Quadratic functions are some of the most important algebraic functions and they need to be thoroughly understood in any modern high school algebra course. 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
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 .
Examples of quadratic functions
a) f(x) = -2x^{ 2} + x - 1
b) 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 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 .
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 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 function f 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)
a) 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.
b) 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
f(x) = x^{ 2} + 2 x - 3
g(x) = -x^{ 2} + 2 x - 1
h(x) = -2^{ 2} + 2 x - 2
Solution
a) To find the x intercepts, we solve
x^{ 2} + 2 x - 3 = 0
discriminant D = 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 two x intercepts are at the points (1,0) and (-3,0) .
b) 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) .
c) 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)
1) 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.
2) 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.
3) 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 ?
4) 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?
5) 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.
a) f(x) = x^{ 2} + 2 x - 3
b) g(x) = 4 x^{ 2} - x + 1
c) h(x) = -x^{ 2} + 4 x + 4
Solution
a) f(0) = -3 . The graph of f has a y intercept at (0,-3) .
b) g(0) = 1 . The graph of g has a y intercept at (0,1) .
c) h(0) = 4 . The graph of h has a y intercept at (0,4) .
Interactive Tutorial (4)
a) Use the applet window to check the y intercept for the quadratic functions in the above example.
b) Use the applet window to check the y intercept is at the point (0,c) for different values of c .