Quadratic Functions

students, have you ever watched a basketball shot arc toward the hoop 🏀, seen a fountain stream rise and fall, or noticed a bridge cable forming a curve? These shapes are often modeled by quadratic functions. In this lesson, you will learn how to recognize, describe, and use quadratic functions in IB Mathematics Analysis and Approaches SL. By the end, you should be able to explain key terminology, interpret graphs, and apply quadratics to real situations.

What is a quadratic function?

A quadratic function is a function that can be written in the form $f(x)=ax^2+bx+c$, where $a\neq 0$. The highest power of $x$ is $2$, so the graph is a parabola. This is one of the most important families of functions in mathematics because it connects algebra, graphing, and problem solving.

The coefficients $a$, $b$, and $c$ each affect the graph in different ways. The value of $a$ controls whether the parabola opens upward or downward. If $a>0$, the graph opens upward and has a minimum point. If $a<0$, the graph opens downward and has a maximum point. The coefficient $c$ gives the $y$-intercept, because when $x=0$, $f(0)=c$.

The graph of a quadratic is symmetric about a vertical line called the axis of symmetry. For $f(x)=ax^2+bx+c$, the axis of symmetry is $x=-\frac{b}{2a}$. This line passes through the vertex, which is the turning point of the parabola. The vertex is important because it gives the maximum or minimum value of the function.

For example, consider $f(x)=x^2-4x+3$. Here, $a=1$, $b=-4$, and $c=3$. Since $a>0$, the parabola opens upward. The axis of symmetry is $x=-\frac{-4}{2(1)}=2$. Substituting $x=2$ gives $f(2)=2^2-4(2)+3=-1$, so the vertex is $(2,-1)$.

Forms of a quadratic function

Quadratic functions can appear in several forms, and each form is useful for a different purpose. One common form is the standard form $f(x)=ax^2+bx+c$. This form is useful for identifying the coefficients and the $y$-intercept. It is also used to apply formulas such as the quadratic formula.

Another important form is the vertex form $f(x)=a(x-h)^2+k$. In this form, the vertex is $(h,k)$ and the axis of symmetry is $x=h$. This form is especially helpful when sketching graphs or describing transformations. For example, $f(x)=2(x-3)^2-5$ has vertex $(3,-5)$, opens upward, and is vertically stretched by a factor of $2$.

A third useful form is factorized form, such as $f(x)=a(x-r_1)(x-r_2)$. This form shows the roots, also called zeros or $x$-intercepts, directly. If $f(x)=0$, then $x=r_1$ or $x=r_2$. For example, $f(x)=(x-1)(x-5)$ has roots at $x=1$ and $x=5$.

These forms are connected. A quadratic may be rewritten from one form to another depending on the information you need. In IB Mathematics, this flexibility is important because many questions ask you to interpret, transform, or solve using the most suitable representation.

Key graph features and terminology

When studying a quadratic function, students, you should be able to identify several key features:

The vertex: the maximum or minimum point.
The axis of symmetry: the vertical line through the vertex.
The intercepts: where the graph crosses the axes.
The direction of opening: up or down.
The range: the set of possible output values.

The domain of every quadratic function is all real numbers, written as $x\in\mathbb{R}$. That is because you can substitute any real number into a polynomial. The range depends on the vertex. For example, if $f(x)=(x-2)^2-7$, the minimum value is $-7$, so the range is $f(x)\ge -7$.

To find intercepts, use substitution. The $y$-intercept occurs when $x=0$. For $f(x)=x^2-6x+5$, the $y$-intercept is $f(0)=5$, so the graph crosses the $y$-axis at $(0,5)$. To find $x$-intercepts, set $f(x)=0$ and solve. For the same function, $x^2-6x+5=0$ factors to $(x-1)(x-5)=0$, giving roots $x=1$ and $x=5$.

A quadratic can have two real roots, one repeated root, or no real roots. This depends on whether the parabola crosses, touches, or misses the $x$-axis. In terms of the discriminant $b^2-4ac$, if $b^2-4ac>0$ there are two distinct real roots, if $b^2-4ac=0$ there is one repeated real root, and if $b^2-4ac<0$ there are no real roots.

Solving quadratic equations

Quadratic equations are equations of the form $ax^2+bx+c=0$. IB AA SL expects you to solve them using several methods. The method you choose depends on the equation.

If the quadratic factors neatly, factorization is often the fastest method. For example, solve $x^2-9x+20=0$. Factor to get $(x-4)(x-5)=0$, so $x=4$ or $x=5$.

If factorization is not easy, you can use the quadratic formula: $x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}.$ This formula works for every quadratic equation with $a\neq 0$. For example, solve $2x^2+3x-2=0$. Here, $a=2$, $b=3$, and $c=-2$. Substituting gives $x=\frac{-3\pm\sqrt{3^2-4(2)(-2)}}{2(2)}=\frac{-3\pm\sqrt{25}}{4}=\frac{-3\pm 5}{4}.$ So the solutions are $x=\frac{1}{2}$ and $x=-2$.

Completing the square is another important technique. It is especially useful for converting a quadratic from standard form into vertex form. For example, rewrite $x^2+6x+1$ by completing the square:

$$x^2+6x+1=(x^2+6x+9)-9+1=(x+3)^2-8.$$

This shows the vertex directly as $(-3,-8)$.

These solution methods are connected to the graph. The roots are the $x$-coordinates where the graph crosses the $x$-axis, and the vertex form shows the turning point. This is a good example of how function language and representation work together in the course.

Transformations of quadratic graphs

Quadratic functions are often studied through transformations of the basic graph $y=x^2$. The function $y=a(x-h)^2+k$ tells you exactly how the graph changes.

If $|a|>1$, the parabola is vertically stretched and becomes narrower. If $0<|a|<1$, it is vertically compressed and becomes wider. If $a<0$, the graph is reflected in the $x$-axis.

The value of $h$ shifts the graph horizontally. If $h>0$, the graph moves right; if $h<0$, it moves left. The value of $k$ shifts the graph vertically. If $k>0$, the graph moves up; if $k<0$, it moves down.

For example, compare $y=x^2$ with $y=-3(x+2)^2+4$. The second graph is reflected in the $x$-axis, stretched by a factor of $3$, shifted left $2$ units, and shifted up $4$ units. Its vertex is $(-2,4)$.

Understanding transformations helps you sketch graphs quickly and accurately. It also helps when interpreting real-world situations where a model is changed by a constant shift or scaling factor.

Quadratic models in real life

Quadratic models appear in many contexts. A ball thrown upward follows a path that is often modeled by a quadratic function because gravity causes a constant downward acceleration. A company may use a quadratic profit model to describe how revenue changes with the number of products sold. Engineers may use quadratics to design curved structures.

Suppose the height of a ball after $t$ seconds is given by $h(t)=-4.9t^2+19.6t+1.5$. The coefficient $-4.9$ indicates downward curvature, matching the effect of gravity in metres per second squared. The initial height is $1.5$ metres because $h(0)=1.5$. The maximum height occurs at the vertex, which can be found using $t=-\frac{b}{2a}=-\frac{19.6}{2(-4.9)}=2$. Substituting gives $h(2)=21.1$, so the ball reaches a maximum height of $21.1$ metres.

This type of interpretation is essential in IB Mathematics because you are often asked not only to calculate, but also to explain what the answer means in context. A numerical result should be linked to the situation it describes.

Conclusion

Quadratic functions are a central part of the topic of Functions because they show how algebraic expressions, graphs, and real situations are connected. students, by understanding forms such as $ax^2+bx+c$, $a(x-h)^2+k$, and $a(x-r_1)(x-r_2)$, you can move between representations with confidence. You should also be able to identify the vertex, axis of symmetry, intercepts, domain, range, and roots. These ideas are not isolated facts; they form a toolkit for solving problems, interpreting graphs, and modeling real events. Mastering quadratics will also support your later work with transformations, composite functions, and inverses.

Study Notes

A quadratic function has the form $f(x)=ax^2+bx+c$ with $a\neq 0$.
Its graph is a parabola.
If $a>0$, the graph opens upward and has a minimum; if $a<0$, it opens downward and has a maximum.
The axis of symmetry is $x=-\frac{b}{2a}$.
The vertex is the turning point and can be seen directly in $f(x)=a(x-h)^2+k$.
The $y$-intercept is $c$, because $f(0)=c$.
The roots are found by solving $f(x)=0$.
A quadratic may be solved by factorization, completing the square, or the quadratic formula $$x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}.$$
The discriminant $b^2-4ac$ tells how many real roots there are.
Quadratic models are used for motion, finance, and design in real life 📈