Download Graph Rational Functions

W2D1 Intro Graphs of Rational Expressions
Warm Up
1. Evaluate
x+1
x2 −6x+8
2. Evaluate
x+3
x2 −2x−35
when x = -1
when x = -5
3. What value of x that will make each denominator equal to zero.
1
x
1
x−1
1
x−2
1
x+3
1
(x−3)(x+1)
Lesson 4 Graph Rational Expressions
Inquiry WS Graph Rat
Do #1 with them to show the shape and highlight the asymptotes.
Asymptote: where the function is undefined. The function nears the asymptote as it approaches infinity
The general form of a rational equation y =
a
+k
x−h
h shifts the graph left and right. In this case the key to graphing is the vertical asymptote shifts left or right.
k shifts the graph up or down. In this case the key to graphing is that the horizontal asymptote shifts up or down.
a is the scale factor or causes a reflection.
The rational function can take the form y =
axn + bxn−1 ...k
cxm + dxm−1 .....p
In this case, solve the denominator to find the vertical asymptotes.
Do long division (ignore the remainder) to find the equation of the horizontal asymptote.
1
EX 1:
Make a table and graph y = x−2
+1
First draw the vertical asymptote. It is x = Whatever makes the denominator zero
x=2
Then draw the horizontal asymptote. it is equal to k if the numerator is a constant.
y=1
Then plug in two points on each side of the vertical asymptote
x = 1 and x = 3
x
y
1
0
3
2
0
1
2
4
3
2
Plot the points. Sketch the curves.
5
4
3
2
1
−2 −1
−1
1
2
3
4
5
6
7
8
−2
−3
−4
−5
Figure 1: Ex 1
x - intercept (1, 0)
1
y-intecept (0, )
2
Domain All Real Numbers except x 6= 2
Range: All Real Numbers except y 6= 1
EX 2:
(YOU TRY if needed) graph y=
−4
x+3
−2
First draw the vertical asymptote. It is x = Whatever makes the denominator zero
VA : x = -3
Then draw the horizontal asymptote. it is equal to k if a is a constant.
HA : y = -2
Then plug in 2 points on each side of the vertical asymptote
x - int (-5,0)
y- int (0, -3.33)
x
y
-5
0
-4
2
-2
-6
1
-3
Domain R x 6= −4 Range R y 6= −2
6
5
4
3
2
1
−8 −7 −6 −5 −4 −3 −2 −1
−1
−2
−3
−4
−5
−6
−7
−8
−9
−10
1
2
3
4
5
Figure 2: Ex 2
EX 3:
Graph y =
x−3
x+4
VA: x = -4
HA: y = 1
x-intercept: (3, 0)
3
y-intercept: (0, - )
4
Domain: R, x 6= -4
Range: R, y 6= 1
Horizontal Asymptote visualization Using Example 3
x
f(x)
1000
1.006
1000000
1.000006
-1000
.994
-1000000
.999994
1
x+4
x−3
−x−4
−7
x
y
3
0
-3
-6
-5
8
-6
9
2
10
9
8
7
6
5
4
3
2
1
−12
−11
−10−9−8−7−6−5−4−3−2−1
−1
−2
−3
−4
−5
−6
−7
1 2 3 4 5
Figure 3: Ex 3
EX 4:
Graph
3x + 1
x−5
VA: x = 5
3
x−5
3x + 1
− 3x + 15
16
HA: y = 3
1
, 0)
3
1
y-intercept: (0, - )
5
Domain: R, x 6= 5
Range: R, y 6= 3
x-intercept: ( -
3
x−5
3x + 1
− 3x + 15
16
x
y
6
19
7
11
4
-13
3
-5
12
10
8
6
4
2
−14 −10 −6
−2
−2
2
6
10
14
18
−4
Figure 4: Ex4
3x+1
x−5
EX 5:
Graph y=
x2
2x2 − 1
− 4x − 12
To find what makes the denominator zero, factor to solve:
y=
2x2 − 1
(x − 6)(x + 2)
2
2
x − 4x − 12
2
2x
−1
− 2x2 + 8x + 24
8x + 23
VA: x=6, x= -2
HA : y=
√2
2
x-int (
, 0) = (1.71, 0)
2
1
y-int (0,
) = (0, .083)
12
Domain: R, x 6= 6, x6= -2
Range: R, y 6= 2
Must plug in 6 points because there are 2 asymptotes now!
