Question

regular algebra 2 cba 4 review

1. describe the effects on the graph of the rational parent function when it is changed to the fo

$f(x) = \frac{-5}{(x-3)} + 7$

1. what are the restrictions to the domain in $y = \frac{3x+12}{4x+16}$?
2. simplify the following: $\frac{2x}{x-2} \div \frac{x}{x^2-4}$
3. an automobiles velocity, starting from a complete stop, is v(t) (shown below), where v is m feet per second. after what elapsed time would the automobile reach a velocity of 84 feet second?

$v(t) = \frac{140t}{5t-6}$
a 1.3 seconds b 1.8 seconds c 1.5 seconds d 1.4 seconds
simplify the following: $\frac{x^2-x-12}{x^2-2x-15}$

Explanation:

Step1: Solve Q1: Identify parent function

The rational parent function is $f(x)=\frac{1}{x}$.

Step2: Q1: Analyze transformations

1. Horizontal shift: $\frac{1}{x-3}$ shifts right 3 units.
2. Reflection & vertical stretch: $\frac{-5}{x-3}$ reflects over x-axis, stretches by 5.
3. Vertical shift: $\frac{-5}{x-3}+7$ shifts up 7 units.

Step3: Q2: Find domain restrictions

Set denominator to 0: $4x+16=0$
Solve: $4x=-16 \implies x=-4$

Step4: Q3: Rewrite division as multiplication

$\frac{2x}{x-2} \div \frac{x}{x^2-4} = \frac{2x}{x-2} \times \frac{x^2-4}{x}$

Step5: Q3: Factor & simplify

Factor $x^2-4=(x-2)(x+2)$. Cancel common terms:
$\frac{2x}{x-2} \times \frac{(x-2)(x+2)}{x} = 2(x+2)=2x+4$
Restrictions: $x
eq2, x
eq-2, x
eq0$

Step6: Q4: Set velocity equal to 84

$84=\frac{140t}{5t-6}$

Step7: Q4: Solve for t

Cross multiply: $84(5t-6)=140t$
$420t-504=140t$
$280t=504$
$t=\frac{504}{280}=1.8$

Step8: Q5: Factor numerator & denominator

Numerator: $x^2-x-12=(x-4)(x+3)$
Denominator: $x^2-2x-15=(x-5)(x+3)$

Step9: Q5: Simplify

Cancel common $(x+3)$: $\frac{x-4}{x-5}$
Restrictions: $x
eq5, x
eq-3$

Answer:

1. The graph of the parent rational function $f(x)=\frac{1}{x}$ is shifted 3 units to the right, reflected across the x-axis, vertically stretched by a factor of 5, and shifted 7 units up.
2. $x

eq -4$

1. $2x+4$, with restrictions $x

eq -2, 0, 2$

1. B 1.8 seconds
2. $\frac{x-4}{x-5}$, with restrictions $x

eq -3, 5$