Unit 6 – Lesson 1
Introducing Quadratic Functions
Minds On:
Use your graphing calculator to sketch the graphs of each of the following
functions and then state whether they are linear or non-linear functions.
a) y  2 x 2  4 x  5
b) y  3x  6
c) y  2 x 3  4 x 2  7
d) y  3x 2  4
e) y  x 2  5x  3
f) y  x 4  2 x 3  4 x 2  8x
How do you know by looking at the graph of a function that it is linear?
How do you know by looking at the equation of a function that it is linear?
What do you notice about the graphs of a, d and e?
What do you notice about the equations of a, d and e?
The functions in a, d and e are called a quadratic functions and their graphs
are called parabolas.
We can tell that a function is quadratic just by looking at the graph if it is Ushaped and symmetrical.
We can tell that a function is quadratic just by looking at the equation if the
largest exponent on the independent variable (x) is a 2.
Examples:
1.
Are each of the following functions linear, quadratic or neither?
a) y = 3x2 – 4x + 7
b) y = -x2 – 6
c) y = 3x – 3
d) y = 2(x – 3)2 + 8
e) y = -x2
f) y = x3 – 4x2
You can also tell whether a function is linear, quadratic or neither using
finite differences. Finite differences are the differences between the yvalues in tables with evenly spaced x-values.
First differences are calculated by subtracting consecutive y-values. If
first differences are constant, the relation is linear.
Second differences are calculated by subtracting consecutive first
differences. If second differences are constant, the function is
quadratic.
2.
Complete the table and then do first and seconds differences for the
following quadratic functions
a)
b)
c)
x
y
x
y
x
y
0
1
-2
-4
3
9
1
3
-1
-2
4
7
2
7
0
0
5
-3
3
13
1
2
6
4
4
21
2
4
7
0
Worksheet
Answer the following questions on a lined piece of paper: