Free Printable 2-12 Properties of Rational Functions

1.

Identify the domain of the function.
$f\left(x\right)=\frac{1}{x+4}+3$

a)

$\left(-\infty,\ -4\right)\cup\left(-4,\ \infty\right)$

b)

$\left(-\infty,\ \infty\right)$

c)

$\left(-\infty,\ 3\right)\cup\left(3,\ \infty\right)$

d)

$\left(-\infty,\ -4\right)\cup\left(-4,\ 3\right)\cup\left(3,\ \infty\right)$

2.

Identify the range of the function.
$f\left(x\right)=\frac{1}{x+4}+3$

a)

$\left(-\infty,\ -4\right)\cup\left(-4,\ \infty\right)$

b)

$\left(-\infty,\ \infty\right)$

c)

$\left(-\infty,\ 3\right)\cup\left(3,\ \infty\right)$

d)

$\left(-\infty,\ -4\right)\cup\left(-4,\ 3\right)\cup\left(3,\ \infty\right)$

3.

Identify the range of the function.
$f\left(x\right)=-\frac{2}{x-3}+1$

a)

$\left(-\infty,\ 1\right)\cup\left(1,\ \infty\right)$

b)

$\left(-\infty,\ \infty\right)$

c)

$\left(-\infty,\ 3\right)\cup\left(3,\ \infty\right)$

d)

$\left(-\infty,\ -2\right)\cup\left(-2,\ \infty\right)$

4.

Identify the domain of the function.
$f\left(x\right)=-\frac{2}{x-3}+1$

a)

$\left(-\infty,\ 1\right)\cup\left(1,\ \infty\right)$

b)

$\left(-\infty,\ \infty\right)$

c)

$\left(-\infty,\ 3\right)\cup\left(3,\ \infty\right)$

d)

$\left(-\infty,\ -2\right)\cup\left(-2,\ \infty\right)$

5.

Describe the transformation of the function from the basic function $y=\frac{1}{x}$ .
$f\left(x\right)=-\frac{2}{x-3}+1$

a)

Shift 3 units right, vertical stretch by factor 2, reflection over the x-axis, shift 1 unit up

b)

Shift 3 units right, horizontal stretch by factor 2, reflection over the x-axis, shift 1 unit up

c)

Shift 1 unit right, vertical stretch by factor 2, reflection over the y-axis, shift 3 units down

d)

Shift 1 unit left, horizontal stretch by factor 2, reflection over the y-axis, shift 3 units up

6.

Describe the transformation of the function from the basic function $y=\frac{1}{x}$ .
$f\left(x\right)=\frac{5x+9}{x+1}$

a)

Shift 1 unit left, vertical stretch by factor 4, shift 5 units up

b)

Shift 1 unit right, vertical stretch by factor 4, shift 5 units up

c)

Shift 1 unit left, vertical stretch by factor 4, reflection over the x-axis, shift 5 units up

d)

Shift 5 units right, horizontal stretch by factor 4, shift 1 unit up

7.

Describe the transformation of the function from the basic function $y=\frac{1}{x}$ .
$f\left(x\right)=\frac{2x+8}{x+3}$

a)

Shift 3 units right, vertical stretch by factor 2, shift 2 units up

b)

Shift 2 units right, vertical stretch by factor 2, shift 3 units up

c)

Shift 3 units right, horizontal stretch by factor 2, shift 2 units down

d)

Shift 3 units left, vertical stretch by factor 2, shift 2 units up

8.

Identify the domain.
$f\left(x\right)=\frac{x^2+5x+6}{2x^2+4x}$

a)

$\left(-\infty,\ -2\right)$

b)

$\left(-\infty,\ -2\right)\cup\left(-2,\ 0\right)\cup\left(0,\ \infty\right)$

c)

$\left(-\infty,\ \infty\right)$

d)

$\left(-\infty,\ 1\right)\cup\left(1,\ 4\right)\cup\left(4,\ \infty\right)$

9.

