Finding maximum and minimum using quadratic functions and not too much algebra

1. Maximum and Minimum using Quadratic Functions
and not too much algebra!
These slides were prepared for management
mathematics students, some of whom had no algebra
at the start of the course.
y = x²
y = x²
10
10
y = x²
9
8 10
8
7
6 6 5
5 4
4 0
2
3 -5 0 5
2 0 -5
1 -5 -2 0 5
0
-4 -2 -1 0 2 4

2. Quadratic Functions
The graphs of many equations are called functions.
In economics and finance we are often interested in where the
minimum value (turning point) is (the X ordinate) and what the
minimum value is (the Y ordinate).
If the graph is inverted then we are looking for the maximum
value.

3. The basic equation for a quadratic function is y = x2.
The minimum value is found at x = 0 and the minimum
value is y = 0
y = x²
10
9
8
7
6
5
4
3
2
1
0
-4 -2 -1 0 2 4

4. A negative in front of the x2 inverts the graph.
The maximum value is found at x = 0 and the
maximum value is y = 0.
y = -x²
1
0
-4 -3 -2 -1 -1 0 1 2 3 4
-2
-3
-4
-5
-6
-7
-8
-9
-10

5. y = x² - 4
6
5
4
3
2
1
0
-4 -3 -2 -1 0 1 2 3 4
-1
-2
-3
-4
-5

6. A constant inserted inside the bracket slides the
graph along the X axis in the Opposite direction.
The minimum value is found at x = 2 and the
minimum value is y = 0.
y = (x - 2)²
10
9
8
7
6
5
4
3
2
1
0
-2 0 2 4 6
-1

7. y = (x + 3)² - 2
8
7
6
5
4
3
2
1
0
-7 -5 -3 -1 1 3
-1
-2
-3