How to Graph Quadratic Functions?

Table Of Content

Introduction

A quadratic function is of form y = ax² + bx + c where a ≠ 0 and a, b, c are real number. Graph of quadratic functions are always a parabola either opening upwards or downwards.

Parabola opening upwards and parabola opening downwards

Parabola opening upwards and parabola opening downwards

To plot graph of any quadratic function, we need answers of these question

  1. What is sign of ‘a’ or coefficient of x² in quadratic function?
  2. Whether the graph of quadratic function intersects with x-axis? And if it does at what point does it intersect?
  3. Whether the graph of quadratic function intersects with y-axis?
  4. What is the greatest or least value of quadratic function?

We shall plot graph of different quadratic function by finding answers to above questions and improve our graph on basis of their answers.

Steps to plot Quadratic Function

Step1: Check out the sign of ‘a’

If ‘a’ is greater than zero then graph of quadratic function will be a parabola opening upwards.

Parabola opening upwards for positive value of ‘a’

Parabola opening upwards for positive value of ‘a’

If ‘a’ is less than zero then graph of quadratic function will be a parabola opening downwards.

Parabola opening downwards for negative value of ‘a’

Parabola opening downwards for negative value of ‘a’

By checking out the sign of ‘a’ we can easily determine whether the parabola for particular quadratic function will open upwards or downwards.

Step 2: Check Intersection with x-axis

To check whether quadratic function intersects with x-axis, find the discriminant (D) of the function. Discriminant, D is given by formula,

D² = b² – 4ac

Depending upon, whether discriminant is greater than, less than or equal to zero, three cases are possible.

Case 1: If D < 0, the graph of quadratic function does not intersects with x-axis. Depending upon the sign of ‘a’, the graph can be a parabola opening downwards or parabola opening upwards but does not intersect with x-axis in any case.

Parabola opening upwards but does not intersect with x-axis for negative value of D

Parabola opening upwards but does not intersect with x-axis for negative value of D

Parabola opening downwards but does not intersect with x-axis for negative value of D

Parabola opening downwards but does not intersect with x-axis for negative value of D

Case 2: If D = 0, the graph of quadratic function will intersects at one and only one point given by coordinate.

Coordinate

Parabola opening upwards or downwards and intersects with x-axis at one point for D=0

Case 3: If D > 0, the graph of quadratic function will intersects at two points on axis. The coordinate of two points are given by

Points of intersection of quadratic function with X-axis when D>0

Parabola opening downwards for negative value of ‘a’ and intersecting X-axis at two points

Parabola opening upwards for positive value of ‘a’ and intersecting X-axis at two points

Step 3: Check Intersection with y-axis

To find point intersection of graph with y-axis, we substitute x = 0 in expression
y = ax²+ bx + c. Therefore,

y = a (0) + b (0) + c
y = c

The graph of quadratic function intersects with y-axis at point with co-ordinate (0, c).

Step 4: Find Vertex of quadratic function

Vertex of a quadratic function (which is a parabola) is a point at which curve changes its nature.

Vertex is the minimum value of quadratic function for a<0

A downward falling curve (a>0) will change its nature and start rising upward at vertex. In this case vertex will give the lowest point of curve or least value of quadratic function.

Or,

Vertex is the maximum value of quadratic function for a<0

An upward rising curve (a<0) will change its nature and start falling at the vertex. In this, vertex will give the highest point of the curve and greater value of quadratic function.

The vertex of ax² + bx + c is given by a point with given co-ordinates

Coordinates

Where D is discriminant and given by b² – 4ac.

After knowing the nature of the graph and finding crucial points like intersection with Y-axis, X-axis and Vertex, we can easily plot the curve. Following examples will demonstrate the method of graphing quadratic functions.

Examples to graph Quadratic Functions

Example 1

Plot the quadratic function

Y = x²+ 6x + 5

This expression is of form ax² + bx + c i.e.  a = 1, b = 6 and c = 5 in above case.

Step 1 . Check sign of ‘a’

Now a (= 1) is greater than zero. Therefore graph of function will be a parabola opening upwards.

Check Sign of 'a'

Step 2 . Find discriminant

Find Discriminant

Since D> 0, so Case 3 of Step 2 is satisfied, curve will intersects x-axis at points

Ponits at X axis

Simplifying the two points we get points of intersection with x-axis at points (-1, 0) and (-5, 0). Improving upon the curve formed in step 1, we draw x-axis cutting the curve at two points.

Curve according to Points

Step 3 . Find intersection with y-axis

The graph of quadratic function intersects with y-axis at point (0, c). The value of parameter ‘c’ in this case is 5. Draw y axis at point (0, 0) which cuts the y axis at point (0, 5).

Intersection on Y axis

Step 4 . Find the least value or the Vertex of the graph

Coordinates of Vertex

Complete graph of quadratic function Y = x² + 6x + 5 is

Graph of quadratic function Y = x² + 6x + 5

Graph of quadratic function Y = x² + 6x + 5

Example 2

Plot the quadratic function

Y= -x² + 4x + 5

Here a = -1, b =4, c =5

Step 1 . Check the sign of ‘a’

a = (-1) is less than zero. Therefore graph of quadratic function will be a parabola opening downward.

Check Sign of 'a'

Step 2 . Find Discriminant

Find Discriminant

Since D > 0, Case 3 of Step 2 is satisfied. Curve will intersects x-axis at point

Points at X axis

The graph of quadratic function intersects x-axis at point (-1, 0) and (5, 0)

Now draw the x-axis and let the graph intersects with it at two points.

Graph according to X axis Points

Step 3 . Find intersection with y-axis

Curves

The curve cuts y-axis at point with coordinate (0, 5). Draw the y-axis which cuts the y axis at point (0, 5).

Intersection on Y axis

Step 4 . Find the coordinates of vertex given by formula

Coordinates of Vertex

Complete graph of quadratic function Y = -x² + 4x + 5 is

Graph of quadratic function Y = -x² + 4x + 5

Graph of quadratic function Y = -x² + 4x + 5