Demystify College Algebra with Step-by-Step Examples and Expert Guidance

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.

For the function that is described by the provided characteristics, write an equation in terms of x and y. A sine curve with a period of $\pi$, an amplitude of 3, a right phase shift of $\frac{\pi }{2}$, and a vertical translation up 2 units.

The two linear equations shown below are said to be dependent and consistent: $2x-5y=3$
$6x-15y=9$
What happens if you use a graphical method?

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?

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

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

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

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 $C\mathrm{\infty }[a,b]$ 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?

h is related to one of the six parent functions.

(a) Identify the parent function f.

(b) Describe the sequence of transformations from f to h.

(c) Sketch the graph of h by hand.

(d) Use function notation to write h in terms of the parent function f. ${\displaystyle h\left(x\right)={(x-2)}^3+5}$

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?

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