The graph of a quadratic function is "U" shaped and is called a parabola. The exploration is carried by changing the values of all 3 coefficients $a$, $h$ and $k$. Once you finish the present tutorial, you may want to go through another tutorial on graphing quadratic functions.
This form of the quadratic function is also called the vertex form.
A  Vertex, maximum and minimum values of a quadratic function
f(x) = a(x  h)^{2} + k
The term (x  h)^{2} is a square, hence is either positive or equal to zero.
(x  h)^{2} ≥ 0
If you multiply both sides of the above inequality by coefficient a, there are two possibilities
to consider, a is positive or a is negative.
case 1: a is positive
a(x  h)^{2} ≥ 0.
Add k to the left and right sides of the inequality
a(x  h)^{2} + k ≥ k.
The left side represents f(x), hence f(x) ≥ k. This means that k is the minimum value of function f.
case 2: a is negative
a(x  h)^{2} ≤ 0.
Add k to the left and right sides of the inequality
a(x  h)^{2} + k ≤ k.
The left side represents f(x), hence f(x) ≤ k. This means that k is the maximum value of function f.
Note also that k = f(h), hence point (h,k) represents a minimum point when a is positive and a
maximum point when a is negative. This point is called the vertex of the graph of f.
Example: Find the vertex of the graph of each function and identify it as a minimum or maximum point.
a) f(x) = (x + 2)^{2}  1
b) f(x) = x^{2} + 2
c) f(x) = 2(x  3)^{2}
a) f(x) = (x + 2)^{2}  1 = (x  (2))^{2}  1
a = 1 , h = 2 and k = 1. The vertex is at (2,1) and it is a maximum point since a is negative.
b) f(x) = x^{2} + 2 = (x  0)^{2} + 2
a = 1 , h = 0 and k = 2. The vertex is at (0,2) and it is a maximum point since a is negative.
c) f(x) = 2(x  3)^{2} = 2(x  3))^{2} + 0
a = 2 , h = 3 and k = 0. The vertex is at (3,0) and it is a minimum point since a is positive.
Interactive Tutorial
Use the html 5 (better viewed using chrome, firefox, IE 9 or above) applet below to explore the graph of a quadratic function in vertex form: f(x)=a (xh)^{2} + k where the coefficients a, h and k may be changed in the applet below. Enter values in the boxes for a, h and k and press draw.
1  Use the boxes on the left panel of the applet to set a to 1, h to 2 and k to 1. Check the
position of the vertex and whether it is a minimum or a maximum point. Compare to part a) in the example above.
2  Set a to 1, h to 0 and k to 2. Check the position of the vertex and
whether it is a minimum or a maximum point. Compare to part b) in the example above.
3  Set a to 2, h to 3 and k to 0. Check the position of the vertex and whether it is a minimum
or a maximum point. Compare to part c) in the example above.
4  Set h and k to some values and a to positive values only. Check that the vertex is always a minimum point.
5  Set h and k to some values and a to negative values only. Check that the vertex is always a maximum point.
B  x intercepts of the graph of a quadratic function in standard form
The x intercepts of the graph of a quadratic function f given by
f(x) = a(x  h)^{2} + k
are the real solutions, if they exist, of the quadratic equation
a (x  h)^{2} + k = 0
add k to both sides
a(x  h)^{2} = k
divide both sides by a
(x  h)^{2} = k / a
The above equation has real solutions if  k / a is positive or zero.
The solutions are given by
x_{1} = h + √( k / a)
x_{2} = h  √( k / a)
Example: Find the x intercepts for the graph of each function given below
a) f(x) = 2(x  3)^{2}+ 2
b) g(x) = (x + 2)^{2}
c) h(x) = 4(x  1)^{2} + 5
a) To find the x intercepts, we solve
2(x  3)^{2} + 2 = 0
2(x  3)^{2} = 2
(x  3)^{2} = 1
two real solutions: x_{<1} = 3 + √1 = 4 and x_{2} = 3  √1 = 2
The graph of function in part a) has two x intercepts are at the points (4,0) and (2,0)
b) We solve
(x + 2)^{2} = 0
one repeated real solution x_{1} =  2
The graph of function in part b) has one $x$ intercept at (2,0).
c) We solve
4(x  1)^{2} + 5 = 0
 k / a =  5 / 4 is negative. The above equation has no real solutions and the graph of function h has
no x intercept.
Interactive Tutorial
1  Go back to the applet window and set the values of a, h and k for each of the examples in
parts a, b and c above and check the the x intercepts of the corresponding graphs.
2  Use the applet window to find any x intercepts for the following functions. Use the
analytical method described in the above example to find the x intercepts and compare the
results.
a) f(x) = 5(x  3)^{2} + 3
b) g(x) = (x + 2)^{2} + 1
c) h(x) = 3(x  1)^{2}
3  Use the applet window and set a and k to values such that k / a < 0.
How many xintercepts the graph of f has ?
4  Use the applet window and set k to zero.
How many xintercepts the graph of f has ?
5  Use the applet window and set a and k to values such that  k / a > 0.
How many xintercepts the graph of f has ?
C  From vertex form to general form with a, b and c.
Rewriting the vertex form of a quadratic function into the general form is carried out by expanding the square in the vertex form and grouping like terms.
Example: Rewrite f(x) = (x  2)^{2}  4 into general form with coefficients a, b and c.
Expand the square in f(x) and group like terms
f(x) = (x  2)^{2}  4 = (x^{2} 4 x + 4)  4
=  x^{2} + 4 x  8
A tutorial on how to find the equation of a quadratic function given its graph can be found in this site.
More references on quadratic functions and their properties.
