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
x1 = h + √(- k / a)
x2 = 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 x2 = 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 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 has ?
4 - Use the applet window and set k to zero.
How many x-intercepts the graph of f 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 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.