The shape and the basic parameters

A quadratic function is a polynomial function in which the highest power of the input variable is $2$. We usually write it in standard form as

In a purely algebraic question, these quantities have no physical unit. In a modelling question, their units depend on whatever is being modelled. Unless the question restricts the domain, the largest possible domain of a quadratic function is $\mathbb{R}$.

A parabola is the plane curve you get as the graph of a quadratic function. If $a>0$, the parabola opens upwards and has a minimum point. If $a<0$, it opens downwards and has a maximum point. The $y$-intercept is the point where the graph crosses the $y$-axis; for $f(x)=ax^2+bx+c$, this is $(0,c)$, since $f(0)=c$.

The axis of symmetry is a straight line that splits the parabola into two mirror-image halves. In standard form, its equation is

$x=-\frac{b}{2a}$

Think of the axis of symmetry as a line, not a point: it is the vertical line $x=-\frac{b}{2a}$, and the turning point always lies on it.

Three equivalent forms

An equivalent representation is an algebraic form that gives the same output for every allowed input, while making different features easier to spot. Quadratics are a good example. The same function can be written in standard form, factorized form, or vertex form.

Quadratic forms and the graph feature each one highlights best.

The standard form is

$f(x)=ax^2+bx+c$

which quickly shows the $y$-intercept $(0,c)$ and gives the axis of symmetry through $x=-\frac{b}{2a}$.

The factorized form is

This form shows the $x$-intercepts $(p,0)$ and $(q,0)$ straight away, because substituting $x=p$ or $x=q$ makes $f(x)=0$.

The vertex form is

The vertex is the point where the parabola reaches its maximum or minimum value, so this form gives the vertex as $(h,k)$. The axis of symmetry is then $x=h$.

Changing from one form to another

You are expected to move between these forms. Expanding turns factorized or vertex form into standard form. Factorizing turns standard form into factorized form when the quadratic has suitable factors. Completing the square turns standard form into vertex form.

Completing the square is worth practising because it shows where vertex form comes from:

$ax^2+bx+c=a\left(x+\frac{b}{2a}\right)^2+c-\frac{b^2}{4a}$

So the vertex has

$h=-\frac{b}{2a}, \qquad k=c-\frac{b^2}{4a}$

For example,

$2x^2-8x+3=2(x^2-4x)+3=2(x-2)^2-5$

The standard form $2x^2-8x+3$ is useful for reading off the $y$-intercept $(0,3)$, while the vertex form $2(x-2)^2-5$ shows the vertex $(2,-5)$ and the axis $x=2$.

Quadratics as models

Quadratics often appear when a quantity changes with a squared term, especially in kinematics and projectile motion. For instance, a vertical displacement model might be written as

The vertex would represent the greatest height, and the roots would represent times when the object is at ground level, if those times are in the physical domain.

This is also where the link to transformations matters. In $f(x)=a(x-h)^2+k$, changing $h$ and $k$ shifts the parabola, while changing $a$ affects its opening direction and width. The parameters are not just letters. They match geometrical features of the graph, and in applications they may represent physical quantities.

A graphing calculator window can hide the important part of a parabola. If you cannot see the vertex or the intercepts, the function has not disappeared; your spatial frame of reference is simply poorly chosen. Adjust the window so the features you are discussing are visible.

The TOK link here is useful: a quadratic model is not the physical event itself. It is a mathematical representation of selected features of the event. That is one reason mathematics can feel different from experimental subjects. Once the model is accepted, the algebraic consequences follow by deduction, while the suitability of the model is judged against context and data.

Quadratic equations, roots and zeros

A quadratic equation is an equation that can be rearranged into the form

A root is a solution of an equation. A zero is an input value that makes a function output $0$. With quadratics, the two terms usually point to the same $x$-values: when you solve $ax^2+bx+c=0$, you are finding the zeros of $f(x)=ax^2+bx+c$.

There are three standard analytic methods.

• Factorization rewrites the quadratic as a product. If $(x-r)(x-s)=0$, then $x=r$ or $x=s$, since a product can be zero only if at least one factor is zero. Here $r$ and $s$ are real constants representing possible roots.
• Completing the square rewrites the quadratic in vertex form. From $a(x-h)^2+k=0$, isolate the squared term and then take square roots when possible.
• The quadratic formula gives the roots of any quadratic equation when real roots exist:

$x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$

The symbol $\pm$ tells you to consider both the plus and minus versions. On a non-calculator paper, check whether the quadratic factorizes before using the formula; IB often gives neat factors when that is the intended method.

