In everyday life, one comes across various types of motions of objects, such as the trajectory of a basketball or a volleyball, etc. Such trajectories are in the shape of a ‘Parabola’, a curve that can be modeled using a Quadratic Equation of single variable, and such equations of single variables are Quadratic Functions.
Quadratic equations are applied to a variety of practical problems, such as monitoring the path of a projectile, predicting a financial stock model, statistical mechanics, and so on. The part of the equation \(ax^2\) is known as the quadratic part, \(bx\) as the linear part, and \(c\) as the constant part of the function.
Graphing Quadratic Functions - the Parabola
The graph of every Quadratic function is called a Parabola.
A parabola is a set of points equidistant from a point and a line.
where the point is called the Focus of a parabola and the line is known as the directrix. Another important point on the parabola is called the Vertex of the parabola. It is the point where the axis of symmetry of a parabola meets the parabola.
Here the axis of symmetry is an imaginary line and the function replicates itself on either side of the line. The graph of a parabola is like a mirror image of a curve, below is a diagram to illustrate this,
Fig. 1. The graph of a quadratic equation.
Here is what the graph of a quadratic function looks like, which is the quadratic function. It can be seen that the curve on the right side of the blue line and the other side of that line, are exactly the same. In mathematical terms, we say that the graph is symmetrical along that blue line. That is why that line is called the axis of symmetry. It is important to note that the axis of symmetry is an imaginary axis, it is not a part of the graph plotted.
Fig. 2. Graph of a Parabola.
It can be seen that the axis of symmetry is parallel to the y-axis and so we say that the parabola is symmetrical to the y-axis. And the point where the parabola meets the axis of symmetry is known as the Vertex of the parabola. It is also the minima of the function. In other words, a vertex is a point where the value of the quadratic function is minimum, hence the name, minima. In the above diagram, point A is the vertex of the parabola.
And for the parabola \(y=ax^2+bx+c\), the axis of symmetry turns out to be \(x=-\dfrac{b}{2a}\) which is symmetrical to y-axis.
There is another crucial point on the parabola, which is the y-intercept of the parabola. It is the point where the parabola meets the y-axis, i.e. where it intercepts the y-axis. Hence, the word, y-intercept. In the above diagram, point \(C\) is the y-intercept of the parabola. To find out the coordinates of \(C\), all we need to do is calculate y at \(x=0\). We get,
$$y=a(0)^2+b(0)+c$$
which gives \(y=c\). Hence, the coordinates of \(C\) are \((0,c)\).
Quadratic Function Equations
We can write quadratic function equations in 3 different forms. Let's look at them in more detail
There are three commonly used Forms of Quadratic Functions.
- Standard or General Form: \(y=ax^2+bx+c\)
- Factored or Intercept Form: \(y=(bx+c)(dx+e)\)
- Vertex Form: \(y=a(x-h)^2+k\)
Each of these forms can be used to determine different information about the path of a projectile. Understanding the benefits of each form of a quadratic function will be useful for analyzing different situations that come your way.
As the name suggests, the general form is what most quadratic functions are in. The intercept form is useful to easily read off the x and y intercepts of the given curve. The vertex form is especially used when the vertex of the curve has to be read off and determine the related properties.
Standard Form of a Quadratic Function
Quadratic equations in one variable are equations that can be expressed in the form
$$f(x)=ax^2+bx+c$$
This is the shape of a parabola, as seen in the image below.
Fig. 3. Graph of a standard parabola.
Essentially, these are the equations that have a degree more than Linear equations. Linear equations have a degree of one and quadratic equations have a degree of \(2\). Here \(a\), \(b\), and \(c\) are constants where \(a\neq 0\). If \(a=0\), then we would only have \(f(x)=bx+c\), which is a linear equation.
So the condition to form a quadratic equation should be that the coefficient of \(x^2\) should be non-zero. The other constants \(b\) and \(c\) can be zero as they won’t affect the degree of the equations.
Vertex Form of a Quadratic Function
The general form of a quadratic \(y=ax^2+bx+c\) may not be the most convenient form to work with, and so we have the Vertex form of a Quadratic Equation. As the name suggests, it is a form based upon the vertex of the parabola formed by the quadratic equation. The vertex is the most important point of a parabola, using which, we can construct the parabola.
The Vertex Form of a Quadratic Equation is given as follows:
$$y=a(x-h)^2+k$$
where the vertex of the parabola lies at the point \((h,k)\). This form is especially useful when we are given the coordinates of the vertex and are asked to find the equation of the parabola.
Factored Form of a Quadratic Equation
The Factored Form of a Quadratic Equation is a form where the quadratic is factored into its linear factors. Just as we had the vertex form to identify the vertex of a parabola formed by the quadratic equation, the factored form is used to identify the intercepts of the parabola formed.
The Factored or Intercept Form of a Quadratic Equation is given as follows:
$$y=a(bx+c)(dx+e)$$
where the two x-intercepts are given by \(x=-\dfrac{c}{b}\) and \(x=-\dfrac{e}{d}\). This can be easily verified by setting \(y=0\) and finding the roots of the quadratic equation. Alternatively, one can use the given x-intercepts and a point on the parabola to figure out the quadratic equation.
Examples of Quadratic Functions
Let's practice identifying quadratic functions!
Which of the following are quadratic functions?
(i) \(f(x)=qx^{3/2}+px\) (ii) \(g(y)=5y^2+2y+9\) (iii) \(h(\theta)=\theta^3+\theta^2\)
Solution:
Recognize the highest degree of each of the functions, if the highest degree is 2 then only it is a quadratic function.
(i) \(f(x)=qx^{3/2}+px\)
It can be seen that the highest degree of this function is \(\dfrac{3}{2}\) and it is trivial that \(\dfrac{3}{2}\neq 2\) and so it is NOT a quadratic function.
(ii) \(g(y)=5y^2+2y+9\)
It is clear that the highest degree of this function is \(2\) and hence it is a Quadratic function.
(iii) \(h(\theta)=\theta^3+\theta^2\)
One can see that the second term has a degree \(2\) but only the highest degree should be taken into consideration which is \(3\), and so it is NOT a quadratic function.
Solving Quadratic Equations
Quadratic functions are a generalized form of quadratic equations. When \(f(x)=d\) for the quadratic function defined earlier, for some real constant \(d\) then the equation formed is known as a Quadratic equation. In general form, a quadratic equation has the form,
$$px^2+qx+r=0$$
where \(p\neq 0\) and \(p, q, r \in \mathbb R\) where \(\mathbb R\) represents the set of real numbers. The solution of a quadratic equation is the value of \(x\) for which the equation is satisfied. In other words, the solution of a quadratic function is the value of \(x\) for which \(f(x)=0\).
We already know that a linear equation has a unique solution, in the case of quadratic equations, there are always two solutions. The solutions need not be unique, they can be the same and solutions may even be complex. However, we will be looking at real solutions and not complex one.
The solutions are also called the zeros of a function. They should not be confused as they are the same thing. To find the zeros, we can simply solve the quadratic using the quadratic formula for zeros, and we get
$$x=\dfrac{-q\pm \sqrt{q^2-4pr}}{2p}$$
For practice on how to solve quadratic equations, see our article on Solving quadratic equations and Graph and solve quadratic equations.
The Inverse of a Quadratic Function
Given that a function is Bijective (Injective and Surjective), the inverse exists. For a quadratic function, which is bijective, the inverse of it can be easily calculated. Every inverse is related to the function as follows,
$$f^{-1}(f(x))=x$$
To find the inverse of \(f(x)=ax^2+bx+c\), we first equate the RHS to y,
$$y=ax^2+bx+c$$
The aim is to solve the above quadratic equation in terms of \(x\), i.e., solve for \(x\) and express \(x\) in terms of \(y\). The above equation can be rearranged to get,
$$ax^2-bx+(c-y)=0$$
which is quadratic in \(x\), and we can find its roots using the quadratic formula, which gives us,
$$x=\dfrac{-b\pm \sqrt{b^2-4ac+4y}}{2a}$$
which is the inverse of \(y\),
$$f^{-1}=\dfrac{-b\pm \sqrt{b^2-4ac+4ay}}{2a}$$
Now replacing the variable \(y\) with \(x\), we get the inverse in \(x\)
$$f^{-1}=\dfrac{-b \pm \sqrt{b^2-4ac+4ax}}{2a}$$
where \(b^2+4ax > 4ac\) for real values of the function.
Quadratic functions - Key takeaways
- A quadratic function is a function whose highest power is \(2\). That is the highest degree of the equation is \(2\).
- The graph of a quadratic function is called a parabola, with a parent equation of \(f(x)=x^2\).
- The solutions (zeros or roots) of a quadratic equation can be calculated using the quadratic formula or factoring the equation into its linear factors.
- Each quadratic equation has two zeros (they need not be unique). They can be real or imaginary.
- The graph of a quadratic function is a parabola that can have its axis of symmetry on the y-axis or x-axis.
- A parabola is defined as the set of points equidistant from a point and a line.
- One can find the axis of symmetry and the coordinates of the vertex by setting the \(y=0\) and \(x=0\) respectively.
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