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Pearson Algebra 2 Common Core, 2011

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Pearson Algebra 2 Common Core, 2011 View details

3. Rational Functions and Their Graphs

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Exercise 32 Page 521

Division by zero is not defined. This means that the denominator cannot be zero.

Practice makes perfect

To graph the given rational function, we will find its domain, asymptotes, and intercepts. Then, we will find points using a table of values. Finally, we will plot and connect those points.

Domain

Consider the given function. y=x(x+1)/x+1 Recall that division by zero is not defined. Therefore, the rational function is undefined where x+1=0. x+1=0 ⇔ x=- 1 This means that x=- 1 is not included in the domain. Domain All real numbers except x=- 1

Asymptotes

Asymptotes can be vertical or horizontal lines.

Vertical Asymptotes

Once again, let's consider the given function. y=x(x+1)/x+1 Notice that we can cancel out common factor between the numerator and the denominator.

y=x(x+1)/x+1

CancelCommonFac

Cancel out common factors

y=x/1

We canceled out the factor x+1. Therefore, the graph has hole at x=-1. The graph does not have a vertical asymptote.

Horizontal Asymptotes

To find the horizontal asymptotes, we can use a set of rules. To properly use these rules, in the following table m and n must be the highest degree of the numerator and denominator.

y=ax^m/bx^n	Asymptote
m	y=0
m>n	none
m=n	y=a/b

We must expand the numerator of the function.

y=x(x+1)/x+1

Distribute x

y=x^2+x/x+1

Let's look at the degrees of the numerator and denominator for our function. y=x^2+x/x^1+1 We can see that the degree of the denominator is lower than the degree of the numerator. Therefore, there is no horizontal asymptote.

Intercepts

The intercepts of the function are the points at which the graph intersects the axes.

x-intercepts

The x-intercepts are the points where the graph intersects the x-axis. At these points, the value of the y-coordinate is zero. Let's substitute 0 for y in the given function and solve for x.

y=x(x+1)/x+1

y= 0

0=x(x+1)/x+1

▼

Solve for x

LHS * x+1=RHS* x+1

0=x(x+1)

Rearrange equation

x(x+1)=0

We found that if y=0, the value of x is 0 or -1. However, x=-1 is not included in the domain of the function. Therefore, there is an x-intercept at (0,0).

y-intercept

The y-intercept is the point where the graph intersects the y-axis. At this point, the value of the x-coordinate is zero. Let's substitute 0 for x in the given function and solve for y.

y=x(x+1)/x+1

x= 0

y=0( 0+1)/0+1

▼

Solve for x

Zero Property of Multiplication

y=0/0+1

Identity Property of Addition

y=0/1

Calculate quotient

y=0

There is a y-intercept at (0,0).

Graph

Let's make a table of values to graph the given function. Make sure to only use values included in the domain of the function.

x	x(x+1)/x+1	y=x(x+1)/x+1
-3	- 3( -3+1)/- 3+1	-3
-2	- 3( -3+1)/- 2+1	-2
0	- 0( 0+1)/0+1	0
1	1( 1+1)/1+1	1
2	2( 2+1)/2+1	2

Finally, let's plot and connect the points. Do not forget to draw the asymptotes and to plot the intercepts and holes, if any.

Subchapter links

Lesson Check p.521

Practice and Problem-Solving Exercises p.521-523

Standardized Test Prep p.523