Mastering Quadratics: Transform Your Function with Our Examples

${\displaystyle f:R\to R}$
f(x)=7x-3
Prove that f is onto.

g is related to one of the six parent functions. (a) Identify the parent function f. (b) Describe the sequence of transformations from f to g. (c) Sketch the graph of g by hand. (d) Use function notation to write g in terms of the parent function f. ${\displaystyle g\left(x\right)=-{(x+3)}^3-10}$

Tabular representations for the functions f, g, and h are given below. Write g(x) and h(x) as transformations of f (x).

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

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

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

Sketch a graph of the function. Use transformations of functions whenever possible.
$f(x)=\{\begin{array}{lll}1-x& if& x<0\\ 1& if& x\ge 0\end{array}$

g(x)
${\displaystyle =\frac{2x^2-3x-20}{x-4},\text{if}x\ne 4}$
and
=kx-15, if x=4
Evaluate the constant k that makes the function continuous.

Given:
n=3;
4 and 2i are zeros;
f(-1)=75
Find an nth-degree polynomial function with real coefficients

Given an indicated variable:
S=180n-360
Solve it for n

g is related to one of the parent functions. Describe the sequence of transformations from f to g. g(x) = -x^3 - 1

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. ${\displaystyle h\left(x\right)=\mid 2x+8\mid -1}$

Tabular representations for the functions f, g, and h are given below. Write g(x) and h(x) as transformations of f(x).

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

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

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

$f:R->R$
$f(x)=8x-2$
$f(a)=f(b)$
${\displaystyle a,b\in R}$
Prove that f is one-to-one.

g is related to one of the six parent functions. (a) Identify the parent function f. (b) Describe the sequence of transformations from f to g. (c) Sketch the graph of g by hand. (d) Use function notation to write g in terms of the parent function f. ${\displaystyle g\left(x\right)=\frac{1}{2}\mid x-2\mid -3}$

If ${\displaystyle f\left(x\right)=\sqrt{2x-x^2}}$ , graph the following functions in the viewing rectangle [-5,5] by [-4,4] . How is each graph related to the graph in part (a)?

Describe how the given functions can be obtained from their basic (or parent) function f using transformations.
${\displaystyle f\left(x\right)=\sqrt{x},g\left(x\right)=\sqrt{2x-10}}$

The function g is related to one of the parent functions described in an earlier section.
${\displaystyle g\left(x\right)=\frac{1}{6}\sqrt{x}}$
a) Identify the parent function f.
b)Describe the sequence of transformations from f to g.
c)Use function notation to write g in terms of f.
$g(x)=(?)f(x)$

Graph the function by hand, not by plotting points, but by starting with the graph of one of the standard functions and then applying the appropriate transformations.
${\displaystyle y=\left|x\right|-2}$

Graph the function by hand, not by plotting points, but by starting with the graph of one of the standard functions and then applying the appropriate transformations.
${\displaystyle y=1-2x+x^2}$

Write an equation for the graphed function by using transformations of the graphs of one of the toolkit functions.
y=?

By using the transformation of function ${\displaystyle y=x^2}$,
sketch the function of y=${\displaystyle \frac{{(x-2)}^2}{3}+4}$

By using the transformation of function y=|x|, sketch the function of y=|x-3|+2