Mastering Quadratics: Transform Your Function with Our Examples

Describe the transformations and sketch the graphs of the following trigonometric functions. $$\begin{gathered} {\displaystyle a.f\left(x\right)=-2\cos \left(2x\right)+3} \\ b.g\left(x\right)=3\sin (x-\pi )-1 \end{gathered}$$

Begin by graphing
$f(x)={\log }_{2}x.$
Then use transformations of this graph to graph the given function. What is the graph's x-intercept? What is the vertical asymptote? Use the graphs to determine each functions domain and range.
${\displaystyle h\left(x\right)=-1+{\log }_{2}x}$

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?

Prove that $(1+\cos \theta )(1-\cos \theta )=({\sin }^2)\theta$

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

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

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