Notes
Roots and exponents
- $\displaystyle \sqrt[n]{a} = a^{1/n}$ and $\displaystyle a^{m/n} = (\sqrt[n]{a})^m$. Negative exponents give reciprocals.
- Simplify radicals by factoring out perfect powers: $\displaystyle \sqrt{72} = 6\sqrt{2}$.
Radical functions
- $\displaystyle f(x) = a\sqrt{x - h} + k$ starts at $(h, k)$. Its domain is $x \geq h$. A negative $a$ reflects it over the $x$-axis.
- $\displaystyle g(x) = a\sqrt[3]{x - h} + k$ is centered at $(h, k)$ and has domain and range of all real numbers.
Solving radical equations
- Isolate the radical, raise both sides to the matching power, solve, then check.
- Squaring can create extraneous solutions, so always substitute back into the original equation. A square root can never equal a negative number.
Topics in this unit
Each topic has its own short notes and practice.
Practice
$\displaystyle \sqrt[5]{32} = 2$ and $2^3 = 8$.
Set $5 - x \geq 0$, so $x \leq 5$.
Square both sides: $2x + 1 = 49$.
Squaring gives $x^2 + 2x + 1 = x + 3$, so $x^2 + x - 2 = 0$ and $x = 1$ or $-2$. For $x = -2$ the left side is $-1$ but the right side is $1$, so it is extraneous.
A cube root function has center $(h, k) = (3, 4)$ and no domain or range restriction.
$\displaystyle \sqrt{\frac{L}{9.8}} = 1$, so $L = 9.8$.
Free response
Write your answer first, then open the scoring guide.
Free-response question
The distance to the horizon from a height of $h$ feet is modeled by $d = \sqrt{1.5h}$ miles.
(a) How far can a person see from the top of a 150 foot tower?
Show the answer and scoring guide
$\displaystyle d = \sqrt{1.5 \cdot 150} = \sqrt{225} = 15$ miles.
Check off the points you earned:
(b) How high must an observer be to see 12 miles to the horizon?
Show the answer and scoring guide
$\displaystyle 12 = \sqrt{1.5h} \Rightarrow 144 = 1.5h \Rightarrow h = 96$ feet.
Check off the points you earned:
(c) Does doubling the height double the distance? Use $h = 96$ and $h = 192$ to explain.
Show the answer and scoring guide
At $h = 96$, $d = 12$. At $h = 192$, $\displaystyle d = \sqrt{288} \approx 16.97$, about 17 miles. This is only about $1.41$ times as far, not 2 times, because $d$ grows with the square root of $h$.
Check off the points you earned: