Quadratic Functions

Welcome, students 🌟 In this lesson, you will explore one of the most important families of functions in mathematics: quadratic functions. Quadratics show up in sports, engineering, business, and physics, so understanding them is useful both in exams and in real life. By the end of this lesson, you should be able to recognize quadratic functions, describe their key features, apply transformations, solve quadratic equations, and connect quadratics to the bigger study of functions.

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 it is called a degree-2 polynomial function. The graph of a quadratic function is a parabola.

The values of $a$, $b$, and $c$ control the shape and position of the graph. If $a>0$, the parabola opens upward like a smile 😊. If $a<0$, it opens downward like a frown ☹️. The number $c$ is the $y$-intercept, because $f(0)=c$.

Quadratic functions are part of the broader family of polynomial functions. In the study of functions, this matters because it helps you compare quadratics with other models such as linear, exponential, rational, and logarithmic functions. A quadratic is especially important because it often appears when modeling area, motion, and optimization.

Forms of a quadratic function

Quadratic functions can be written in different forms, and each form is useful for a different reason.

The standard form is $f(x)=ax^2+bx+c$. This form is useful for seeing the coefficients and for expanding expressions.

The vertex form is $f(x)=a(x-h)^2+k$. This form makes the vertex easy to identify. The vertex is $(h,k)$, and the axis of symmetry is $x=h$. This form is very helpful when studying transformations.

The factored form is $f(x)=a(x-r_1)(x-r_2)$, where $r_1$ and $r_2$ are the roots. This form is useful when solving equations and finding the $x$-intercepts.

For example, consider $f(x)=x^2-6x+5$. This is in standard form. It can also be written as $f(x)=(x-1)(x-5)$, which shows the roots $x=1$ and $x=5$. Completing the square gives $f(x)=(x-3)^2-4$, which reveals the vertex $(3,-4)$.

Notice how one function can be represented in several ways. This is a major idea in IB Mathematics: Analysis and Approaches HL: different representations give different kinds of information.

Key features of the graph

A quadratic graph has several features that you should be able to identify and interpret.

The vertex is the turning point of the parabola. If $a>0$, the vertex is the minimum point. If $a<0$, the vertex is the maximum point. The vertex tells you the extreme value of the function.

The axis of symmetry is the vertical line $x=h$ for a parabola in vertex form. It splits the graph into two mirror-image halves. This symmetry is why parabolas are so visually neat ✨.

The $y$-intercept occurs when $x=0$, so it is the point $(0,c)$ in standard form.

The $x$-intercepts are the solutions to $f(x)=0$. These are also called the roots or zeros of the function. A quadratic may have two real roots, one repeated real root, or no real roots. This depends on the discriminant $\Delta=b^2-4ac$.

If $\Delta>0$, there are two distinct real roots. If $\Delta=0$, there is one repeated real root. If $\Delta<0$, there are no real roots.

For example, for $f(x)=x^2+4x+5$, we have $\Delta=4^2-4(1)(5)=16-20=-4$. Since $\Delta<0$, the graph has no $x$-intercepts.

These features are important because they help you sketch graphs and solve equations without guessing.

Transformations of quadratic functions

Quadratic functions are often studied as transformations of the basic graph $y=x^2$.

If you start with $y=x^2$ and replace it with $y=a(x-h)^2+k$, the graph is shifted horizontally by $h$, shifted vertically by $k$, and stretched or compressed by factor $|a|$. If $a<0$, the graph is also reflected in the $x$-axis.

For example, $y=2(x-1)^2-3$ has the same basic parabola as $y=x^2$, but it is moved right 1, down 3, and stretched vertically by a factor of 2. Its vertex is $(1,-3)$.

A real-world example comes from the path of a ball thrown into the air. If $h(t)$ gives the height of the ball after time $t$, then $h(t)$ is often modeled by a quadratic function. The vertex may represent the maximum height, and the zeros may represent when the ball leaves the ground and when it hits the ground again.

This is one reason quadratics are so useful: they describe motion under constant acceleration, like gravity.

Solving quadratic equations and inequalities

Quadratic functions are closely connected to equations and inequalities.

To solve $ax^2+bx+c=0$, you can factor, complete the square, use the quadratic formula, or read the roots from the graph if possible. The quadratic formula is

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

This formula works for every quadratic equation with $a\neq 0$.

For example, solve $x^2-5x+6=0$. Factoring gives $(x-2)(x-3)=0$, so $x=2$ or $x=3$.

Quadratic inequalities ask where a quadratic is positive or negative. For instance, to solve $x^2-5x+6>0$, first find the roots $x=2$ and $x=3$. Since the parabola opens upward, the expression is positive outside the roots. So the solution is $x<2$ or $x>3$.

Graphing is especially useful here. The sign of $f(x)$ depends on whether the graph lies above or below the $x$-axis.

In IB questions, you may need to combine algebra and reasoning. For example, you may be asked to solve $f(x)\geq 0$ and explain your answer using both a graph and an inequality statement. This shows understanding of function language, not just calculation.

Connections to inverse and composite functions

Quadratic functions also connect to inverse and composite functions, which are part of the wider functions topic.

A quadratic function does not usually have an inverse that is itself a function on all real numbers, because most quadratic graphs fail the horizontal line test. For example, the function $f(x)=x^2$ gives the same output for $x=2$ and $x=-2$.

However, if the domain is restricted, an inverse function can be defined. For example, if $f(x)=x^2$ with domain $x\geq 0$, then the inverse is $f^{-1}(x)=\sqrt{x}$.

Composite functions can also involve quadratics. If $f(x)=x^2-1$ and $g(x)=2x+3$, then

$$f(g(x))=(2x+3)^2-1$$

This simplifies to a new function. Composite functions help you understand how one function acts after another, which is useful in modelling and in algebraic manipulation.

These ideas show that quadratics are not isolated facts. They connect to the full function toolkit: domain, range, graphing, transformation, inverse functions, and composition.

Summary and exam-style thinking

When solving problems with quadratic functions, always think about the representation you are given. If the equation is in standard form, you may use the discriminant or the quadratic formula. If it is in vertex form, you can identify the vertex quickly. If it is in factored form, the roots are immediate.

IB Mathematics: Analysis and Approaches HL often expects you to explain reasoning clearly. For example, if a question asks for the maximum value of $f(x)=-2(x-4)^2+7$, you should recognize from vertex form that the vertex is $(4,7)$ and that the maximum value is $7$ because the parabola opens downward.

You should also be comfortable moving between algebra and graph. A graph is not just a picture; it is a representation of a function’s behavior. The vertex, intercepts, and symmetry all tell a mathematical story.

Conclusion

Quadratic functions are a core part of the Functions topic because they combine algebraic structure, graphical behavior, and real-world modeling. They help you understand how functions are represented, transformed, solved, and interpreted. students, if you can move confidently between standard form, vertex form, graph, and inequalities, you have a strong foundation for more advanced function topics in IB Mathematics: Analysis and Approaches HL 🚀

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 parabola opens upward; if $a<0$, it opens downward.
The vertex form is $f(x)=a(x-h)^2+k$, and the vertex is $(h,k)$.
The axis of symmetry is $x=h$.
The standard form gives the $y$-intercept $(0,c)$.
The roots are the solutions of $f(x)=0$.
The discriminant is $\Delta=b^2-4ac$.
If $\Delta>0$, there are two real roots; if $\Delta=0$, one repeated real root; if $\Delta<0$, no real roots.
The quadratic formula is $x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$.
Quadratic inequalities are often solved by finding roots and testing intervals.
Quadratic functions connect to transformations, inverse functions, composite functions, and real-world modeling.