# What transformations of the parent graph of f(x)=sqrt{x} produce the graphs of the following functions? a) m(x)=sqrt{7x - 3.5} - 10 b) j(x)=-2sqrt{12x} + 4

Question
Transformations of functions
What transformations of the parent graph of
$$\displaystyle{f{{\left({x}\right)}}}=\sqrt{{{x}}}$$
produce the graphs of the following functions?
a) $$\displaystyle{m}{\left({x}\right)}=\sqrt{{{7}{x}\ -\ {3.5}}}\ -\ {10}$$
b) $$\displaystyle{j}{\left({x}\right)}=-{2}\sqrt{{{12}{x}}}\ +\ {4}$$

2021-01-07
Step 1
First we are going to graph the prent function and the first transformation and second transformation of parent function.

Step 2
As we can see from the graph the first transformation of parent function has horizontal shift of 0.5 units to the right, and vertical shift of 10 units down. Vertical stretch was also preformed on the first function.
The second transformation of parent function has been reflected across x-axis and vertically shifted up 4 units. Horizontal compression was also preformed on the for the second function.

### Relevant Questions

In the following items, you will analyze how several transformations affect the graph of the function $$\displaystyle{f{{\left({x}\right)}}}={\frac{{{1}}}{{{x}}}}$$. Investigate the graphs of $$\displaystyle{f{{\left({x}\right)}}}={\frac{{{1}}}{{{x}}}},{g{{\left({x}\right)}}}={f{{\left({x}\right)}}}={\frac{{{1}}}{{{x}+{2}}}},{h}{\left({x}\right)}={\frac{{{1}}}{{{x}-{2}}}},{p}{\left({x}\right)}={\frac{{{1}}}{{{x}-{4}}}}\ \text{and}\ {z}{\left({x}\right)}={\frac{{{1}}}{{{x}^{{{2}}}+{1}}}}$$. If you use a graphing calculator, select a viewing window $$\displaystyle\pm{23.5}$$ for x and $$\displaystyle\pm{15.5}$$ for y. At what values in the domain did vertical asymptotes occur for each of the functions? Explain why the vertical asymptotes occur at these values.
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.
$$\displaystyle{y}={1}-{2}\sqrt{{{x}}}+{3}$$
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.
$$\displaystyle{g{{\left({x}\right)}}}=\ -{2}{\left|{x}\ -\ {1}\right|}\ -\ {4}$$
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.$$\displaystyle{g{{\left({x}\right)}}}={\frac{{{1}}}{{{3}}}}{\left({x}-{2}\right)}^{{{3}}}$$
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.
$$\displaystyle{h}{\left({x}\right)}={\left(-{x}\right)}^{{{2}}}-{8}$$
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. $$\displaystyle{y}={x}^{{{2}}}-{4}{x}+{5}$$
Begin by graphing $$\displaystyle{f{{\left({x}\right)}}}={{\log}_{{{2}}}{x}}$$ 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 function's domain and range. $$\displaystyle{r}{\left({x}\right)}={{\log}_{{{2}}}{\left(-{x}\right)}}$$
Begin by graphing $$\displaystyle{f{{\left({x}\right)}}}={{\log}_{{{2}}}{x}}$$ Then use transformations of this graph to graph the given function. What is the vertical asymptote? Use the graphs to determine each functions domain and range. $$\displaystyle{g{{\left({x}\right)}}}=\ -{2}{{\log}_{{{2}}}{x}}$$
Begin by graphing $$\displaystyle{f{{\left({x}\right)}}}=\ {{\log}_{{{2}}}{x}}$$ Then use transformations of this graph to graph the given function. What is the vertical asymptote? Use the graphs to determine each function's domain and range. $$\displaystyle{g{{\left({x}\right)}}}={\frac{{{1}}}{{{2}}}}\ {{\log}_{{{2}}}{x}}$$
Begin by graphing $$\displaystyle{f{{\left({x}\right)}}}=\ {{\log}_{{{2}}}{x}}$$ Then use transformations of this graph to graph the given function. What is the vertical asymptote? Use the graphs to determine each function's domain and range. $$\displaystyle{g{{\left({x}\right)}}}=\ -{2}\ {{\log}_{{{2}}}{x}}$$