called a parabola.
Shape and axis of symmetry
When $a>0$ the parabola opens upward and has a minimum point; when $a<0$ it opens downward and has a maximum point. The curve is symmetric about the vertical line $x=-\frac{b}{2a}$, called the axis of symmetry.
Key formula
Axis of symmetry: $x=-\frac{b}{2a}$. The $x$-coordinate of the vertex (turning point) is also $-\frac{b}{2a}$; substitute it back to find the $y$-coordinate.
Intercepts
The $y$-intercept is found by putting $x=0$, giving $y=c$. The $x$-intercepts are the roots of $ax^2+bx+c=0$; a parabola may cut the $x$-axis at two points, touch it at one point, or not meet it at all.
Worked example
For $y=x^2-4x+3$: $a=1>0$ so it has a minimum. Axis of symmetry $x=-\frac{-4}{2}=2$. At $x=2$, $y=4-8+3=-1$, so the vertex is $(2,-1)$. The $y$-intercept is $3$; the $x$-intercepts are $x=1$ and $x=3$.
Remember
- $a$ sets the direction: up for $a>0$, down for $a<0$.
- The vertex lies on the axis of symmetry $x=-b/(2a)$.
- The $y$-intercept is always $c$.