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As x approaches a constant, c, the y coordinates approach – or + .
As x approaches – or + the y coordinates approach a constant c.
As x approaches – or + the y coordinates approach a function q(x). Slant asymptote is another name because the q(x) is usually linear.
xy
1
Translate 1f x
x
Graph 13f x
x
The plus 3 is on the outside, so add 3 to each y coordinate. Also move the horizontal asymptote y = 0 to y = 3.
x = 0
y = 0
y = 3
Translate 1f x
x
Graph
The minus 2 is on the inside, so add 2 to each x coordinate. Also move the vertical asymptote x = 0 to x = 2.
x = 0
y = 0
1
2f x
x
x = 2
Translate 1f x
x
Graph
The power of 2 will mean that there will be no negative y coordinates and the y coordinates will increase faster, when 0 < x <1, and decrease faster, when x > 1.
x = 0
y = 0
2
1f x
x
Translate 1f x
x
Graph
The negative will flip the graph over the x-axis and the plus 2 will shift it up 2 units.
x = 0
y = 0y = 2
12f xx
21
x
xf
Translate
Graph
We will just shift the previous example down 2.
x = 0
y = 0
2
1f x
x
21
2
xxf
y = -2
Translate
Graph
Manipulate the function until you have the original function somewhere in the new function.
x = 0
y = 0
2
1f x
x
2
2 9
x
xxf
y = 1
22
2 9
xx
xxf
2
191x
xf
11
92
x
xf
The negative 9 will flip over the x-axis (multiply the y coordinates by -9) and shift up 1 unit. We will introduce a new set of points when x = 3 and -3. This will cancel the -9 to -1.
FACTOR THE NUMERATOR & DENOMINATOR. We will work with both forms of R(x).
Set the factors in the DENOMINATOR equal to zero and solve for x. This will be the equations of the Vertical Asymptotes.
Compare the leading terms of N(x) and D(x). CASE 1. n < m : The degree of the N(x) is less than the degree of the D(x).
The HORIZONTAL ASYMPTOTE is y = 0. CASE 2. n = m : The degree of the N(x) is equal to the degree of the D(x).
The HORIZONTAL ASYMPTOTE is y = a/b . CASE 3. n > m : The degree of the N(x) is greater than the degree of the D(x).
NO HORIZONTAL ASYMPTOTE.Need to find OBLIQUE ASYMPTOTE.
xqyxNxD
Set the factors in the NUMERATOR equal to zero and solve for x. This will be the x-intercepts in the form ( __ , 0 ).
The y-intercept is ( 0, c/d )
Graph with an open circle on the x-axis.Graph with a closed circle on the x-axis.
Mainly the RHB because it is the same for even and odd powers. Divide the leading coefficients’ signs for the sign of the RHB
Holes occur when there is a binomial factor that is the same in both the top and bottom. This will eliminate a vertical asymptote and an x–intercept.Example.
32
312
xx
xxxR The Hole is located at ( 3, R(3) )
after you cancel the binomials.
5,31
16,3
23
132,3
2
12
x
x
x = 3 is needed to find the hole.
1.
2.
H.A. and y-intercept are the easiest to find before factoring. 1
Case 1. y = 0
( 0, -3/-2 ) = ( 0, 3/2 )
12
3
xx
xFACTOR
x = -2 x = 13.
4. x = 3; ( 3, 0 )
3-2 10
Graphing Order.1. V.A. & x-intercepts.
3. H.A.
2. Positive & Negative regions. RHB
++
& y-intercepts.
1.
2.
H.A. and y-intercept are the easiest to find before factoring.
Case 2. y = 1
( 0, -4/0 ) = undefinedNO Y-INTERCEPT
3
22
xx
xx FACTOR
x = 0 x = -33.
4. x = 2, -2; ( 2, 0 ), ( -2, 0 )
-2-3 0
Graphing Order.1. V.A. & x-intercepts.
3. H.A.
2. Positive & Negative regions. RHB
++
& y-intercepts.
1
1 + 0
2
+
1
1.
2.
1
FACTOR
3.
4.
3-2 1
Graphing Order.1. V.A. & x-intercepts.
3. O.A.
2. Positive & Negative regions. RHB
++
& y-intercepts.
21 6 6
1 1 1
x x x x
x x x
3 2
1
x x
x
Case 3. No H.A., Oblique Asymptote
x = 3; ( 3, 0 )
( 0, -6/-1 ) = ( 0, 6 )
x = 1
1 -1 -6
1 1 0
1
X
X
y = x x = -2; ( -2, 0 )
No remainder in Oblique Asymptotes
6
1.
2.
H.A. and y-intercept are the easiest to find before factoring.
Case 2. y = 1
( 0, -9/-6 ) = ( 0, 3/2 )
32
33
xx
xxFACTOR
x = 23.
4. x = 3; ( 3, 0 )2-3
Graphing Order.1. V.A. & x-intercepts.
3. H.A.
2. Positive & Negative regions. RHB
++
& y-intercepts.
3
y = 1
2
3
x
xWe have a HOLE at (-3, __). Find R(-3).
5
6
23
333
R
5
6,3
4. Graph the HOLE
5
6,3
2
3,0
Draw in all asymptotes, intercepts, and label them.
y = 0
x = 4 x = 0
( 2, 0 )
What does the Horizontal Asymptote tell us?
The top degree is less than the bottom degree. Case 1 xf
What does the x-int. tell us?
The factor on the top.
What does the Vertical Asymptotes tell us?
The factors on the bottom.
2x 4xx
Draw in all asymptotes, intercepts, and label them.
y = 2
x = -2
( -4, 0 )
What does the Horizontal Asymptote tell us?
The top degree is equal to the bottom degree. Case 2
xf
What does the x-int. tell us?
The factors on the top.
What does the Vertical Asymptotes tell us?
The factors on the bottom.
4xx
2x
( 0, 0 )
Both sides of the V.A. are going down…multiplicity of 2.
2The HA is y = 2. This means the leading coefficient ratio is 2
1 2
Draw in all asymptotes, intercepts, HOLES, and label them.
y = 1
x = 2
( 3, 0 )
( -2, ? )
( 0, 1.5 )
What does the Horizontal Asymptote tell us?
The top degree is equal to the bottom degree. Case 2
What does the x-int. tell us?
xf
The factors on the top.
3x
What does the Vertical Asymptotes tell us?
The factors on the bottom.
2x
What does the HOLE tell us?Repeat factors on the top and the bottom.
2x
2x
We can find y-coord. of the HOLE.
25.1
4
5
22
322
R
( -2, 1.25 )
We can verify the y-int. 5.1
2
3
2
3
Label all asymptotes, intercepts, and multiplicity.
y = 1
x = -1
x = 2
( 3, 0 )
( 0, 0. ? )
( 1, 0 )
(2)
(2)
xf 3x
2x
1x
1x
What does the x-int. tell us?
The factors on the top.
What does the Vertical Asymptotes tell us?
The factors on the bottom.
What does the Horizontal Asymptote tell us?
The top degree is equal to the bottom degree. Case 2
We can verify the y-int. 5.1
2
3
12
13
TIME OUT! Y-int is POSITIVE.
2
2
ccc
75.4
3
12
1322
( 0, 0.75 )
cx 2
Best if c were to equal 1.(x2 + c) is used because this will not create an x – intercepts, x = + i
1
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