4.3 Properties of Rational Functions

Example 1: Use the graph to find the domain and range

A vertical asymptote occurs where there is division by zero, and this will not be included in the domain. The domain is all numbers not including -6 and 1 . The interval notation is (−∞,−6) ∪ (−6,1/2) ∪ (1/2,∞). The range is all the y-values that are used. In this case the graph will never use the y-value of 0 since there is a horizontal asymptote. Therefore the range is all numbers not including 0: (−∞,0) ∪ (0,∞).

Asymptotes

An asymptote is defined as a line or curve that approaches a given curve arbitrarily closely, as illustrated in the below diagram

Example 7: Graph y = 4x^2 - 2 / x^2 using transformations

First divide by everything by x. This tells us to stretch the graph of y = 1 / x^2 by a factor of 2, reflect it about the x axis and move it up 4 units

Example 6: Graph y = x - 1 / x using transformations

First divide everything by x to get y = x / x - 1 / x. This tells us to move the graph of y = 1 / x up one unit and then reflect it over the x axis

Example 3: Find the intercept

First favorite to simplify. For the y intercept put a 0 for x. When you do you will get a zero in the denominator which is undefined. So there is no y intercept. To find the x intercept we need to set the top equal to zero. So we have x - 3 = 0. We get x = 3, so (3,0).

Example: Find the x and y intercept for

First find the y intercept by putting a zero for x. So our point is (0,6). Next find x intercept. Set the numerator equal to zero. You get x = 6. Our points is (6,0)

Example 5: Graph y = -1 / x^2 + 6x + 9 using transformation

First we will factor: y = −1 / (x + 3)(x + 3) which can be written as y = −1 / (x + 3)^2. The x + 3 tells us that we need to move y = 1 / x^2 three places to the left. The negative tells us we need to flip the graph horizontally.

Example 2: Find the intercepts.

First you want to factor to see if it can be simplified further, this can't be simplified so let's find the y intercept by putting a 0 for x. Y intercept is (0, 1/6). To find x intercept set the top equal to zero, so we have (1-x)(1+x) = 0.

m

Highest power (degree) of the denominator

n

Highest power (degree) of the numerator

To find the y intercept for a rational function

Put in a zero for x

Finding the vertical asymptote

Set the denominator equal to zero and solve for x

To find the x intercept for a rational function

Set the numerator equal to zero

Example 2: Find the domain

The domain is all numbers expect x = 2 and x = 3. the interval notation is (−∞,2) ∪ (2,3) ∪ (3, ∞)

Example 6: Find the domain and range of y = x - 1 / x

The domain would be (−∞,0)∪(0,∞) since zero caused division by zero. The range would be (−∞,1)∪(1,∞) since the graph never crosses the H.A.

Example 7: Find the domain and range of y = 4x^2 - 2 / x^2

The domain would be (−∞,0)∪(0,∞) since zero caused division by zero. The range would be (−∞,4) since the graph never goes higher than the H.A.

Example 1: Use the graph to find the asymptotes

The horizontal asymptote is y = 0 The vertical asymptote are x = -6 and x = 1/2 There are no oblique asymptotes

Example 6: Find the asymptotes of y = x - 1 / x

The vertical asymptote would be x = 0. The horizontal asymptote would be y = 1. There is no oblique (slant) asymptote.

Example 7: Find the asymptotes of y = 4x^2 - 2 / x^2

The vertical asymptote would be x = 0. The horizontal asymptote would be y = 4. There is no oblique (slant) asymptote.

Example 5: Graph using transformation y = 1 / x - 3

The x - 3 says we need to move the graph of y = 1/x three places to the right.

Example 1: Use the graph to find the intercept of the graph if any

There is an x intercept at (-3,0) but no y intercept

Example 2: Find the asymptotes

To find the vertical asymptote set the bottom equal to zero. So we have (x - 2)(x - 3) = 0 so x = 2 and x = 3. To find the horizontal asymptote look at the original equation. The highest power on the top is the same as the highest power on the bottom, so we use rule 2 again. Our a^n = -1 and b^m = 1, so the horizontal asymptote is y = -1/1 so y = -1.

Example 3: Find the asymptotes

To find the vertical asymptote set the bottom equal to zero. So we have 2x(1 + 2x) = 0, so x = 0 and x = -1/2. To find the horizontal asymptote look at the original equation. The highest power on the top is less than the highest power on the bottom, so we use rule 1. This says that horizontal asymptote is automatically y = 0.

Example 4: Find the asymptotes

To find the vertical asymptote we need to set the bottom equal to zero. So we have x + 2 = 0 so x = −2. To find the horizontal asymptote let's look at our original equation. The highest power on the top is more than the highest power on the bottom, so now we have rule 3. This tells us there is no horizontal asymptote, but there is an oblique asymptote. For oblique asymptote use long division. We get 3x + 4.

Finding the oblique asymptote

Use long division

Finding the horizontal asymptote

We need to look at n and m. If n< m then the equation is automatically y = 0 If n = 0 then the equation is y = a^n/b^m If n > m then there is no horizontal asymptote there is an oblique asymptote

Horizontal asymptote

a description of the rational function as the input values, x, go to positive infinity or negative infinity Had an equation that starts with y =

Rational function

a function whose numerator and denominator are polynomials

vertical asymptote

when there is a zero in the denominator and x approaches that number. Had an equation that starts with x =