Quadratic Relations and Their Properties

A quadratic relation is one that can be written in the form $y=a{x}^2+bx+c$, where $a$, $b$ and $c$ are numbers and $a\ne 0$. The highest power of $x$ is 2, so it is called a second-degree relation. The condition $a\ne 0$ matters: if $a=0$, the $x^2$ term disappears and $y=bx+c$ is linear. Examples of quadratic relations: $y=x^2$, $y=2x^2-5$ (here $b=0$), and $y=(x-1)(x+3)$, because expanding gives $y=x^2+2x-3$. Not quadratic: $y=4x-1$ (no $x^2$ term) and $y=x^3+x$ (the highest power is 3). Every quadratic relation has a parabola as its graph.

You can spot a quadratic relation in a table of values without graphing. First make sure the $x$-values go up by equal steps. Then find the first differences: subtract each $y$-value from the one after it (later minus earlier). For a linear relation, the first differences are all the same. For a quadratic relation they are not, but if you subtract each first difference from the next one, you get the second differences, and those are all the same. Here is the table for $y=x^2-4x$, where each first difference compares one row with the row below it.

Using Finite Differences to Find Type of Relation (Linear, Quadratic, Neither)

• Vertex: the turning point of the parabola, the lowest point if it opens up or the highest point if it opens down. In the figure it is $(3,-4)$.
• Axis of symmetry: the vertical line through the vertex. Fold the graph along it and the two halves match. Name it with an equation: here $x=3$.
• Zeros (also called x-intercepts): the $x$-values where the parabola crosses or touches the $x$-axis, so $y=0$. Here they are $x=1$ and $x=5$. A parabola can have two zeros, one zero (the vertex sits on the $x$-axis) or no zeros (it never reaches the $x$-axis).
• y-intercept: the $y$-value where the parabola crosses the $y$-axis, found by setting $x=0$. In $y=a{x}^2+bx+c$ it is $c$, because $a{(0)}^2+b(0)+c=c$. Here it is $5$. A parabola of this form has exactly one $y$-intercept.
• Minimum or maximum value: the $y$-coordinate of the vertex. A parabola that opens up has a minimum value (here $-4$); one that opens down has a maximum value.

Symmetry is the most useful property of all. Any two points on a parabola at the same height are mirror images across the axis, so the axis lies exactly halfway between them. In the table for $y=x^2-4x$, the value $y=0$ appears at $x=0$ and at $x=4$, so the axis is $x=\frac{0+4}{2}=2$. The vertex is on the axis, so it is $(2,-4)$, which matches the lowest $y$-value in the table. In the figure, the $y$-intercept $(0,5)$ has a partner $(6,5)$ on the other side: both are 3 units from the axis $x=3$.