Question

transformations and asymptotes in rational and radical equations
assessment

1. in the equation $y = \sqrt{x - 5}$, what transformation is applied to the parent function $y = \sqrt{x}$?

a. shift 5 units left
b. shift 5 units right
c. shift 5 units up
d. shift 5 units down

1. which transformation is represented by the equation $y = \frac{1}{x + 3}$?

a. shift 3 units left
b. shift 3 units right
c. stretch vertically by a factor of 3
d. compress horizontally by a factor of 3

1. what transformation is shown in the equation $y = \sqrt{\frac{1}{2}x}$?

a. stretch horizontally by a factor of 2
b. compress horizontally by a factor of 2
c. stretch vertically by a factor of 2
d. compress vertically by a factor of 2

1. how would you describe the transformation in $y = -2\sqrt{x}$?

a. reflect over the x - axis and stretch vertically by a factor of 2
b. reflect over the y - axis and stretch horizontally by a factor of 2
c. shift 2 units down
d. compress vertically by a factor of 2

1. what transformation is applied to $y = \frac{1}{x}$ to get $y = \frac{1}{x}-1$?

a. shift 1 unit right
b. shift 1 unit left
c. shift 1 unit up
d. shift 1 unit down

1. for the rational function $f(x)=\frac{2}{x + 1}$, what is the vertical asymptote?

a. $x = 0$
b. $x = 1$
c. $x = 2$
d. $x=-1$

1. what is the horizontal asymptote of the function $g(x)=\frac{4}{x - 2}$?

a. $y = 0$
b. $y = 2$
c. $y = 4$
d. $y=-2$

Explanation:

Question 1

Step1: Recall horizontal shift rule

For a function \( y = f(x - h) \), it's a shift of \( h \) units right (if \( h>0 \)) or left (if \( h<0 \)) from \( y = f(x) \). Here, parent function is \( y=\sqrt{x} \), new function is \( y=\sqrt{x - 5} \), so \( h = 5>0 \), shift 5 units right.

Step1: Recall horizontal shift for rational functions

For \( y=\frac{1}{x + a} \), it's a shift of \( a \) units left (if \( a>0 \)) from \( y=\frac{1}{x} \). Here, \( y=\frac{1}{x+3} \), so shift 3 units left.

Step1: Recall horizontal stretch/compression for square root functions

For \( y=\sqrt{bx} \), if \( 0 < b<1 \), it's a horizontal stretch by factor \( \frac{1}{b} \). Here, \( b=\frac{1}{2} \), so stretch factor is \( \frac{1}{\frac{1}{2}} = 2 \), horizontal stretch by factor 2.

Answer:

b. Shift 5 units right

Question 2