Download Identifying Analyzing graphs of Functions. Vertical Line Test

MA 11C
Arcadia Valley Career Technology Center
Topic: Relations and Functions
Show-Me Standards: MA3, G1-8,MA4
Tech Math I Credit
Last Update: March 2008
Focus: Identifying/Analyzing graphs of
Functions. Vertical Line Test.
MO Grade Level Expectations: D1C10,
D3A9
NCTM Standards: 14C, 14D
OBJECTIVE: The students will be able to graph an equation and tell whether it’s a
function by using the Vertical Line Test.
To start this lesson, let’s go back to Lesson 11A and remind ourselves what makes a relation a
function. Each number in the domain of a relation must map to or pair with exactly one number in the
range. You also hear a function described as “every x value has a unique y value”. Remember
that equations are generators of relations that sometimes turn out to be functions.
y = 2x + 1 generates several pairs: (0,1) (1,3) (2,5) etc. Let’s look at what happens when we
graph these pairs on a coordinate grid.
y^
7
6
5
4
3
2
1
0
- if we connect the points, we form a line.
- if we generated more pairs, they
would graph along the same line as the
one you see.
●
●
●
1 2 3 4 5 6 7
>
x
This lesson is going to concentrate on picking out functions by looking at graphs instead of sets of
pairs. We are going to look at some graphs that are connected pairs of some equation and be able to
tell if the equation is a function. The connected pairs make all kinds of shapes depending on how the
equation pairs the numbers.
Vertical Line Test- The graph represents a function if any vertical line drawn on the graph will
cross the graph in at most one place or exactly one place. This proves that each x-value(domain)
has only one y-value(range).
Look at these two graphs:
y
x
y
x
The first graph is a function because it passes the Vertical Line Test. No vertical line drawn on the
graph crosses the graph more than once. The equation that generates the points that make the
shape is a function. The second graph is not a function. You can see that at least one vertical line
drawn on the graph crosses the graph more than once. The equation that generates the pairs for this
shape is not a function.
Sometimes you have to look at shapes, or picture in your mind shapes, without the coordinate axes
drawn. Picture a circle. Would an equation that generates pairs that form a circle be a function?
-
If you picture vertical lines drawn over the top of this
circle as it sits on a coordinate grid, you can see that
a circle equation is not a function. A vertical line
would definitely cross the circle more than once.
Example:
EX1. Generate pairs with the equation y = x2 + 2. Picture the graph and see if the equation forms a
function.
Randomly choose values for x .
Let’s start with 0.
y = (0)2 + 2 = 2
that gives us (0,2)
Let’s try 1 .
y = (1)2 + 2 = 3
that gives us (1,3)
Let’s try -1 .
y = (-1)2 + 2 = 3
that gives us (-1,3)
Let’s try 2 and -2 .
y = (2)2 + 2 = 6
y = (-2)2 + 2 = 6
(2,6)
(-2,6)
You can create as many pairs as you want to. It is sometimes tough to tell if you have enough pairs
to form the true shape of the equation. Usually after 3 to 4 pairs have been found, you get the
general idea of the shape. Still, be cautious.
(-2,6)
(2,6)
(-1,3)
(1,3)
(0,2)
This is the general shape of EX1. It is called a “parabola”.
It passes the Vertical Line Test so we can say that
y = x2 + 2 is a function and therefore note the equation in
function notation, f(x) = x2 + 2 .
PROBLEMS:
1. If graphed, tell which of these are functions.
a. a horizontal line.
b. a square.
c.
d.
e.
f.
g.
2. Generate pairs with the following equations. Picture the graphs of each and see if the equation is
a function. If so, write the equation using function notation.
a. y = 3x – 6
b. y = 2x2 + 1
c. y = x3
d. y = ½ x + 4
e. y = √ x – 1
, all values for x must be positive.