Question

What transformations of the parent graph of f(x) = \sqrtc

Transformations of functions
ANSWERED
asked 2021-08-14
What transformations of the parent graph of \(\displaystyle{f{{\left({x}\right)}}}=\sqrt{{c}}\) produce the graphs of the following functions?
a) \(\displaystyle{m}{\left({x}\right)}=\sqrt{{{7}{x}-{3.5}}}-{10}\)
b) \(\displaystyle{j}{\left({x}\right)}=-{2}\sqrt{{{12}{x}}}+{4}\)

Answers (1)

2021-08-15

a)First, we have to rewrite the function m(x) in the form \(\displaystyle{m}{\left({x}\right)}=\sqrt{{{x}—{h}}}+{k}\)
So, the steps are: factor the radicand, use the Product Property of Radicals and than simplify, so, we have next:
\(\displaystyle{m}{\left({x}\right)}=\sqrt{{{7}{\left({x}-{0.5}\right)}}}-{10}\)
\(\displaystyle=\sqrt{{7}}\cdot\sqrt{{{x}-{0.5}}}-{10}\)
\(\displaystyle={2.65}\sqrt{{x}}={0.5}-{10}\)
So, the graph of m(x) is a vertical stretch of a parent function by a factor of 2.65 or \(\displaystyle\sqrt{{7}}\), translated right 0.5 units and translated down 10 units. On the following picture there is a graph of \(\displaystyle{y}=\sqrt{{x}}{\quad\text{and}\quad}{m}{\left({x}\right)}=\sqrt{{{7}{x}-{3.5}}}-{10}\)
image

b)First, we have to rewrite the function j(x) in the form: \(\displaystyle{j}{\left({x}\right)}={a}\sqrt{{{x}-{h}}}+{k}\) So, the steps are: use the Product Property of Radicals and than simplify, so we have next:
\(\displaystyle{j}{\left({x}\right)}=-{2}\sqrt{{{4}\cdot{3}\cdot{x}}}+{4}\)
\(\displaystyle=-{2}\sqrt{{4}}\cdot\sqrt{{3}}\cdot\sqrt{{x}}+{4}\)
\(\displaystyle=-{2}\cdot{2}\cdot\sqrt{{3}}\cdot\sqrt{{x}}+{4}\)
\(\displaystyle=-{4}\sqrt{{3}}\cdot\sqrt{{x}}+{4}\)
So, the graph of j(x) is a vertical stretch of a parent function by a factor of \(\displaystyle-{4}\sqrt{{3}}\), translated up 4 units.

image
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