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 functionf 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 .

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 = 1 -10+10 b = 0 -10+10 c = 1 -10+10
>

• Use the boxes on the left panel of the applet window to set coefficients a, b and c to the values in the examples above, 'draw' and observe the graph obtained. Note that the graph corresponding to part a) is a parabola opening down since coefficient a is negative and the graph corresponding to part b) is a parabola opening up since coefficient a is positive. You may change the values of coefficient a, b and c and observe the graphs obtained.

## 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 .

quadratic function in general form and complete the square to rewrite it in standard form.
• Given function f(x)
f(x) = ax 2 + bx + c

• factor coefficient a out of the terms in x 2 and x

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

• add and subtract (b / 2a) 2 inside the parentheses
f(x) = a ( x 2 + (b/a) x + (b/2a) 2 - (b/2a) 2 ) + c

• Note that
x 2 + (b/a) x + (b/2a) 2

• can be written as
(x + (b/2a)) 2

• We now write f as follows
f(x) = a ( x + (b / 2a) ) 2 - a(b / 2a) 2 + c

• which can be written as
f(x) = a ( x + (b / 2a) ) 2 - (b 2 / 4a) + c

• This is the standard form of a quadratic function with
h = - b / 2a

k = c - b 2 / 4a

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 functionf given by f(x) = -2 x 2 + 4 x + 1 in standard form and find the vertex of the graph.

Solution
• given function
f(x) = -2 x 2 + 4x + 1

• factor -2 out
f(x) = -2(x 2 - 2 x) + 1

• We now divide the coefficient of x which is -2 by 2 and that gives -1.
f(x) = -2(x 2 - 2x + (-1) 2 - (-1) 2) + 1

• add and subtract (-1) 2 within the parentheses
f(x) = -2(x 2 - 2x + (-1) 2) + 2 + 1

• group like terms and write in standard form
f(x) = -2(x - 1) 2 + 3

• The above gives h = 1 and k = 3.

• h and k can also be found using the formulas for h and k obtained above.
h = - b / 2a = - 4 / (2(-2)) = 1

k = c - b 2 / 4a = 1 - 4 2/(4(-2))= 3

• The vertex of the graph is at (1,3).

Interactive Tutorial (2)

• Go back to the applet window and set a to -2, b to 4 and c to 1 (values used in the above example). Check that the graph opens down (a < 0) and that the vertex is at the point (1,3) and is a maximum point.

• Use the applet window and set a to 1, b to -2 and c to 0, f(x) = x 2 - 2 x. Check that the graph opens up (a > 0) and that the vertex is at the point (1,-1) and is a minimum point.

## 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)

• Go to the applet window and set the values of a, b and c for each of the examples in parts a, b and c above and check the discriminant and the x intercepts of the corresponding graphs.
• Use the applet window to find any x intercepts for the following quadratic functions.
a) f(x) = x 2 + x - 2
b) g(x) = 4 x 2 + x + 1
a) h(x) = x 2 - 4 x + 4
Use the analytical method described in the above example to find the x intercepts and compare the results.
• Use the applet window and set a, b and c to values such that b 2 - 4 a c < 0. How many x-intercepts does the graph of f(x) have ?
• Use the applet window and set a, b and c to values such that b 2 - 4 a c = 0. How many x-intercepts the does the graph of f(x) have?
• Use the applet window and set a, b and c to values such that b 2 - 4ac > 0. How many x-intercepts does the graph of f(x) have ?

## 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)
• Use the applet window to check the y intercept for the quadratic functions in the above example.
• Use the applet window to check the y intercept is at the point (0,c) for different values of c.

## 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.

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

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.

More on quadratic functions and related topics

Step by Step Math Worksheets SolversNew !
Linear ProgrammingNew ! Online Step by Step Calculus Calculators and SolversNew ! Factor Quadratic Expressions - Step by Step CalculatorNew ! Step by Step Calculator to Find Domain of a Function New !
Free Trigonometry Questions with Answers -- Interactive HTML5 Math Web Apps for Mobile LearningNew ! -- Free Online Graph Plotter for All Devices
Home Page -- HTML5 Math Applets for Mobile Learning -- Math Formulas for Mobile Learning -- Algebra Questions -- Math Worksheets -- Free Compass Math tests Practice
Free Practice for SAT, ACT Math tests -- GRE practice -- GMAT practice Precalculus Tutorials -- Precalculus Questions and Problems -- Precalculus Applets -- Equations, Systems and Inequalities -- Online Calculators -- Graphing -- Trigonometry -- Trigonometry Worsheets -- Geometry Tutorials -- Geometry Calculators -- Geometry Worksheets -- Calculus Tutorials -- Calculus Questions -- Calculus Worksheets -- Applied Math -- Antennas -- Math Software -- Elementary Statistics High School Math -- Middle School Math -- Primary Math
Math Videos From Analyzemath
Author - e-mail

Updated: February 2015