Download - Graphing General Rational Functions
![Page 1: Graphing General Rational Functions](https://reader033.vdocuments.us/reader033/viewer/2022061416/56813a30550346895da21863/html5/thumbnails/1.jpg)
Graphing General Rational Functions
![Page 2: Graphing General Rational Functions](https://reader033.vdocuments.us/reader033/viewer/2022061416/56813a30550346895da21863/html5/thumbnails/2.jpg)
What’s the big idea?We’re always looking for shortcuts to make sketching easier. The next two questions shed some light on a function’s graph:
What happens to f(x) as x as gets closer and closer to where the denominator is zero?
What happens to f(x) as x goes way right (positive infinity) or way left (negative infinity)?
The answer to the first question deals with asymptotes. The answer to the second question deals with the end behavior of a function
![Page 3: Graphing General Rational Functions](https://reader033.vdocuments.us/reader033/viewer/2022061416/56813a30550346895da21863/html5/thumbnails/3.jpg)
Talkin’ bout Asymptotes.Let’s look at f(x) = 1/(x-2)
We know that x ≠ 2. But what happens as x gets close to 2?
If x = 1.9999, f(x) = -10,000
If x = 2.0001, f(x) = 10,000.
-3 8
-4
4
x
y
Ah! As x approaches the asymptote from the left, the function goes
to -∞And as x approaches the asymptote from the
right, the function goes to ∞
We write that as:x 2- f(x) -∞x 2+ f(x) ∞
![Page 4: Graphing General Rational Functions](https://reader033.vdocuments.us/reader033/viewer/2022061416/56813a30550346895da21863/html5/thumbnails/4.jpg)
Vertical Asymptotes
-3 8
-3
8
x
y
![Page 5: Graphing General Rational Functions](https://reader033.vdocuments.us/reader033/viewer/2022061416/56813a30550346895da21863/html5/thumbnails/5.jpg)
Let p(x) and q(x) be polynomials with no mutual
factors.
p(x) = amxm + am-1xm-1 + ... + a1x
+ a0
Meaning: p(x) is a polynomial of degree m
Example: 3x2+2x+5; degree = 2
q(x) = bnxn + bn-1x
n-1 + ... + b1x + b0
Meaning: q(x) is a polynomial of degree n
Example: 7x5-3x2+2x-1; degree = 5
Rational Function: f(x) = p(x)/q(x)
![Page 6: Graphing General Rational Functions](https://reader033.vdocuments.us/reader033/viewer/2022061416/56813a30550346895da21863/html5/thumbnails/6.jpg)
x-intercepts are the zeros of p(x) (the top)
Meaning: Solve the equation: p(x) = 0
Vertical asymptotes occur at zeros of q(x)
Meaning: Solve the equation: q(x) = 0 (bottom)
Horizontal Asymptote depends on the degree of p(x), which is m, and the degree of q(x), which is n.
If m < n (bottom wins), then y = 0
If m = n (tie), divide the leading coefficients
If m > n (top wins), then NO horizontal asymptote
Key Characteristics
![Page 7: Graphing General Rational Functions](https://reader033.vdocuments.us/reader033/viewer/2022061416/56813a30550346895da21863/html5/thumbnails/7.jpg)
Graph y =
x-intercepts: Set top = 0 and solve!
3x2 = 0 x2 = 0 x = 0.
Vertical asymptotes: Set bottom = 0 and solve
x2 - 4 = 0 (x - 2)(x+2) = 0 x= ±2
Degree of p(x) = degree of q(x) top and bottom tie divide the leading coefficients
Y = 3/1 = 3.
Horizontal Asymptote: y = 3
Graphing a rational function where m = n
3x2
x2-4
![Page 8: Graphing General Rational Functions](https://reader033.vdocuments.us/reader033/viewer/2022061416/56813a30550346895da21863/html5/thumbnails/8.jpg)
Here’s the picture!x y
-4 4
-3 5.4
-1 -1
0 0
1 -1
3 5.4
4 4 You’ll notice the three branches. This often happens with overlapping horizontal and vertical asymptotes.
The key is to test points in each region!
Domain: x ≠ ±2Range: y > 3 & y ≤ 0
![Page 9: Graphing General Rational Functions](https://reader033.vdocuments.us/reader033/viewer/2022061416/56813a30550346895da21863/html5/thumbnails/9.jpg)
Example: Graph y =
State the domain and range.
x-intercepts: None; p(x) = 4 and 4 ≠ 0
Vertical Asymptotes: None; q(x) = x2 + 1.
But if x2 + 1 = 0 x2 = -1. No real solutions.
Degree p(x) < Degree q(x) --> Horizontal Asymptote at y = 0 (x-axis)
Graphing a Rational Function where m < n
4
x2 + 1
![Page 10: Graphing General Rational Functions](https://reader033.vdocuments.us/reader033/viewer/2022061416/56813a30550346895da21863/html5/thumbnails/10.jpg)
We can see that the domain is ALL REALS while the range is 0 < y ≤ 4
Let’s look at the picture!
![Page 11: Graphing General Rational Functions](https://reader033.vdocuments.us/reader033/viewer/2022061416/56813a30550346895da21863/html5/thumbnails/11.jpg)
Graph y =
x-intercepts: x2- 2x - 3 = 0
(x - 3)(x + 1) = 0 x = 3, x = -1
Vertical asymptotes: x + 4 = 0 x = -4
Degree of p(x) > degree of q(x)
No horizontal asymptote
Graphing a Rational Function where m > n
x2- 2x - 3x + 4
![Page 12: Graphing General Rational Functions](https://reader033.vdocuments.us/reader033/viewer/2022061416/56813a30550346895da21863/html5/thumbnails/12.jpg)
Not a lot of pretty points on this one. This graph actually has a special type of asymptote called “slant.” It’s drawn in purple. Don’t worry, I’ll show you how it works.
Picture time!x y
-12 -20.6
-9 -19.2
-6 -22.5
-2 2.5
0 -0.75
2 -0.5
6 2.1
![Page 13: Graphing General Rational Functions](https://reader033.vdocuments.us/reader033/viewer/2022061416/56813a30550346895da21863/html5/thumbnails/13.jpg)
Slant AsymptotesThese happen when the top polynomial overpowers the bottom polynomial.
“Overpowers” means has a higher degree.
The equation of the slant asymptote is just the result when you divide the two polymomials out.
Prior example:
Equation of Asymptote!
Remainder is
Irrelevant
![Page 14: Graphing General Rational Functions](https://reader033.vdocuments.us/reader033/viewer/2022061416/56813a30550346895da21863/html5/thumbnails/14.jpg)
The Big IdeasAlways be able to find:
x-intercepts (where numerator = 0)
Vertical asymptotes (where denominator = 0)
Horizontal asymptotes:
If bottom wins: x-axis asymptote
If they tie: divide leading coeff.
If top wins: No horizontal asymp.
Sketch branch in each region (plot points)
![Page 15: Graphing General Rational Functions](https://reader033.vdocuments.us/reader033/viewer/2022061416/56813a30550346895da21863/html5/thumbnails/15.jpg)
Practice ProblemsGraph y =
6
x2 + 3
No x-intercepts No vertical asymptoteH.A.: y = 0
x y
0 2
-1 1.5
1 1.5
-2 6/7
2 6/7
![Page 16: Graphing General Rational Functions](https://reader033.vdocuments.us/reader033/viewer/2022061416/56813a30550346895da21863/html5/thumbnails/16.jpg)
Practice ProblemsGraph y =
x2 - 4
x + 1
x y
0 -4
-3 -5/2
x-int: x2 – 4 = 0; x = ±2V.A.: x + 1 = 0 x = -1
H.A.: Top wins: No H.A.Slant: y = x - 1
![Page 17: Graphing General Rational Functions](https://reader033.vdocuments.us/reader033/viewer/2022061416/56813a30550346895da21863/html5/thumbnails/17.jpg)
Practice ProblemsGraph y =
2x2
x2 - 1 x-int: 2x2 = 0 x = 0V.A.: x2 – 1 = 0 x = ±1
H.A.: Tie y = 2/1 = 2
x y
2 8/3
3 9/4
-2 8/3
-3 9/4
Bounce
back, baby!