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Linear & Quadratic
Functions
PPT 2.1.1
What is a function?
In order for a relation to be a function, for
every input value, there can only be one
output value.
To test to see if the relation is a function,
we perform the vertical line test.
Vertical Line Test
10
If a vertical line were to go 8
through a graph, would it cross 6
4
through the graph more than 2
once at any spot? - 10 - 8 - 6 -4 -2
-2
2 4 6 8 10
-4
-6
Click your mouse and watch -8
the vertical line go through the - 10
linear graph and then do it
again to go through the
parabola. 10
8
6
Since the vertical line only 4
2
crosses each graph once, - 10 - 8 -6 -4 -2 2 4 6 8 10
each one is considered to be a -2
function. Every input value (x) -4
-6
has only one output value (y). -8
- 10
Thinking??? How would you change the graphs so that they were not functions?
Not a Function
10
8
6
4
2
-10 -8 -6 -4 -2 2 4 6 8 10
-2
-4
-6
-8
-10
The graph above is clearly not a function since
the vertical line crosses the graph more than
once.
Every input value (x) has two output values (y)
except for one. Which one?
Linear Functions
Recall from your previous math classes that linear
functions can be written in the following way:
y = mx + b
where m is the slope of the line and b is the y-intercept
of the line
If m is positive, the line climbs from left to right.
The leading coefficient is
the coefficient of the
If m is negative, the line falls fromterm in a polynomialfirst
left to right. in
descending order by
degree (value of
exponents).
The value of m, would be considered the leading
coefficient of a linear function. In future lessons, we will
be using a to represent the leading coefficient for all
polynomial functions.
Domain and Range
Domain: is the set of x-values (or input
values) that exist within the graph or the
equation of a function.
Range: is the set of y-values (or output
values) that exist within the graph or the
equation of a function.
Domain and Range of
Linear Functions
10
8
Most linear functions will have the same
6 domain and range as the example but not all.
4 What type of linear functions would have a
2 different domain and/or range?
-10 -8 -6 -4 -2 2 4 6 8 10 HORIZONTAL!
-2
-4
A horizontal line will have the same domain but
-6 its range will be just a single number. The y-
-8 value of all the points on the line.
-10
Let’s look at the domain of this function. The value of x can be a very large
positive or negative number and anything in between. If x is not restricted in any
way, we define the domain in the following manner:
Domain: All real numbers or {x| x Є R}
Now let’s look at the range. In the same way, the value of y can be a very
large positive or negative number and anything in between. If y is not
restricted in any way, we define the range in the following manner:
Range: All real numbers or {y| y Є R}
End Behaviour of
Linear Functions
10
8 When we discuss end behaviour, we are looking at
6 what happens to the y-values as x approaches
4
positive infinity () and negative infinity (-), the
2
ends of the function.
-10 -8 -6 -4 -2
-2
2 4 6 8 10
Let’s look at our example. As x approaches infinity
-4 (imagine travelling to the right on the line), what
-6 happens to values of y?
-8
They get larger or we can say that they approach
-10
infinity.
As x approaches negative infinity (imagine travelling
to the left on the line), what happens to values of y?
Is it possible to draw a They get smaller or we can say that they approach
straight line with the negative infinity.
following end behaviour: We summarize the end behaviour in this manner:
As x , y - and as As x , y and as x -, y -.
x -, y -?
Domain and Range of
Quadratic Functions
10
8
Let’s consider the domain. Since our graph is not
6
restricted (we could see very large positive and
4
2
negative numbers and everything in between), the
-10 -8 -6 -4 -2 2 4 6 8 10
domain is:
-2 All real numbers or {x| x Є R}.
-4
-6 The value of y, however, is restricted. The y-values
-8
will not get lower than -5.
-10
Since our y-values are restricted, the range is:
All real numbers greater than and including -5 or {y| y≥-5, y Є R}.
Describe the parabola that has domain {x| x Є R} and range {y| y10, y Є R}.
End Behaviour of
Quadratic Functions
10
8 Let’s now look at the end behaviour of the
6 parabola. As x approaches infinity (imagine
4
travelling to the right on the parabola), what
2
happens to values of y?
-10 -8 -6 -4 -2 2 4 6 8 10
-2 They get larger or we can say that they
-4 approach infinity.
-6 As x approaches negative infinity (imagine
-8
travelling to the left on the parbola), what
-10
happens to values of y?
They get larger again or we can say that they
approach infinity.
We summarize the end behaviour in this
manner:
As x , y and as x -, y .
X - Intercepts
10 10
8 8
6 6
4 4
The x-intercept is 4. 2 2
- 10 - 8 - 6 -4 -2 2 4 6 8 10 - 10 - 8 - 6 -4 -2 2 4 6 8 10
-2 -2
-4 -4
-6 -6
-8 -8
- 10 - 10
The x-intercepts are 3 and -3.
An x-intercept is the x-value where the graph
crosses the x-axis.
Quadratic Functions
and X-Intercepts
10
10
8 This parabola has 8
6 two x-intercepts. 6
4
4
2
2
This parabola has
-10 -8 -6 -4 -2 2 4 6 8 10
-2 no x-intercepts. -8 -6
-10 -4 -2 2 4 6 8 10
-2
-4
-4
-6
-6
-8 10
-8
-10 8
-10
6
4
2
-10 -8 -6 -4 -2 2 4 6 8 10
-2
-4 This parabola has one
-6 x-intercept.
-8
-10
