Quadratic Functions and Equations

A quadratic function is a polynomial function. Since the highest degree of a quadratic equation is 2, the root values that result in solving the equations are also 2. Any solution of a quadratic equation gives a parabola when plotted in a graph. Solving such quadratic equations is employed in various mathematical and engineering platforms, such as the trajectory of a rocket, the path of a cricket ball while bowling and after batting, etc. A quadratic equation has different methods for finding a solution as well as has different forms.

Quadratic Functions and Equations

• Any algebraic equation with the maximum exponential on the variable as 2 is a quadratic function. 

• It contains at least one variable with the power as 2.

• On solving a quadratic equation, 2 values are obtained, which give the root values of the equation.

• It infers when these values of x are substituted in the equation. The equation becomes zero.

• The standard format of a quadratic equation is given by f(x) = ax² + bx + c. 

Different Forms of a Quadratic Equation

The standard form of a quadratic equation is ax² + bx + c. However, there are two forms of quadratic equations. Though the quadratic equations exist in different forms, they are always inter-convertible. The various forms of a quadratic equation are given below.

1. Standard form

• The standard form of a quadratic equation as mentioned above is f(x) = ax² + bx + c.

• In such equations, the value of ‘a’, the coefficient of x², is never zero.

• If the coefficient of x² is zero, then that means that there is no x²  in the equation at all.

• The coefficient of x and the constant can be zero.

2. Vertex form

• The vertex form of a quadratic equation is in the format f(x) = a(x-h)² + k.

• The values of ‘h’ and ‘k’ are the coordinate points of the vertex of the quadratic equation.

• A vertex is a point in the quadratic graph where the parabolic curve meets the parabola’s axis of symmetry.

• It can either be the maximum or the minimum value of the parabola. The value is maximum when the curve opens downwards. The value is minimum when the curve faces upwards.

3. Intercept form 

• The intercept or factored form of a quadratic equation is in the format f(x) = a(x – p)(x – q).

• In such an equation, the values of ‘p’ and ‘q’ give the values of ‘x’.

• It means that p and q are the values of x-coordinates in the parabola.

• The coordinates are written as (p,0) and (q,0).

Graph of a Quadratic Function

• The parabolic curve opens upwards when the value of ‘a’ is greater than zero, which is a positive value.

• The curve opens downwards when the value of ‘a’ is less than zero. That is, the ‘a’ value is negative.

• When the parabola is facing upwards, like a ‘U-shape’, the value of the vertex will be the minimum value of the equation.

• When the curve faces downwards, like a ‘dome’, the value of the vertex will be the maximum value of the function.

• The axis of symmetry divides the parabolic curve into two halves.

• The graph is plotted when the root values of the quadratic equation are real. Thus, the range of a quadratic graph lies within the Real Numbers (R), -∞ to +∞.

• The root values provide the x-intercept of the parabola that makes a point on the x-axis.

• The y-intercept is the point where the parabola cuts the y-axis. 

• The quadratic equation can have only one such y-intercept.

Quadratic Function Formula

• The quadratic equations are solved to obtain the root values by three different methods.

• Among these, using the quadratic formula is one.

• Any quadratic equation in the format ax² + bx + c can be solved using the quadratic formula.

               x = [-b  ±(b² – 4ac)2a]

In this formula,

‘a’ is the coefficient of x² in the equation.

‘b’ is the coefficient of x in the equation.

‘c’ is the constant in the equation.

Discriminant (D) of the quadratic equation is given by (b² – 4ac) from the quadratic formula. The discriminant dictates the nature of the root values.

• The root values are real and unique if D is greater than zero.

• The root values are imaginary if D is less than zero.

• The root values are equal, and one if D equals zero.

Conclusion 

A quadratic function has two root values since its highest degree is 2. Any quadratic equation can be solved by three different methods to find the roots. These values give the x-intercept for plotting the graph, which results in a parabolic curve. The roots lie on the x-axis as the y-intercept is zero. The coefficient of x² in a quadratic equation can never be zero, and it determines the nature of the parabola.