Quadratic Functions(General Form)

Quadratic functions and the properties of their graphs such as vertex and x and y intercepts are explored interactively using an html5 applet.

You can also use this applet to explore the relationship between the x intercepts of the graph of a quadratic function f(x) and the solutions of the corresponding quadratic equation f(x) = 0. The exploration is carried by changing values of 3 coefficients a, b and c included in the definition of f(x).

Once you finish the present tutorial, you may want to go through tutorials on quadratic functions , graphing quadratic functions and Solver to Analyze and Graph a Quadratic Function

If needed, Free graph paper is available.

A - Definition of a quadratic function


A quadratic function f is a function of the form
f(x) = ax 2 + bx + c

where a, b and c are real numbers and a not equal to zero. The graph of the quadratic function is called a parabola. It is a "U" shaped curve that may open up or down depending on the sign of coefficient a.


Examples of quadratic functions

  1. f(x) = -2x 2 + x - 1
  2. f(x) = x 2 + 3x + 2

Interactive Tutorial (1)

Explore quadratic functions interactively using an html5 applet shown below; press "draw' button to start

a =
-10+10

b =
-10+10

c =
-10+10

>


B - Standard form of a quadratic function and vertex


Any quadratic function can be written in the standard form

f(x) = a(x - h) 2 + k


where h and k are given in terms of coefficients a, b and c.

Let us start with the quadratic function in general form and complete the square to rewrite it in standard form.
When you graph a quadratic function, the graph will either have a maximum or a minimum point called the vertex. The x and y coordinates of the vertex are given by h and k respectively.

Example : Write the quadratic function f given by f(x) = -2 x 2 + 4 x + 1 in standard form and find the vertex of the graph.

Solution
Interactive Tutorial (2)


C - x intercepts of the graph of a quadratic function



The x intercepts of the graph of a quadratic function f given by
f(x) = a x 2 + b x + c

are the real solutions, if they exist, of the quadratic equation
a x 2 + b x + c = 0


The above equation has two real solutions and therefore the graph has x intercepts when the discriminant D = b^2 - 4 a c is positive. It has one repeated solution when D is equal to zero. The solutions are given by the quadratic formulas

x 1 = (-b + √ D)/(2 a)

and
x 2 = (-b - √ D)/(2 a)


Example: Find the x intercepts for the graph of each function given below

  1. f(x) = x 2 + 2 x - 3
  2. g(x) = -x 2 + 2 x - 1
  3. h(x) = -2 2 + 2 x - 2

Solution
  1. To find the x intercepts, we solve

    x 2 + 2 x - 3 = 0

    discriminant D = 2 2 - 4 (1)(-3) = 16

    two real solutions:
    x1 = (-2 + √16) / (2 * 1) = 1
    and
    x2 = (-2 - √16) / (2 * 1) = -3

    The graph of function in part a) has two x intercepts are at the points (1,0) and (-3,0).

  2. We solve -x 2 + 2 x - 1 = 0

    discriminant D = 2 2 - 4(-1)(-1) = 0

    one repeated real solutions x_1 = -b / 2a = -2 / -2 = 1

    The graph of function in part b) has one x intercept at (1,0).

  3. We solve -2 x 2 + 2 x - 2 = 0

    discriminant D = 2 2 - 4(-2)(-2) = -12

    No real solutions for the above equation

    No x intercept for the graph of function in part c).

Interactive Tutorial (3)

Answers


D - y intercepts of the graph of a quadratic function



The y intercept of the graph of a quadratic function is given by f(0) = c.

Example: Find the y intercept of the graph of the following quadratic functions.
  1. f(x) = x 2 + 2 x - 3
  2. g(x) = 4 x 2 - x + 1
  3. h(x) = -x 2 + 4 x + 4
Solution
  1. f(0) = -3. The graph of f has a y intercept at (0,-3).
  2. g(0) = 1. The graph of g has a y intercept at (0,1).
  3. h(0) = 4. The graph of h has a y intercept at (0,4).
Interactive Tutorial (4)

E - Exercises: Find the equation of a quadratic function given its graph



As an exercise you are asked to find the equation of a quadratic function whose graph is shown in the applet and write it in the form f(x) = a x 2 + b x + c.

USE this applet to Find Quadratic Function Given its Graph

Example: Find the graph of the quadratic function f whose graph is shown below.

graphical solution to check


Solution

There are several methods to answer the above question but all of them have one idea in common: you need to understand and then select the right information from the graph.

method 1:

The above graph has two x intercepts at (-3,0) and (-1,0) and a y intercept at (0,6). The x coordinates of the x intercepts can be used to write the equation of function f as follows:

f(x) = a(x + 3)(x + 1)

We now use the y intercept f(0) = 6

6 = a(0 + 3)(0 + 1)

and solve for a to find a = 2. The formula for the quadratic function f is given by :

f(x) = 2(x + 3)(x + 1) = 2 x 2 + 8 x + 6

method 2:

The above parabola has a vertex at (-2 , -2) and a y intercept at (0,6). The standard (or vertex) form of a quadratic function f can be written

f(x) = a(x + 2) 2 - 2

We use the y intercept f(0) = 6

6 = a(0 + 2) 2 - 2. Solve for a to find a = 2. The formula for the quadratic function f is given by :

f(x) = 2(x + 2) 2 - 2 = 2 x 2 + 8 x + 6

method 3:

Since a quadratic function has the form

f(x) = a x 2 + b x + c

we need 3 points on the graph of f in order to write 3 equations and solve for a, b and c.

The following points are on the graph of f

(-3 , 0) , (-1 , 0) and (0 , 6)

point (0 , 6) gives

f(0) = 6 = a * 0 2 + b * 0 + c = c
solve for c to obtain c = 6
The two other points give two more equations

(-3 , 0) gives f(-3) = a * (-3) 2 + b * (-3) + 6

which leads to 9 a - 3 b + 6 = 0

and (-1 , 0) gives f(-3) = a (-1) 2 + b * (-1) + 6

which becomes a - b + 6 = 0

Solve the last two equations in a and b to obtain

a = 2 and b = 8 and gives

f(x) = 2 x 2 + 8 x + 6

Go back to the applet above, generate a graph and find its equation. You can generate as many graphs, and therefore questions, as you wish.

NEXT

More on quadratic functions and related topics