x
y
-4
31
≈ 1.5
20
-3
17
≈2
9
-1
- 17
5
-7
7
97
9
8
127
20
≈ 11
≈6
11
10
9
8
7
6
5
4
3
2
1
−11−9 −7 −5 −3−1
−1 1 3 5 7 9 11 13 15
−2
−3
−4
−5
−6
−7
Figure 5: Ex 5
2x2 −1
x2 −4x−12
EX 6:
graph y=
2x
−9
x2
When you do long division x2 can’t go into 2x so the asymptote is y = 0
VA : x= 3, x= -3
HA : y=0
x-int (0, 0)
y-int (0, 0)
Domain: R, x 6= ±3
Range: R, y 6= 0
x
y
-5
-
5
≈ −.5
8
-4
-
8
≈ −1
7
-1
1
2
2
-
4
8
7
≈1
5
5
8
≈ .5
4
5
8
7
6
5
4
3
2
1
−6 −5 −4 −3 −2 −1
−1
−2
−3
−4
−5
−6
−7
−8
1
2
3
4
5
6
Figure 6: Ex 6
2x
x2 −9
EX 7:
graph y=
x2
x+3
+ 5x + 6
When you do long division x2 can’t go into 2x so the asymptote is y = 0
VA : x= -2, x= -3
HA : y=0
x-int (-3, 0)
1
y-int (0, )
2
Domain: R, x 6= −2x 6= −3
Range: R, y 6= 0
x
y
-5
-
1
3
-4
-
1
2
-2.5
-2
-1
1
1
1
3
6
5
4
3
2
1
−6 −5 −4 −3 −2 −1
−1
−2
−3
−4
−5
−6
1
2
3
4
Figure 7: Ex 7
x+3
x2 + 5x + 6
1. Graph
x+2
x2 −3x−10
2. Graph
x+4
x−2
3. Graph
−x+1
x+2
4. Graph
x+3
2x+8
5. Graph
2x+1
x+3
+ 4 6. Graph
4
x−2
5
8
4
6
3
2
4
1
2
−5−4−3−2−1
−1
1 2 3 4 5 6 7 8 9 10
−10 −8 −6 −4 −2
−2
−2
−3
2
4
6
8
10
6
8
−4
−4
−5
−6
Figure 8: 1.
x+2
x2 −3x−10
2.
x+4
x−2
8
6
6
4
4
2
2
−14−12−10 −8 −6 −4 −2
−14−12−10 −8 −6 −4 −2
2
4
6
8
−4
−4
−x+1
x+2
Domain: x 6= -2
4
−2
−2
Figure 9: 3.
2
Range: y 6= -1
4.
x+3
2x+8
Domain: x 6= -4
Range: y 6=
1
2
14
5
12
4
3
10
2
8
1
6
−5 −4 −3 −2 −1
−1
4
2
3
4
5
−2
2
−14−12−10 −8 −6 −4 −2
−2
1
−3
2
4
Figure 10: 5.
6
−4
8
−5
2x+1
x+3
+ 4 Domain: x 6= -3
Range: y 6= 6.
4
x−2
Exit Pass
2x − 3
1. Graph f(x) =
x−4
List the Vertical Asymptote, Horizontal Asymptote, x-intercept, y-intercept
2. What are the Domain and Range
1.
VA : x= 4
HA : y= 2
x-int (1.5, 0)
y-int (0, .75)
2 . D x 6= 4
R: y 6= 2
7
6
5
4
3
2
1
−5−4−3−2−1
−1
1 2 3 4 5 6 7 8 9 10
−2
−3
Figure 11: Exit Pass