Vertex Form | Richard Crooks's Website

The vertex form of the quadratic expression is in the form:

$$y = a(x+h)^2 + k$$

The vertex form of the quadratic expression describes the location of the vertex of the parabola. The vertex is the point in the parabola where the slope is 0, and is also the point where the line of symmetry across the parabola. Aside the $x$, the vertex form contains 3 coefficients, $a$, $h$ and $k$.

The $a$ coefficient

The a coefficient describes the degree of stretch or compression in the parabola (Figure 1). The $a$ coefficient manipulates $a$ squared term, thus the same exponential trend is produced, but is stretched or compressed according to the value of $a$.

An animated GIF of a quadratic function in the vertex form showing how the shape of the parabola changes as the a value changes — Figure 1: Changing the $a$ value of the vertex form of the quadratic expression. The $a$ value of the vertex form directly describes the degree of stretch or compression of the parabola, with the parabola becoming more stretched the further the value is from 0. Positive $a$ values have parabolas that open upwards, while negative $a$ values have parabolas that open downwards.

If the $a$ is greater than 1, then the parabola is stretched. An $a$ value of less than 1 is compressed. If the $a$ value is negative then the parabola is upside down.

The $h$ coefficient

The $h$ coefficient describes where on the $x$ axis the vertex is (Figure 2). The $x$ coordinate of the vertex is the negative of the $h$ value, so an h value of + 1 means an $x$ coordinate of -1, while an $h$ value of - 1 means an x coordinate of +1.

An animated GIF of a quadratic function in the vertex form showing how location of the vertex changes horizontally as the h value changes — Figure 2: Changing the $h$ value of the vertex form of the quadratic expression. The $h$ value describes the $x$ coordinate of the vertex. The $h$ value is the negative of the $x$ coordinate of the vertex, where positive values of $h$ refer to negative $x$ coordinates of the vertex and vice versa.

The vertex of a parabola is at the point where $x = 0$, as 0 squared is 0, while positive and negative values either side of 0 have increasingly large outputs when squared. As the $h$ value modifies the $x$ value prior to being squared, this has the effect of changing where this midpoint 0 is. This is also why the $x$ coordinate is the negative of the $h$ value. An $h$ value of –3 would change an x value of +3 to be effectively 0 prior to squaring, thus moving the entire parabola to $x = 3$.

The $k$ coefficient

The $k$ coefficient changes the $y$ coordinate of the vertex. Unlike the other coefficients which need some interpretation, the $y$ coordinate of a vertex is directly the $k$ coefficient in the vertex form (Figure 3).

An animated GIF of a quadratic function in the vertex form showing how location of the vertex changes vertically as the k value changes — Figure 3: Changing the $k$ value of the vertex form of the quadratic expression. The $k$ value of the vertex form describes the $y$ coordinate of the vertex. The number is directly equivalent, so the $k$ value of a vertex form can be derived simply from knowing the location vertex without any adjustments or calculations.

The $k$ coefficient as an addition term applied after the other terms in the expression has the effect of directly moving the parabola along the $y$ axis.

The vertex form is a useful for of a quadratic expression if you wish to analyze features of the vertex, such as if the parabola describes a flight path, the maximum height reached.