Completing the square is more than another solving trick. It links the algebra to the graph. For example, if

$f(x)=3(x-4)^2-12$

then $f(x)=0$ gives $3(x-4)^2=12$, so $(x-4)^2=4$, and hence $x=2$ or $x=6$. In vertex form, the symmetry is clear: the roots sit the same distance from the axis $x=4$.

Quadratic inequalities

A quadratic inequality is an inequality that can be rearranged so that a quadratic expression is compared with zero, such as

$ax^2+bx+c>0, \qquad ax^2+bx+c\le0$

A clean method is to rearrange everything to one side, solve the corresponding quadratic equation, and then use the graph or a sign chart to choose the interval. Don’t solve these by casually taking square roots or dividing by an expression involving $x$; sign errors often start there.

For a parabola opening upwards, the quadratic is positive outside two distinct roots and negative between them. For a parabola opening downwards, the signs reverse. With a strict inequality, such as $>$ or $<$, the roots are not included. With an inclusive inequality, such as $\ge$ or $\le$, the roots are included.

For example, to solve

$x^2-x-20>0$

first solve $x^2-x-20=0$. Factorizing gives $(x-5)(x+4)=0$, so the boundary values are $x=5$ and $x=-4$. Since the parabola opens upwards, the expression is positive outside the roots:

$x<-4 \quad \text{or} \quad x>5$

If the question restricts the answer set, apply that restriction at the end. A solution over $\mathbb{R}$ may need to be cut down if the question asks for natural numbers, positive values, times, lengths, or another contextual domain.

On a calculator paper, a graph can be the quickest route. You may graph $y=ax^2+bx+c$ and identify where it lies above or below the $x$-axis, or graph both sides of the original inequality and compare which graph is higher. Still write the mathematical conclusion using inequalities or interval notation; calculator notation is not a substitute for a solution.

What the discriminant tells you

The discriminant is the expression under the square root in the quadratic formula. We write it as

In pure algebra, it has no physical unit. It tells you which real-root case you have before you solve the equation.

The sign of $\Delta$ gives the nature of the roots:

• If $\Delta>0$, the equation has two distinct real roots. Graphically, the parabola cuts the $x$-axis twice.
• If $\Delta=0$, the equation has two equal real roots. Graphically, the vertex lies on the $x$-axis, so the parabola just touches it.
• If $\Delta<0$, the equation has no real roots. Graphically, the parabola does not meet the $x$-axis.

Here the graph and the algebra are saying the same thing. In the quadratic formula, the square root is positive, zero, or not real. On the graph, that shows up as two crossings, one touching point, or no crossing.

Parameter questions

A parameter is a constant in a family of equations whose value changes the particular member of the family. Discriminant questions often ask for parameter values that give different root cases.

For example, consider

Start by checking that the equation is actually quadratic: we need $3k\ne0$, so $k\ne0$. The coefficients are $a=3k$, $b=2$ and $c=k$, giving

$\Delta=2^2-4(3k)(k)=4-12k^2$

For two distinct real roots,

$4-12k^2>0$

so $k^2<\frac{1}{3}$. Therefore

$-\frac{1}{\sqrt{3}}<k<\frac{1}{\sqrt{3}}, \qquad k\ne0$

For two equal real roots,

$4-12k^2=0$

so

$k=\pm\frac{1}{\sqrt{3}}$

For no real roots,

$4-12k^2<0$

so

$k<-\frac{1}{\sqrt{3}} \quad \text{or} \quad k>\frac{1}{\sqrt{3}}$

That first check is small, but it matters: if the coefficient of $x^2$ can become zero, the equation may stop being quadratic. Don’t make the discriminant do a job it wasn’t meant to do.

Links and historical note

In physics, the same discriminant idea helps decide whether a projectile reaches a height, whether an energy equation gives possible positions, or whether a simple harmonic motion equation gives real times. In chemistry, equilibrium equations can lead to quadratics, and the real, physically meaningful root has to be chosen from the algebraic possibilities.

Quadratic methods are much older than modern symbolic notation. One Babylonian multiplication identity is

Ancient Indian sources such as the Sulba Sutras and the Bakhshali Manuscript also contain methods equivalent to solving quadratic equations. The notation has changed enormously; the underlying relationships have been studied for a very long time.