📝 Instructions: More Complex Rational Functions

In this set, you will determine the characteristics of functions with squares or higher powers. Use the following rules:

Zeros (Roots): Numerator = 0 (provided the denominator is not also 0 there).
Vertical Asymptote (VA): Denominator = 0.
Horizontal Asymptote (HA): Look at the behavior for an extremely large x.
y-intercept: Calculate f(0).

Block 1: Basic Powers

1)

y = \frac{1}{x}

2)

y = \frac{1}{x^{2}}

3)

y = \frac{x}{x^{2}}

4)

y = \frac{x + 1}{x^{2}}

5)

y = \frac{2}{x - 3}

Show solutions Block 1

1) Zero: none; VA: x=0; HA: y=0; y-int: none.
2) Zero: none; VA: x=0; HA: y=0; y-int: none.
3) Simplify to 1/x: see problem 1.
4) Zero: x=-1; VA: x=0; HA: y=0; y-int: none.

5) Zero: none; VA: x=3; HA: y=0; y-int: (0,

- \frac{2}{3}

).

Block 2: Quadratic

6)

y = \frac{x^{2}}{x - 1}

7)

y = \frac{x^{2} - 1}{x}

8)

y = \frac{x}{x^{2} - 4}

9)

y = \frac{3}{x^{2} - 9}

10)

y = \frac{x^{2} + 1}{x}

Show solutions Block 2

6) Zero: x=0; VA: x=1; HA: none; y-int: (0, 0).
7) Zeros: x=±1; VA: x=0; HA: none; y-int: none.
8) Zero: x=0; VA: x=±2; HA: y=0; y-int: (0, 0).

9) Zero: none; VA: x=±3; HA: y=0; y-int: (0,

- \frac{1}{3}

).

10) Zero: none; VA: x=0; HA: none; y-int: none.

Block 3: Mix & Analyze

11)

y = \frac{x^{2}}{x^{2} - 1}

12)

y = \frac{x^{2} - 4}{x^{2}}

13)

y = \frac{x^{2} + 2 x}{x - 1}

14)

y = \frac{1}{x^{3}}

15)

y = \frac{x^{2}}{x^{2} + 1}

Show solutions Block 3

11) Zero: x=0; VA: x=±1; HA: y=1; y-int: (0, 0).
12) Zeros: x=±2; VA: x=0; HA: y=1; y-int: none.
13) Zeros: x=0, x=-2; VA: x=1; HA: none; y-int: (0, 0).
14) Zero: none; VA: x=0; HA: y=0; y-int: none.

15) Zero: x=0; VA: none (

x^{2} + 1 \neq 0

); HA: y=1; y-int: (0, 0).