Quadratic Functions(General Form)
Quadratic functions and the properties of their graphs such as vertex and
You can also use this applet to explore the relationship between the
Once you finish the present tutorial, you may want to go through tutorials on quadratic functions and graphing quadratic functions.
If needed, Free graph paper is available.
A  Definition of a quadratic function
A quadratic functionf is a function of the form
where
Examples of quadratic functions
f(x) = 2x^{ 2} + x  1 f(x) = x^{ 2} + 3x + 2
Interactive Tutorial (1)
Explore quadratic functions interactively using an html5 applet shown below; press "draw' button to start

Use the boxes on the left panel of the applet window to set coefficients
a ,b andc 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
where
Let us start with the 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 inx^{ 2} andx
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
Example : Write the quadratic function
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 is2 by 2 and that gives1 .
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 andk = 3 .

h andk can also be found using the formulas forh andk 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) .

Go back to the applet window and set a to
2 ,b to4 andc to1 (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 to2 andc to0 ,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
are the real solutions, if they exist, of the quadratic equation
The above equation has two real solutions and therefore the graph has
and
Example: Find the
f(x) = x_{ 2} + 2 x  3 g(x) = x_{ 2} + 2 x  1 h(x) = 2_{ 2} + 2 x  2
Solution

To find the
x intercepts, we solve
x^{ 2} + 2 x  3 = 0
discriminantD = 2^{ 2}  4 (1)(3) = 16
two real solutions:
x_{1} = (2 + √16) / (2 * 1) = 1
and
x_{2} = (2  √16) / (2 * 1) = 3
The graph of function in part a) has twox intercepts are at the points(1,0) and(3,0) .

We solve
x^{ 2} + 2 x  1 = 0
discriminantD = 2^{ 2}  4(1)(1) = 0
one repeated real solutionsx_1 = b / 2a = 2 / 2 = 1
The graph of function in part b) has onex intercept at(1,0) .

We solve
2 x^{ 2} + 2 x  2 = 0
discriminantD = 2^{ 2}  4(2)(2) = 12
No real solutions for the above equation
No x intercept for the graph of function in part c).

Go to the applet window and set the values of
a ,b andc for each of the examples in partsa ,b andc above and check the discriminant and thex 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 andc to values such thatb^{ 2}  4 a c < 0 . How manyx intercepts does the graph off(x) have ? 
Use the applet window and set
a ,b andc to values such thatb^{ 2}  4 a c = 0 . How manyx intercepts the does the graph off(x) have? 
Use the applet window and set
a ,b andc to values such thatb^{ 2}  4ac > 0 . How manyx intercepts does the graph off(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
Example: Find the
f(x) = x^{ 2} + 2 x  3 g(x) = 4 x^{ 2}  x + 1 h(x) = x^{ 2} + 4 x + 4

f(0) = 3 . The graph off has ay intercept at(0,3) .

g(0) = 1 . The graph ofg has ay intercept at(0,1) .

h(0) = 4 . The graph ofh has ay intercept at(0,4) .

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 ofc .
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
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
We now use the
and solve for
method 2:
The above parabola has a vertex at
We use the
method 3:
Since a quadratic function has the form
we need 3 points on the graph of
The following points are on the graph of
point
solve for
The two other points give two more equations
which leads to
and
which becomes
Solve the last two equations in a and b to obtain
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
 Derivatives of Quadratic Functions: Explore the quadratic function f(x) = ax ^{ 2} + b x + c and its derivative graphically and analytically.
 Match Quadratic Functions to Graphs. Excellent activity where quadratic functions are matched to graphs.
 Find Vertex and Intercepts of Quadratic Functions  Calculator: An applet to solve calculate the vertex and x and y intercepts of the graph of a quadratic function.

Tutorial on Quadratic Functions (1).

Quadratic Functions  Problems (1).

graphing quadratic functions .

quadratic functions in vertex form .
 Quadratic Functions Transformations
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