Identify the domain.
$f\left(x\right)=\frac{x^2+4x+3}{x^2-\ x\ -\ 6}$

a)

$\left(-\infty,\ 6\right)$

b)

$\left(-\infty,\ -2\right)\cup\left(-2,\ 3\right)\cup\left(3,\ \infty\right)$

c)

$\left(-\infty,\ \infty\right)$

d)

$\left(-\infty,\ -3\right)\cup\left(-3,\ -1\right)\cup\left(-1,\ \infty\right)$

10.

Identify any vertical asymptotes.
$f\left(x\right)=\frac{1}{x^2-\ 3x\ }$

a)

$x=0$
$x=3$

b)

$x=0$
$x=-3$

c)

$y=0$
$y=3$

d)

There are no vertical asymptotes.

11.

Identify any horizontal asymptotes.
$f\left(x\right)=\frac{1}{x^2-\ 3x\ }$

a)

$y=0$

b)

$y=1$

c)

$y=0$
$y=3$

d)

There are no horizontal asymptotes.

12.

Identify any horizontal asymptotes.
$f\left(x\right)=\frac{2x^2+2x-12}{x^2-\ 4\ }$

a)

$x=2$
$x=-2$

b)

$y=2$

c)

$y=0$

d)

There are no horizontal asymptotes.

13.

Identify any vertical asymptotes.
$f\left(x\right)=\frac{2x^2+2x-12}{x^2-\ 4\ }$

a)

$x=-2$

b)

$x=2$
$x=-3$

c)

$x=2$
$x=-2$

d)

There are no vertical asymptotes.

14.

Identify the y-intercepts.
$y=\frac{x^2-1}{2x^2+8x+6}$

a)

$\left(1,\ 0\right)$ and $\left(-1,\ 0\right)$

b)

$\left(0,\ 1\right)$ and $\left(0,\ -1\right)$

c)

$\left(0,\ -\frac{1}{6}\right)$

d)

$\left(\frac{1}{6},\ 0\right)$

15.

Identify the x-intercepts.
$y=\frac{x^2-1}{2x^2+8x+6}$

a)

$\left(1,\ 0\right)$

b)

$\left(0,\ 1\right)$ and $\left(0,\ -1\right)$

c)

$\left(0,\ -\frac{1}{6}\right)$

d)

$\left(\frac{1}{6},\ 0\right)$

16.

Identify the x-intercepts.
$y=\frac{x^2+2x-8}{-3x+3}$

a)

$\left(-4,\ 0\right)$ and $\left(2,\ 0\right)$

b)

$\left(0,\ -4\right)$ and $\left(0,\ 2\right)$

c)

$\left(0,\ -\frac{8}{3}\right)$

d)

$\left(-\frac{8}{3},\ 0\right)$

17.

Identify the y-intercepts.
$y=\frac{x^2+2x-8}{-3x+3}$

a)

$\left(-4,\ 0\right)$ and $\left(2,\ 0\right)$

b)

$\left(0,\ -4\right)$ and $\left(0,\ 2\right)$

c)

$\left(0,\ -\frac{8}{3}\right)$

d)

$\left(-\frac{8}{3},\ 0\right)$

18.

Identify the end behavior asymptote.
$y=\frac{1}{x^2-3x}$

a)

$y=0$

b)

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

c)

$y=1$

d)

$y=-3x$

19.

Identify the end behavior asymptote.
$y=\frac{x^2-1}{2x^2+8x+6}$

a)

$y=\frac{1}{2}$

b)

$y=-\frac{1}{6}$

c)

$y=-1$
$y=-3$

d)

$y=-1$
$y=1$

20.

Identify the end behavior asymptote.
$y=\frac{x^2+2x+8}{x-3}$

a)

$y=x+5$

b)

$y=x^2+5x+23$

c)

$y=-4$
$y=2$

d)

$y=3$

2-12 Properties of Rational Functions

Total questions: 20

Worksheet time: 40mins

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