A quadratic function is a polynomial function in which one of the terms has a variable of degree 2 $\left({x}^{2}\right)$, and none of the other terms have exponents larger than 2. The graph of a quadratic function is called a parabola. A quadratic function is written as

 $f\left(x\right)=a{x}^{2}+bx+c$

For now, you’ll only look at quadratic functions with $b=0$ and $c=0$. In these cases, the function will look like the one in the formula box below. The value of $a$ decides if the graph is smiling (minimum) or frowning (maximum).

Formula

### QuadraticFunctionwhere$b=0$and$c=0$

 $f\left(x\right)=a{x}^{2}$

Below you see several examples of quadratic functions as described above. The three graphs in Figure (a) show three different functions where $a>0$, where the pink graph is $y={x}^{2}$. The three graphs in Figure (b) show three different functions where $a<0$, where the pink graph is $y=-{x}^{2}$.

Rule

• If $a>0⇒$ the graph looks like a smile $\bigcup$: Figure (a).

• If $a>1⇒$ the graph still looks like it’s smiling $\bigcup$, but on the inside of the pink graph $\left(a=1\right)$ in Figure (a): See the blue graph.

• If $a>0$, but $a<1⇒$ the graph is still smiling $\bigcup$, but on the outside of the pink graph $\left(a=1\right)$ in Figure (a): See the gray graph.

• If $a<0⇒$ the graph appears to be frowning $\bigcap$: Figure (b).

• If $a<-1⇒$ the graph is still frowning $\bigcap$, but on the inside of the pink graph $\left(a=-1\right)$ in Figure (b): See the blue graph.

• If $a<0$, but $a>-1⇒$ the graph looks like a frown $\bigcap$, but on the outside of the pink graph $\left(a=-1\right)$ in Figure (b): See the gray graph.

• If $a=0$, it’s not a quadratic function, but a constant term equal to 0. The graph is then the $x$-axis.

We are often interested in finding the $x$-values and $y$-values of the maximum or minimum, meaning the point where the graph changes from increasing to decreasing, or from decreasing to increasing. In our case, $f\left(x\right)=a{x}^{2}$, this always happens at the origin $\left(0,0\right)$!

When you start working with more advanced problems, the theory of the quadratic function will expand, and the graphs you work with will move away from the origin and have other maxima or minima.