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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 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) = -x2 + 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) = -x2 + 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
There are two applets to explore quadratic functions in vertex form
1) Aquadratic function HTML5 applet - vertex form .
2) A java applet
The button below starts the applet on a separate large screen.
1 - Click on the button above "click here to start" to start the applet and maximize the window
obtained.
2 - Use the sliders 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.
3 - Use the sliders to 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.
4 - 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.
5 - Set h and k to some values and a to positive values only. Check that the vertex is always a
minimum point.
6 - 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
a(x - h)2 = -k
add -k to both sides
(x - h)2 = -k/a divide both sides by a
The above equation has real solutions if -k/a is positive or zero.
The solutions are given by
x1 = h + sqrt(-k/a)
x2 = h - sqrt(-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 x1 = 3 + sqrt(1) = 4 and x2 = 3 - sqrt(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 x1 = - 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 x-intercepts the graph of f(x) has ?
4 - Use the applet window and set k to zero.
How many x-intercepts the graph of f(x) has ?
5 - Use the applet window and set a and k to values such that -k/a > 0.
How many x-intercepts the graph of f(x) 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 if 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.
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