The two linear equations shown below are said to be dependent and consistent: \(2x−5y=3\)
\(6x−15y=9\)
Explain in algebraic and graphical terms what happens when two linear equations are dependent and consistent.

Write an equation in terms of x and y for the function that is described by the given characteristics. A sine curve with a period of \(\pi\), an amplitude of 3, a right phase shift of \(\pi/2\), and a vertical translation up 2 units.

The graph below expresses a radical function that can be written in the form \(f(x) = a(x + k)^{\frac{1}{n}} + c\). What does the graph tell you about the value of k in this function?

Find the linear equations that can be used to convert an (x, y) equation to a (x, v) equation using the given angle of rotation. \(\displaystyle\theta={70}^{\circ}\)

Tabular representations for the functions f, g, and h are given below. Write g(x) and h(x) as transformations of f (x).\(\begin{array}{|c|c|c|c|c|c|} \hline x & -2 & -1 & 0 & 1 & 2 \\ \hline f(x) & -1 & -3 & 4 & 2 & 1 \\ \hline \end{array}\)

\(\begin{array}{|c|c|c|c|c|c|} \hline x & -3 & -2 & -1 & 0 & 1 \\ \hline g(x) & -1 & -3 & 4 & 2 & 1 \\ \hline \end{array}\)

\(\begin{array}{|c|c|c|c|c|c|} \hline x & -2 & -1 & 0 & 1 & 2 \\ \hline h(x) & -2 & -4 & 3 & 1 & 0 \\ \hline \end{array} \)

Prove that \((1+\cos\theta)(1−\cos\theta)=(\sin^2)\theta\)

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

Is the vector space \(C \infty[a,b]\) of infinitely differentiable functions on the interval [a,b], consider the derivate transformation D and the definite integral transformation I defined by \(D(f)(x)=f′(x)D(f)(x)=f′(x)\ and\ I(f)(x)=∫xaf(t)dt f(t)dt\). (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 the parent function to obtain each of the following functions.
a) \(\displaystyle{f{{\left({x}\right)}}}=-{3}{\log{{\left({2}{x}\right)}}}\)
b) \(\displaystyle{f{{\left({x}\right)}}}={\log{{\left({x}-{5}\right)}}}+{2}\)
c) \(\displaystyle{f{{\left({x}\right)}}}={\left(\frac{{1}}{{2}}\right)}{\log{{5}}}{x}\)
d) \(\displaystyle{f{{\left({x}\right)}}}={\log{{\left(−{\left(\frac{{1}}{{3}}\right)}{x}\right)}}}−{3}\)

Table shows the number of cellular phone subscribers in the United States and their average monthly bill in the years from 2000 to 2010.

\(\begin{matrix} \text{Year} & \text{Subscribers} & \text{Average Local}\\ \text{ } & \text{(millions)} & \text{Monthly Bill ()}\\ \text{2000} & \text{109.5} & \text{45.27}\\ \text{2001} & \text{128.4} & \text{47.37}\\ \text{2002} & \text{140.8} & \text{48.40}\\ \text{2003} & \text{158.7} & \text{49.91}\\ \text{2004} & \text{182.1} & \text{50.64}\\ \text{2005} & \text{207.9} & \text{49.98}\\ \text{2006} & \text{233.0} & \text{50.56}\\ \text{2007} & \text{255.4} & \text{49.79}\\ \text{2008} & \text{262.7} & \text{50.07}\\ \text{2009} & \text{276.6} & \text{48.16}\\ \text{2010} & \text{300.5} & \text{47.21}\\ \end{matrix}\)

In 1995 There were 33.8 million subscribers with an average local monthly bill of $51.00. Add these points to the scatter plots. Do the 1995 points match well with the trends in the rest of the data?