Graphically, a parabola represents a quadratic function f(x) = ax2 + bx + c. The method followed to graph a quadratic function makes use of the characteristics of the graph, like the vertex, axis of symmetry and intercepts etc. The sign of a, the leading coefficient determines whether the parabola opens up or down and whether the vertex is the lowest or the highest point in the graph of the quadratic equation. The graphical method is not always useful in finding the exact solution of quadratic equations. But, it can nevertheless be used in estimating the roots and also display clearly the maximum or minimum function value. The quadratic graphs are also useful in showing the regions representing quadratic inequalities.

The steps followed in graphing a quadratic function are as follows:
  1. Find the vertex of the equation, either using the formula or writing the equation in vertex form.
  2. Note down the axis of symmetry. If the vertex is found as (h,k), then the axis of symmetry is x = h.
  3. Find the x intercepts, if they can be easily found as rational numbers by factoring.
  4. Make a table choosing few x values symmetrically with respect to x = h. Find the y values for the x values chosen, using the function equation. The method of choosing symmetric x values is explained below with the help of a number line. Also, include the y intercepts in the table, if it is close enough to graph along with the other points found.
  5. Plot the vertex. Draw the axis of symmetry. Plot the points found from the table made. Join the points to get a smooth curve, using free hand.

            How to graph a Quadratic Function

Suppose the x coordinate of the vertex is determined as h = 1. We can include pairs of points equidistant and lying on opposite sides of 1 on number line. Like, 0 and 2 make one such pair, which are equidistant from x = 1. You may find the y values corresponding to these x values are equal. We may include 2 or 3 such pairs in the table, which will yield points symmetric to the axis, joining which we get a clear parabolic shape for the graph done,

The vertex form of a Quadratic function is y = a(x - h)2 + k where, the vertex and axis of the graph are given by (h, k) and x = h. Writing a quadratic function in vertex form helps us to find the vertex and the axis of the graph direct. We can then form the table required and graph the function.

Solved Example

Question: Graph the quadratic function f(x) = 2(x - 2)2 -3. Mark the vertex and axis of the parabola. Estimate the x and y intercepts using the graph.
Solution:
The vertex of the parabola is (2,3) and the axis is x = 2.
To make the table of ordered pairs, let us choose x values on either side of x = 2 and calculate the corresponding y values using the equation given. The table so made is shown below along with the graph done.

 
  x  
   y  
  2    3
  1   -1
  3   -1
  0    5
  4    5
        
   Graphing a Quadratic Function in Vertex Form 

You may note here, the symmetric points have the same y values. f(1) = f(3) -1 and f(0) = f(4) = 5. The vertex (2,3) and axis x = 2 are marked on the Graph. The y intercept = 5. Estimated x intercepts are x = 0.8 and x = 3.2.

The graph of the square function f(x) = x2 is considered as the parent graph for the family of Quadratic functions. Any quadratic function can be graphed applying the techniques of transformation on the parent graph. The transformations for the general quadratic equation f(x) = a(x - h)2 + k can be described as follows:
a = The vertical stretch factor.
h = Horizontal shift to the right.
k = Vertical shift up.
Thus, any ordered pair (x, y) for the parent graph y = x  can be transformed using the mapping given to plot the point for the required graph.

Graphing Quadratic Functions using Transformations

Let us graph the quadratic function f(x) = $\frac{1}{2}$ (x + 3)2 - 5.
The transformations involved are vertical shrink by a factor $\frac{1}{2}$ followed horizontal shift to the left by 3 units and vertical shift down by 5 units. Hence, the transformation mapping is (x, y) =>  (x - 3, $\frac{1}{2}$ y -5) .

 y = x2.
  (x,y)
 â‡’    f(x) = $\frac{1}{2}$ (x + 3)2 - 5
   (x - 3, $\frac{1}{2}$ y -5)
     
  Graphing Quadratic Functions using Transformation 
 
 x 
  y 
 0   0
 1   1
 -1   1
 -2   4
 2   4
 3   9
 -3   9
       
X = x -3
 Y = $\frac{1}{2}$ y -5 
   -3
      -5
   -1
     -4.5
   -4
     -4.5
   -5      -3
   -1
     -3
   -6
     -0.5
    0
     -0.5

The Graph of the given function along with the parent graph y = x2 (graph in green) is shown above. The horizontal and vertical shifts are clearly seen. The vertical shrink factor $\frac{1}{2}$ has made the transformed graph wider than the parent graph.
The vertex and axis of Quadratic functions are found using algebraic methods and are generally used in graphing any quadratic function. The solution/s of quadratic equation of the form ax2 + bx + c = 0 is found by reading the x intercepts of the graph of y = ax2 + bx + c. Generally, it is possible only to get an approximated value of the x intercepts from the graph. Hence, the graph can be done for verifying the solutions got using algebraic methods.
Given below are some of the examples in graphing quadratic functions.

Solved Examples

Question 1: Write the quadratic function f(x) = -x2 + 4x + 5 in vertex form and graph the function.
Solution:
The function can be written in the vertex form by completing the squares.
f(x) = -(x2 -4x) + 5                                    (Group the variables together)
      = -(x2 - 4x + 4) +5 + 4                        (Complete the square.)
f(x) = - (x - 2)2 + 9                                    (Function written in vertex form.)  
The vertex of the graph is hence (2,9) and the axis of symmetry is x = 2.
The table made and the graph are shown below. The graph is U turned down as the leading term is negative here.
      
  x     y  
  2  9
  1  8
  3  8
  0  5
  4  5
       
 Graphing Quadratic Functions Examples
 

The x intercepts of the graph can be noted as (-1,0) and (5,0) while, the y intercept is already taken as a point in the table.

Question 2: Find the mapping rule for graphing the function f(x) = -2x2 + 5x -3
Solution:
Writing the function in vertex form, we get f(x) = -2(x - 1.25)2 + 0.125
The transformations involved are a vertical stretch by a factor of 2, reflection over the x axis, Horizontal shift to the right by 1.25 and a vertical shift up by 0.125.