Mastering Quadratics: Transform Your Function with Our Examples

find the exact value of $\tan (\frac{14\pi }{3})$​​​​​​​ using the unit circle

Is the vector space $C\mathrm{\infty }[a,b]$ of infinitely differentiable functions on the interval [a,b], consider the derivate transformation D and the definite integral transformation I defined by $D(f)(x)=f\prime (x)D(f)(x)=f\prime (x)andI(f)(x)=\int xaf(t)dtf(t)dt$. (a) Compute $(DI)(f)=D(I(f))(DI)(f)=D(I(f))$. (b) Compute $(ID)(f)=I(D(f))(ID)(f)=I(D(f))$. (c) Do this transformations commute? That is to say, is it true that $(DI)(f)=(ID)(f)(DI)(f)=(ID)(f)$ for all vectors f in the space?

Describe the transformations that must be applied to $y=x^2$ to create the graph of each of the following functions.
a) ${\displaystyle y=\frac{1}{4}{(x-3)}^2+9}$
b) ${\displaystyle y={\left(\left(\frac{1}{2}\right)x\right)}^2-7}$

A 30-cm-diameter wheel is turning at a speed of 3 radians per second. What is the linear speed, in meters per minute, of a point on its rim?

${\displaystyle f\left(x\right)=2x}$
${\displaystyle \frac{f(x+\mathrm{△}x)-f\left(x\right)}{\mathrm{△}}x}$

Table shows the number of wireless service subscribers in the United States and their average monthly bill in the years from 2000 through 2015

. $\begin{array}{ccc}\text{Year}& \text{Subscribers}& \text{Average Monthly}\\ & \text{(millions)}& \text{Revenue per Subscriber Unit }\\ \text{2000}& \text{109.5}& \text{48.55}\\ \text{2001}& \text{128.4}& \text{49.79}\\ \text{2002}& \text{140.8}& \text{51.00}\\ \text{2003}& \text{158.7}& \text{51.55}\\ \text{2004}& \text{182.1}& \text{52.54}\\ \text{2005}& \text{207.9}& \text{50.65}\\ \text{2006}& \text{233.0}& \text{49.07}\\ \text{2007}& \text{255.4}& \text{49.26}\\ \text{2008}& \text{270.3}& \text{48.87}\\ \text{2009}& \text{285.6}& \text{47.97}\\ \text{2010}& \text{296.3}& \text{47.53}\\ \text{2011}& \text{316.0}& \text{46.11}\\ \text{2012}& \text{326.5}& \text{48.99}\\ \text{2013}& \text{335.7}& \text{48.79}\\ \text{2014}& \text{355.4}& \text{46.64}\\ \text{2015}& \text{377.9}& \text{44.65}\end{array}$

One of the scatter plots suggests a linear model. Use the points at t = 0 and t = 15 to find a model in the form y = mx + b.

A simplified form of $(\sqrt{3^{2}x})+2+3^{-2x}$ that does not involve radicals or fractional exponents can be obtained after factoring the radicand. Find this simplification.

Tabular representations for the functions f, g, and h are given below. Write g(x) and h(x) as transformations of f(x). $\overline{)\begin{array}{c}\\ x& -2& -1& 0& 1& 2\\ \\ f(x)& -2& -1& -3& 1& 2\end{array}}$

$\overline{)\begin{array}{c}\\ x& -1& 0& 1& 2& 3\\ g(x)& -2& -1& -3& 1& 2\end{array}}$

$\overline{)\begin{array}{c}\\ x& -2& -1& 0& 1& 2\\ h(x)& -1& 0& -2& 2& 3\end{array}}$

${\displaystyle 4\sqrt{x}+x=1}$

h is related to one of the parent functions described in this chapter. Describe the sequence of transformations from f to h.
${\displaystyle h\left(x\right)=-\sqrt{x+4}}$