Quadratic Function

A quadratic function is a polynomial function! Read this article to know more about quadratic functions and their types.

Quadratic Function

The meaning “quad” means “square”. Hence, a polynomial of degree \(2\) is called a quadratic function.

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Step by step guide to the quadratic function

The quadratic function is a polynomial function with one or more variables in which the highest power of the variable is two. Since the highest degree sentence in a quadratic function is of the second degree, therefore it is also called the polynomial of degree \(2\). A quadratic function has at least one sentence that is second degree.

Standard form of a quadratic function

The standard form of a quadratic function is \(f(x)= ax^2+ bx+c\), where \(a, b\) and \(c\) are real numbers with \(a≠0\).

Vertex of the quadratic function

The vertex of a quadratic function (which is \(U\)-shaped) is where the function has a maximum value or a minimum value. The axis of symmetry of the quadratic function intersects the function (parabola) at the vertex.

Different forms of a quadratic function

A quadratic function can be in different forms: standard form, vertex form, and intercept form. Here are the general forms of each of them:

  • Standard form: \(f(x)=ax^2+bx+c\), where \(a≠0\).
  • Vertex form: \(f(x)=a(x – h)^2+k\), where \(a≠0\) and \((h, k)\) is the vertex of the parabola representing the quadratic function.
  • Intercept form: \(f(x)=a(x – p)(x – q)\), where \(a≠0\) and \((p, 0)\) and \((q, 0)\) are the \(x\)-intercepts of the parabola representing the quadratic function.

The parabola opens upwards or downwards as per the value of \(a\) different:

  • If \(a>0\), then the parabola opens upward.
  • If \(a<0\), then the parabola opens downward.

We can always convert one form to another. We can easily convert the vertex form or intercept form into standard form by just simplifying the algebraic expressions.

Convert the standard form of a quadratic function to vertex form

A quadratic function \(f(x)=ax^2+bx+c\) can be easily converted into the vertex form \(f(x)=a (x – h)^2+ k \) by using the values \(h=-\frac{b}{2a}\) and \(k= f(-\frac{b}{2a})\). Here is an example.

Example: Convert the quadratic function \(f(x)=2x^2 – 12x+3\) into the vertex form.

  • Step-1: By comparing the given function with \(f(x)=ax^2+bx+c\), we get \(a=2, b= -12\), and \(c = 3\).
  • Step-2: Find \(h\) using the formula: \(h =-\frac{b}{2a}=-\frac{\left(-12\right)}{2\left(2\right)}= 3\).
  • Step-3: Find \(k\) using the formula: \(k = f(-\frac{b}{2a})= f(3)= 2(3)^2- 8(3)+ 3= 18 -24+3=-3\).
  • Step-4: Substitute the values into the vertex form: \(f(x) = 2 (x – 3)^2- 3\).

Convert the standard form of a quadratic function to intercept form

A quadratic function \(f(x)= ax^2+bx+c\) can be easily converted into the vertex form \(f(x)= a (x – p)(x – q)\) by using the values of \(p\) and \(q\) (\(x\)-intercepts) by solving the quadratic equation \(ax^2+bx+c =0\).

Example: Convert the quadratic function \(f(x)=x^2-7x+12\) into the intercept form.

  • Step-1: By comparing the given function with \(f(x)=ax^2+bx+c\), we get \(a=1\).
  • Step-2: Solve the quadratic equation: \(x^2-7x+ 12= 0\)
    By factoring the left side part, we get
    \((x-3) (x-4) = 0\)
    \(x=3, x=4\)
  • Step-3: Substitute the values into the intercept form: \(f(x)=1(x -3)(x – 4)\).

Domain and range of quadratic function

The domain of a quadratic function is the set of all \(x\)-values that define the function, and the range of a quadratic function is the set of all \(y\)-values that the function results in by substituting different \(x\)- values.

Domain of a quadratic function

The quadratic function is a polynomial function defined for all real values of \(x\). So, the domain of a quadratic function is the set of real numbers, that is, \(R\). In interval notation, the domain of any quadratic function is \((-∞, ∞)\).

Range of a quadratic function

The range of the quadratic function depends on the graph’s opening side and vertex. Therefore, to determine the range of the quadratic function, search for the lowest and highest values of \(f(x)\) in the graph of the function.

The range of any quadratic function with vertex \((h, k)\) and the equation \(f(x)= a(x – h)^2+ k\) is:

  • \(y≥k\) (or) \([k, ∞)\) when \(a>0\) (as the parabola opens up when \(a>0\)).
  • \(y≤k\) (or) \((-∞, k]\) when \(a<0\) (as the parabola opens down when \(a<0\)).

Quadratic Function – Example 1:

 Write the quadratic function \(f(x) = (x-12)(x+3)\) in the general form \(ax^2+ bx + c\).

Solution: 

We will just expand \(f(x)=(x-12)(x+3)\) to write it in the general form.

\(f(x)=(x-12)(x+3)\)

\(=x(x+3)-12(x+3)\)

\(= x^2+ 3x -12x-36\)

\(=x^2- 9x – 36\)

Exercises for Quadratic Function

  1. Determine the vertex of the quadratic function \(f\left(x\right)\:=\:2\left(x+4\right)^2\:-\:3\).
  2. Convert the quadratic function \(f(x)=x^2 +6x-5\) into the vertex form.
  3. Find the domain and range of  \(f\left(x\right)=3\left(x-\frac{4}{7}\right)^2+\frac{9}{10}\).
  4. Write the equation \(f(x)=13+x^2−6x\) in standard form.
This image has an empty alt attribute; its file name is answers.png
  1. \(\color{blue}{(-4,-3)}\)
  2. \(\color{blue}{f(x)=(x+3)^2−14}\)
  3. \(\color{blue}{D=\:\left(-\infty \:,\:\infty \:\right), R=\:[\frac{9}{10},\:\infty \:)}\)
  4. \(\color{blue}{f\left(x\right)=\left(x-3\right)^2+4}\)

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