Transcript Linear & Quadratic Functions

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
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If a vertical line were to go
through a graph, would it cross
through the graph more than
once at any spot?
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Click your mouse and watch
the vertical line go through the
linear graph and then do it
again to go through the
parabola.
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Since the vertical line only
crosses each graph once,
each one is considered to be a
function. Every input value (x)
has only one output value (y).
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Thinking??? How would you change the graphs so that they were not functions?
Not a Function
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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
coefficient of the first
If m is negative, the line falls fromthetermleft
to right.
in a polynomial
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
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Most linear functions will have the same
domain and range as the example but not all.
What type of linear functions would have a
different domain and/or range?
HORIZONTAL!
A horizontal line will have the same domain but
its range will be just a single number. The yvalue of all the points on the line.
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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
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Is it possible to draw a
straight line with the
following end behaviour:
As x  , y  - and as
x  -, y  -?
When we discuss end behaviour, we are looking at
what happens to the y-values as x approaches
positive infinity () and negative infinity (-), the
ends of the function.
Let’s look at our example. As x approaches infinity
(imagine travelling to the right on the line), what
happens to values of y?
They get larger or we can say that they approach
infinity.
As x approaches negative infinity (imagine travelling
to the left on the line), what happens to values of y?
They get smaller or we can say that they approach
negative infinity.
We summarize the end behaviour in this manner:
As x  , y   and as x  -, y  -.
Domain and Range of
Quadratic Functions
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Let’s consider the domain. Since our graph is not
restricted (we could see very large positive and
negative numbers and everything in between), the
domain is:
All real numbers or {x| x Є R}.
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The value of y, however, is restricted. The y-values
will not get lower than -5.
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| y10, y Є R}.
End Behaviour of
Quadratic Functions
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Let’s now look at the end behaviour of the
parabola. As x approaches infinity (imagine
travelling to the right on the parabola), what
happens to values of y?
They get larger or we can say that they
approach infinity.
As x approaches negative infinity (imagine
travelling to the left on the parbola), what
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
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The x-intercept is 4.
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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
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This parabola has
two x-intercepts.
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This parabola has
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This parabola has one
x-intercept.
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