Quadratic function

A quadratic function is a function given by

Its graph is a parabola. The condition $a \neq 0$ is not a formality: were $a$ zero, the formula would collapse to $bx + c$, a linear function, and the graph would straighten into a line.

Arms, vertex, axis of symmetry

A parabola has one turning point — the vertex $(p, q)$ — and is symmetric about the vertical line through it:

The coordinates of the vertex come from the formulas:

where $\Delta = b^2 - 4ac$ — the same delta we computed for quadratic equations. The safest way to get the second coordinate is as $f(p)$: the delta formula gives the same number but invites a sign slip.

The sign of $a$ decides which way the arms point:

Extremum and range

The vertex is the lowest or the highest point of the graph, so its second coordinate is the extreme value of the function:

• $a > 0$ — arms up, a minimum of value $q$ at the vertex, range $\langle q, +\infty)$;
• $a < 0$ — arms down, a maximum of value $q$ at the vertex, range $(-\infty, q\rangle$.

So a quadratic function never takes all real values — half of the vertical axis always stays empty. The domain, in contrast, is every real number: anything can be squared.

The axis of symmetry $x = p$ is also the boundary of monotonicity. For $a > 0$ the function decreases to the left of $p$ and increases to the right; for $a < 0$ it is the other way round.

The three forms

The same function can be written in three ways, and each shows something different without any computation:

• the general one shows the value at zero: $f(0) = c$;
• the canonical one shows the vertex: $(p, q)$;
• the factored one shows the zeros: $x_1$ and $x_2$.

The factored form exists only when there are zeros at all, that is when $\Delta \geq 0$ (at $\Delta = 0$ it takes the shape $a(x - x_0)^2$). The canonical form exists always.

Note that the vertex sits exactly halfway between the zeros: $\frac{1 + 3}{2} = 2$. That is the fastest route to $p$ once the roots are known — and it comes straight from the symmetry of the parabola.

The zeros

The zeros of a quadratic function are the solutions of $ax^2 + bx + c = 0$, that is the points where the parabola crosses the $x$-axis. How many there are depends on the delta:

The whole computation — the formulas for $x_1$ and $x_2$, the incomplete cases, Vieta's formulas — lives in the lesson on quadratic equations. What matters here is the consequence for the graph: when $\Delta < 0$ the parabola lies entirely above the axis (for $a > 0$) or entirely below it (for $a < 0$), so the function has a constant sign.

Where you see it

A parabola is not a school ornament but a shape nature and engineering pick on their own:

• projectile motion — a stone thrown at an angle travels along a parabola (air resistance aside);
• satellite dishes and reflectors — a parabolic dish gathers parallel rays into a single point, the focus;
• optimisation — questions like "the largest area for a given perimeter" lead straight to the vertex.

Exercises

The session covers the four skills of this lesson: the value at a point, the coordinates of the vertex, and rewriting the general form into the canonical and the factored one. For the vertex type two numbers separated by a comma — first $p$, then $q$. For a change of form type the finished formula, e.g. (x-2)^2-1.

Common mistakes

• Losing the minus in $p$ — the formula is $p = -\frac{b}{2a}$, so $b = -4$ gives $+2$.
• Confusing $p$ with $q$ — $p$ lives on the horizontal axis, $q$ is a value of the function.
• The sign of $a$ and the arms — the arms point up when $a > 0$, not "always up".
• A factored form at $\Delta < 0$ — there is none, because there are no zeros.
• Taking $c$ for a zero — $c$ is the value at zero, the crossing with the $y$-axis.
• Calling the range all real numbers — a parabola always stops at $q$.