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# To determine. The correct graph for the function[g(x)=-frac{1}{2}f(x)+1 is B # To determine. The correct graph for the function[g(x)=-frac{1}{2}f(x)+1 is B

Question
Transformation properties asked 2021-02-19
To determine.
The correct graph for the function$$\displaystyle{\left[{g{{\left({x}\right)}}}=-{\frac{{{1}}}{{{2}}}}{f{{\left({x}\right)}}}+{1}\right.}$$ is B

## Answers (1) 2021-02-20

Vertical translation:
For $$\displaystyle{b}\ {>}\ {0}$$,
The graph of $$\displaystyle{y}={f{{\left({x}\right)}}}\ +\ {b}$$ is the graph of $$\displaystyle{y}={f{{\left({x}\right)}}}$$ {shifted up b units}
The graph of $$\displaystyle{y}={f{{\left({x}\right)}}}\ -\ {b}$$ is the graph of $$\displaystyle{y}={f{{\left({x}\right)}}}\ +\ {b}$$ {shifted down b units}
Reflection:
Across the x-axis:
The graph of $$\displaystyle{y}=\ -{f{{\left({x}\right)}}}$$ is the reflection of the graph of $$\displaystyle{y}={f{{\left({x}\right)}}}$$ across the x-axis
Across the y-axis:
The graph of $$\displaystyle{y}=\ {f{{\left(-{x}\right)}}}$$ is the reflection of the graph of $$\displaystyle{y}={f{{\left({x}\right)}}}$$ across the x-axis
Vertically Stretching or Shrinking:
The graph of $$\displaystyle{y}=\ {a}{f{{\left(-{x}\right)}}}$$ can be obtained from the graph of $$\displaystyle{y}={f{{\left({x}\right)}}}$$ by
Stretching verttically for $$\displaystyle{m}{i}{d}{a}{m}{i}{d}\ {>}\ {1}$$ or shrinking vertically for $$\displaystyle{a}\ {>}\ {m}{i}{d}{a}{m}{i}{d}\ {>}\ {1}$$.
For $$\displaystyle{a}{<}{0}$$, the graph is also reflected across the x-axis
. By the properties of transformation, the graph of $$\displaystyle{g{{\left({x}\right)}}}=-{\frac{{{1}}}{{{2}}}}{f{{\left({x}\right)}}}+{1}$$ is,
The transformation of $$\displaystyle{y}={f{{\left({x}\right)}}}$$ and shri
vertically by a factor of $$\displaystyle{\frac{{{1}}}{{{2}}}}$$
Then the graph is reflection about the x-axis and shifted up one unit.
From the given graphs, the graphs B satisfies the all condition's stated above.
Thus, the correct graph for the function $$\displaystyle{g{{\left({x}\right)}}}=-{\frac{{{1}}}{{{2}}}}{f{{\left({x}\right)}}}+{1}$$ is B

### Relevant Questions asked 2021-05-16
Consider the curves in the first quadrant that have equationsy=Aexp(7x), where A is a positive constant. Different valuesof A give different curves. The curves form a family,F. Let P=(6,6). Let C be the number of the family Fthat goes through P.
A. Let y=f(x) be the equation of C. Find f(x).
B. Find the slope at P of the tangent to C.
C. A curve D is a perpendicular to C at P. What is the slope of thetangent to D at the point P?
D. Give a formula g(y) for the slope at (x,y) of the member of Fthat goes through (x,y). The formula should not involve A orx.
E. A curve which at each of its points is perpendicular to themember of the family F that goes through that point is called anorthogonal trajectory of F. Each orthogonal trajectory to Fsatisfies the differential equation dy/dx = -1/g(y), where g(y) isthe answer to part D.
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(b) Find the probability that arandomly selected sample of Pima clay loam will have bulk densityless than $$\displaystyle{0.9}\frac{{g}}{{c}}{m}^{{3}}$$.
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(d) What point has the property that only 10% of the soil samples have bulk density this high orhigher?
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b)The domain of the function $$g{{\left({x}\right)}}={2}^{{{x}-{4}}}$$ in the interval notation.
c)The range of the function $$g{{\left({x}\right)}}={2}^{{{x}-{4}}}$$ in the interval notation.
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