1. An Introduction to Functions and
Their Graphs
Mock Tutoring Lesson Prepared by
Valerie Cavanaugh
June 3, 2013

2. What Is a Function?
A function is a relation between an input and an output.
A function is like a machine.
y = x(x – 1)(x + 1)

3. The Graph/Curve of a Function
-1
-0.5
0
0.5
1
-1.5 -1 -0.5 0 0.5 1 1.5
x
y
y = x(x – 1)(x + 1)
Driving along the curve of the function
Remember! A function is a relation between an input and an output.
Analogy: Imagine a function as a means of describing the shape of a
road on a map.
START

4. Nomenclature
•Function name: Arbitrary; can be any symbol
•“Input”
–Arbitrary
–Placeholder
–Independent variable
–Value is given
•f(x) is classic function notation
–x = distance from y-axis
-1
-0.5
0
0.5
1
-1.5 -1 -0.5 0 0.5 1 1.5
x
y
y = x(x – 1)(x + 1)
Function
name
“Input”
Operation(s) to perform on “input” to obtain “output”
f(x) = x(x – 1)(x + 1)

5. Objective: To find the output value that corresponds to a given input
value.
f(x) = x(x – 1)(x + 1)
1. Denote evaluation with given input value by replacing input
symbol to left of equal sign:
x = 0.5
f(0.5) = x(x – 1)(x + 1)
2. Replace each instance of input symbol to right of equal sign with
input value:
f(0.5) = 0.5(0.5 – 1)(0.5 + 1)
3. Perform operation(s) to right of equal sign to obtain output value:
f(0.5) = 0.5(-0.5)(1.5) = -0.375
Evaluating a Function

6. -1
-0.5
0
0.5
1
-1.5 -1 -0.5 0 0.5 1 1.5
x
y
-1
-0.5
0
0.5
1
-1.5 -1 -0.5 0 0.5 1 1.5
x
y
-1, 0
-0.5,
0.375
0, 0
0.5, -
0.375
1, 0
-1
-0.5
0
0.5
1
-1.5 -1 -0.5 0 0.5 1 1.5
x
y
Evaluating a Function
x y = f(x)
-1.5 -1.875
-1 0
-0.5 0.375
0 0
0.5 -0.375
1 0
1.5 1.875
In order to obtain the graph of a
function, it must be evaluated
using various input values.
f(x) = x(x – 1)(x + 1)

7. -1
-0.5
0
0.5
1
-1.5 -1 -0.5 0 0.5 1 1.5
long
lat
-1
-0.5
0
0.5
1
-1.5 -1 -0.5 0 0.5 1 1.5
long
lat
Longitude
Latitude
Evaluating a Function
long lat = f(long)
-1.5 -1.875
-1 0
-0.5 0.375
0 0
0.5 -0.375
1 0
1.5 1.875
Analogy: Imagine that the latitude
of a point on a road is related to its
longitude in the manner below.
f(long) = long(long – 1)(long + 1)
The road takes shape as the latitude
function is evaluated using various
longitude values.
long = 0.5
f(0.5) = long(long – 1)(long + 1)
f(0.5) = 0.5(0.5 – 1)(0.5 + 1)
f(0.5) = 0.5(-0.5)(1.5) = -0.375

8. -1
-0.5
0
0.5
1
-1.5 -1 -0.5 0 0.5 1 1.5
x
y
y = x(x – 1)(x + 1)
Vertical Line Test
Special Rules
A function must:
1. Work for every possible
input value.
2. Yield only one output value
per input value.
∴ A vertical line may intersect its
graph at most once.
Analogy
If a car begins a trip heading:
•East, then it will never head
west
•West, then it will never head
-1
-0.5
0
0.5
1
-1.5 -1 -0.5 0 0.5 1 1.5
x
y
y = x(x – 1)(x + 1)