Notes: `(4,8) , (8,30)` Write a power function `y=ax^b` whose graph passes through the given points

Thursday, 1 May 2014

$\displaystyle{\left({4},{8}\right)},{\left({8},{30}\right)}$ Write a power function $\displaystyle{y}={a}{{x}}^{{b}}$ whose graph passes through the given points

To determine the power function $\displaystyle{y}={a}{{x}}^{{b}}$ from the given coordinates: $\displaystyle{\left({4},{8}\right)}$ and $\displaystyle{\left({8},{30}\right)}$ , we set-up system of equations by plug-in the values of $\displaystyle{x}$ and $\displaystyle{y}$ on $\displaystyle{y}={a}{{x}}^{{b}}$ .

Using the coordinate $\displaystyle{\left({4},{8}\right)}$ , we let $\displaystyle{x}={4}$ and $\displaystyle{y}={8}$ .

First equation: $\displaystyle{8}={a}\cdot{{4}}^{{b}}$

Using the coordinate $\displaystyle{\left({8},{30}\right)}$ , we let $\displaystyle{x}={8}$ and $\displaystyle{y}={30}$ .

Second equation: $\displaystyle{30}={a}\cdot{{8}}^{{b}}$

Isolate "a" from the first equation.

$\displaystyle{8}={a}\cdot{{4}}^{{b}}$

$\displaystyle\frac{{8}}{{{4}}^{{b}}}=\ldots$

Using the coordinate $\displaystyle{\left({4},{8}\right)}$ , we let $\displaystyle{x}={4}$ and $\displaystyle{y}={8}$ .

First equation: $\displaystyle{8}={a}\cdot{{4}}^{{b}}$

Using the coordinate $\displaystyle{\left({8},{30}\right)}$ , we let $\displaystyle{x}={8}$ and $\displaystyle{y}={30}$ .

Second equation: $\displaystyle{30}={a}\cdot{{8}}^{{b}}$

Isolate "a" from the first equation.

$\displaystyle{8}={a}\cdot{{4}}^{{b}}$

$\displaystyle\frac{{8}}{{{4}}^{{b}}}=\frac{{{a}\cdot{{4}}^{{b}}}}{{{4}}^{{b}}}$

$\displaystyle{a}=\frac{{8}}{{{4}}^{{b}}}$

Plug-in $\displaystyle{a}=\frac{{8}}{{{4}}^{{b}}}$ on $\displaystyle{30}={a}\cdot{{8}}^{{b}}$ , we get:

$\displaystyle{30}=\frac{{8}}{{{4}}^{{b}}}\cdot{{8}}^{{b}}$

$\displaystyle{30}={8}\cdot\frac{{{8}}^{{b}}}{{{4}}^{{b}}}$

$\displaystyle{30}={8}\cdot{{\left(\frac{{8}}{{4}}\right)}}^{{b}}$

$\displaystyle{30}={8}\cdot{{\left({2}\right)}}^{{b}}$

$\displaystyle\frac{{30}}{{8}}=\frac{{{8}\cdot{{\left({2}\right)}}^{{b}}}}{{8}}$

$\displaystyle\frac{{15}}{{4}}={{2}}^{{b}}$

Take the "ln" on both sides to bring down the exponent by applying the

natural logarithm property: $\displaystyle{\ln{{\left({{x}}^{{n}}\right)}}}={n}\cdot{\ln{{\left({x}\right)}}}$ .

$\displaystyle{\ln{{\left(\frac{{15}}{{4}}\right)}}}={\ln{{\left({{2}}^{{b}}\right)}}}$

$\displaystyle{\ln{{\left(\frac{{15}}{{4}}\right)}}}={b}\cdot{\ln{{\left({2}\right)}}}$

Divide both sides by $\displaystyle{\ln{{\left({2}\right)}}}$ to isolate b.

$\displaystyle\frac{{{\ln{{\left(\frac{{15}}{{4}}\right)}}}}}{{\ln{{\left({2}\right)}}}}=\frac{{{b}\cdot{\ln{{\left({2}\right)}}}}}{{{\ln{{\left({2}\right)}}}}}$

$\displaystyle{b}=\frac{{{\ln{{\left(\frac{{15}}{{4}}\right)}}}}}{{\ln{{\left({2}\right)}}}}{\quad\text{or}\quad}{1.91}$ (approximated value).

Plug-in $\displaystyle{b}={1.91}$ on $\displaystyle{a}=\frac{{8}}{{{4}}^{{b}}}$ , we get:

$\displaystyle{a}=\frac{{8}}{{{4}}^{{1.91}}}$

$\displaystyle{a}\approx{0.566}$ (approximated value)

Plug-in $\displaystyle{a}\approx{0.566}$ and $\displaystyle{b}\approx{1.91}$ on $\displaystyle{y}={a}{{x}}^{{b}}$ , we get the power function as:

$\displaystyle{y}={0.566}{{x}}^{{1.91}}$

Notes

Thursday, 1 May 2014

$\displaystyle{\left({4},{8}\right)},{\left({8},{30}\right)}$ Write a power function $\displaystyle{y}={a}{{x}}^{{b}}$ whose graph passes through the given points

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How are race, gender, and class addressed in Oliver Optic's Rich and Humble?

Thursday, 1 May 2014

Write a power function whose graph passes through the given points

No comments:

Post a Comment

How are race, gender, and class addressed in Oliver Optic&#39;s Rich and Humble?

$\displaystyle{\left({4},{8}\right)},{\left({8},{30}\right)}$ Write a power function $\displaystyle{y}={a}{{x}}^{{b}}$ whose graph passes through the given points

How are race, gender, and class addressed in Oliver Optic's Rich and Humble?