A quadratic function in the form

$f(x)=a{ x }^{ 2 }+bx+x$

is in standard form.

Regardless of the format, the graph of a quadratic function is a parabola.

The graph of $y=x^2-4x+3$ 

The graph of any quadratic equation is always a parabola.

Coefficients and Graphs of Quadratic Function

Each coefficient in a quadratic function in standard form has an impact on the shape and placement of the function's graph. 

Coefficient of $x^2$, $a$

The coefficient $a$ controls the speed of increase (or decrease) of the quadratic function from the vertex. A larger, positive $a$ makes the function increase faster and the graph appear thinner.

The coefficient $a$ controls the speed of increase of the parabola.

The black curve is $y=4x^2$ while the blue curve is $y=3x^2.$ The black curve appears thinner because its coefficient $a$ is bigger than that of the blue curve. 

Whether the parabola opens upward or downward is also controlled by $a$. If the coefficient $a>0$, the parabola opens upward, and if the coefficient $a<0$, the parabola opens downward.

Quadratics either open upward or downward

The blue parabola is the graph of $y=3x^2.$ It opens upward since $a=3>0.$ The black parabola is the graph of $y=-3x^2.$ It opens downward since $a=-3<0.$

The Axis of Symmetry

The coefficients $b$ and $a$ together control the axis of symmetry of the parabola and the $x$-coordinate of the vertex. The axis of symmetry for a parabola is given by:

$x=-\dfrac{b}{2a}$ 

For example, consider the parabola $y=2x^2-4x+4 $ shown below. Because $a=2$ and $b=-4,$ the axis of symmetry is:

 $x=-\frac{-4}{2\cdot 2} = 1$

The vertex also has $x$ coordinate $1$.

The graph of $y=2x^2-4x+4.$

The axis of symmetry is a vertical line parallel to the y-axis at  $x=1$.

The $y$-intercept of the Parabola

The coefficient $c$ controls the height of the parabola. More specifically, it is the point where the parabola intercepts the y-axis. The point $(0,c)$ is the $y$ intercept of the parabola. Note that the parabola above has $c=4$ and it intercepts the $y$-axis at the point $(0,4).$