Quadratic Functions and Equations in One Variable

Quadratic Functions and Equations
  A quadratic expression in one variable is an algebraic expression that has the highest power variable is two.  
  • The basic form of a quadratic expression is \(ax^2 + bx + c\), which is \(a, b\, \text{and} \,c\)  is a constant and \(a ≠ 0\), \(x\) is a variable.
  • \(a\) is the coefficient of \(x^2\)\(b\) is the coefficient of \(x\) and \(c\) is a constant 
  Besides \(x\), other letters can be used to represents variables   

Relationship between a quadratic function and many-to-one relation

  Quadratic function, \(f(x)= ax^2+bx+c \)  
  • All quadratic functions have the same image for two different images 
  • Many-to-one relation
  • Have two shapes of graph 
  Shapes of the graph, \(f(x)= ax^2+bx+c , a \neq0\)  
  •  For the graph \(a<0\)\((x_1,y_1)\) is known as maximum point 
  • For the graph \(a>0\)\((x_2, y_2)\) is known as minimum point
  The curved shape of the graph of a quadratic function is called a parabola   
Axis of symmetry of the graph of a quadratic function 
  Definition: A straight line that is parallel to the \(y-\)axis and divides the graph into two parts of the same size and shape   
  • The axis of symmetry will pass through the maximum and minimum point of the graph of the function as shown in the diagram below 

Equation axis of symmetry , \(x= - \dfrac{b}{2a}\)

Effects of changing the values of \(a,b, \) and \(c\) on graphs of quadratic functions, \(f(x)= ax^2 +bx +c\)
  • The value of \(a\) determines the shape of the graph 
  • The value of \(b\) determines the position of the axis of symmetry 
  • The value of \(c\) determines the position of the \(y-\)intercept
Forming a quadratic equation based on a situation 
  • A quadratic function is written in the form of \(f(x)= ax^2 +bx +c \) while a quadratic equation is written in the general form \(ax^2 +bx +c = 0\)
Roots of a quadratic equation 
  • The root of a quadratic equation \(ax^2 +bx +c = 0\) are the values of the variables, \(x\) which satisfy the equation. 

Relationship between the roots of a quadratic equation and the positions of the roots. 

The roots of equation \(ax^2 +bx +c = 0\) are the points of intersection of the graph of the quadratic functions \(f(x)= ax^2 +bx +c \) and the \(x-\)axis which are also knowns as the \(x-\)intercepts. 
Determine the roots of a quadratic equation by using factorisation method 
  A quadratic equation needs to be written in the form of \(ax^2 +bx +c = 0\) before we carry out factorisation   
  Determine the roots of this quadratic equations by using factorisation method \(x^2 - 5x + 6 = 0\).  


\(\begin{aligned} &\space x^2 - 5x +6 = 0 \\\\& (x-3)(x-2) = 0 \\\\& x =3 \space \text{or} \space x = 2. \end{aligned}\)

Determine the roots of a quadratic equation by using graphical method 
  • The roots of a quadratic equation \(ax^2 +bx +c = 0\) can be obtained by using a graphical method by reading the values of \(x\) which are the points of intersections of the graph