湖北省武汉六中高三数学复习课件：2.4有理函数的空心坐标

f x x2 x 6
x2
第四页，编辑于星期日：十五点 四十一分。
Example 1: Solution
f x x2 x 6
x2
This is a line!
How did that happen?
It has a slope of 1 and a y-intercept of -3
第五页，编辑于星期日：十五点 四十一分。
For example:
f
x
3x3
7x x2
10Leabharlann fxx43x3 2x2 5x x2 x 1
7
f
x
5x3 4x3
3x 3x
1 1
f
x
x4
x2 x 1 3x3 2x2 5x
7
f
x
x3
3x 1 9x2 2x 1
f
x
x 1 x3 1
f
x
x4
6x3
x 1 5x2
x
111
However, it would be unreasonable and impractical to look at all the different possibilities
第十页，编辑于星期日：十五点 四十一分。
Example 2
Consider the function:
f
x
x2
x2 1 3x 4
(a) Determine the location of the hole
(b) Graph the function and state the domain and range
Therefore, the domain and range are:
(-2, -5)
Domain :x / x 2, x
Range :y / y 5, y
The line has a hole!
(yes, that’s what we call it)
The hole is located at (-2, -5)
Example 2: Solution
(b) This function has two asymptotes: The line y = 1
The line x = -4
It also has a hole at
1,
2 5
So, the function cannot have x = -4 and x = 1:
A rational function
Reciprocal of linear function, reciprocal of quadratic function, etc. Will have asymptotes AND holes
There is no limit to the number of holes a rational function can have
No vertical asymptote No horizontal asymptote No “gaps”
So, what makes it special?
第六页，编辑于星期日：十五点 四十一分。
Example 1: Notes
Let’s take a closer look at the function:
Domain :x / x 2, x
第七页，编辑于星期日：十五点 四十一分。
Example 1: Notes
Notice that the numerator can be factored
f x x2 x 6
x2
Factor the simple trinomial
Need two numbers that add to -1 and multiply to -6
Summary
The graph of a rational functions can have holes Occurs whenever a binomial factor cancels out
We find holes the same way we find vertical asymptotes:
A vertical asymptote
A horizontal asymptote A hole
The x-coordinate of a hole affects the domain The y-coordinate of a hole affects the range
第十四页，编辑于星期日：十五点 四十一分。
第十五页，编辑于星期日：十五点 四十一分。
Practice Problems
P. 190 #4
Find the location of the hole(s) Graph using “Graph”
第十六页，编辑于星期日：十五点 四十一分。
Unit 2: Rational Functions
Lesson 4: Holes
第一页，编辑于星期日：十五点 四十一分。
What is a Rational Function?
Any function of the form:
hx
f g
x x
Where f(x) and g(x) are polynomial functions
Exactly what our graph showed 第八页，编辑于星期日：十五点 四十一分。
Example 1: Notes
However…
Our domain is
x / x 2, x
Which means our graph shows the function f(x) = x – 3 (a line) with the point where x = -2 removed
第十一页，编辑于星期日：十五点 四十一分。
Example 2: Solution
(a) Notice that the numerator and denominator can be factored
f
x
x2
x2 1 3x
4
Factor the difference of squares
Factor the simple trinomial Need two numbers that add to 3 and multiply to -4
Let’s take a closer look at our graph
第九页，编辑于星期日：十五点 四十一分。
Example 1: Notes
f x x2 x 6
x2
The hole occurs because we removed the point with x = -2
At this point, y = -5
So, we are going to worry about only those given on the last slide and one more special case
第三页，编辑于星期日：十五点 四十一分。
Example 1
Graph the function given by:
f
x
x 1 x 1 x 4 x 1
The numbers are 4 & -1
f
x
x 1 x 1 x 4 x 1
Cancel the x – 1 ‘s
f x x1
x4
ax b
Rational function of the form
cx d
第十二页，编辑于星期日：十五点 四十一分。
Domain :x / x 4, x 1, x
And the function cannot have y = 1 and y = 2/5:
Range
:
y
/
y
2 5
,
y
1,
y
第十三页，编辑于星期日：十五点 四十一分。
Example 2: Notes
The function in this example had…
Set the denominator equal to zero & solve for x
If a rational function has a hole, it will simplify to… A polynomial function
Lines, parabolas, etc.
Example 1: Notes
As the graph shows, the rational function given is a line
It doesn’t have any of the properties of rational functions we’ve discussed
f x x 2 x 3
x2
f x x 2 x 3
x2
The numbers are 2 & -3 Cancel the x + 2 ‘s
f x x3
This is the equation of a line with a slope of 1 and a y-intercept of -3
The rational functions we have seen are of the form
hx 1
kx c
h
x
ax2
1 bx
c
h x ax b
cx d
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What Comes Next?
We can keep building rational functions by putting different degree polynomial functions in the numerator and denominator
f x x2 x 6
x2
As we’ve been discussing, the denominator CANNOT be zero:
x20 x 2
Because x = -2 makes the denominator zero, it must be removed from the domain: