Question

1. which transformation occurs in $f(x) = \sqrt{x} + 3$?

a. reflection over y - axis
b. shift down 3 units
c. shift up 3 units
d. stretch vertically

1. solve the equation $\frac{2}{x} = 6$

a. $x = 3$
b. $x = \frac{1}{3}$
c. $x = -\frac{1}{3}$
d. $x = -3$

1. solve the equation $\frac{x + 1}{x - 2} = 3$

a. $x = 3.5$
b. $x = -5$
c. $x = 5$
d. $x = -3.5$

1. which is equivalent to $\sqrt{8x^2}$?

a. $2\sqrt{2x}$
b. $2\sqrt{2}x$
c. $4x$
d. $2x\sqrt{2}$

1. what is the vertical asymptote of $f(x) = \frac{3}{x + 5}$?

a. $x = 3$
b. $x = -5$
c. $x = 5$
d. $x = 0$

1. which transformation turns $f(x) = \sqrt{x}$ into $f(x) = 2\sqrt{x}$?

a. reflection over x - axis
b. vertical compression
c. horizontal stretch
d. vertical stretch

1. which expression is equivalent to $\sqrt3{x^6}$?

a. $x^2$
b. $x^3$
c. $x^{18}$
d. $x^{12}$

Explanation:

Question 8

The parent function is \( y = \sqrt{x} \). For a function \( y = f(x)+k \), if \( k>0 \), it is a shift up by \( k \) units. Here \( f(x)=\sqrt{x}+3 \), so \( k = 3>0 \), meaning a shift up 3 units.

Step1: Cross - multiply to solve for \( x \)

Given \( \frac{2}{x}=6 \), cross - multiplying gives \( 6x = 2 \).

Step2: Solve for \( x \)

Divide both sides by 6: \( x=\frac{2}{6}=\frac{1}{3} \). Wait, no, wait. Wait, \( \frac{2}{x}=6\Rightarrow6x = 2\Rightarrow x=\frac{2}{6}=\frac{1}{3} \)? Wait, no, the options: Wait, the equation is \( \frac{2}{x}=6 \). Multiply both sides by \( x \): \( 2 = 6x \), then \( x=\frac{2}{6}=\frac{1}{3} \). So the correct option is b. \( x = \frac{1}{3} \)

Step1: Cross - multiply

Given \( \frac{x + 1}{x-2}=3 \), cross - multiply: \( x + 1=3(x - 2) \)

Step2: Expand the right - hand side

\( x + 1=3x-6 \)

Step3: Move terms with \( x \) to one side and constants to the other

Subtract \( x \) from both sides: \( 1 = 2x-6 \)
Add 6 to both sides: \( 2x=7 \)

Step4: Solve for \( x \)

Divide by 2: \( x=\frac{7}{2}=3.5 \)? Wait, no, wait. Wait, \( x + 1=3(x - 2)\Rightarrow x + 1=3x-6\Rightarrow1 + 6=3x - x\Rightarrow7 = 2x\Rightarrow x = 3.5 \)? But the option a is \( x = 3.5 \). Wait, maybe I made a mistake. Wait, let's check again. \( \frac{x + 1}{x - 2}=3 \). Cross - multiply: \( x + 1=3x-6 \). Then \( 1+6 = 3x - x\Rightarrow7 = 2x\Rightarrow x = 3.5 \). So the correct option is a. \( x = 3.5 \)

Answer:

c. Shift up 3 units

Question 9