Notes: `(3,27) , (5,243)` Write an exponential function `y=ab^x` whose graph passes through the given points.

Friday, 29 September 2017

$\displaystyle{\left({3},{27}\right)},{\left({5},{243}\right)}$ Write an exponential function $\displaystyle{y}={a}{{b}}^{{x}}$ whose graph passes through the given points.

The given
two points of the exponential function are (3,27) and (5,243).

To determine
the exponential function

$\displaystyle{y}={a}{{b}}^{{x}}$

plug-in the given x and
y values.

For the first point (3,27), the values of x and y are x=3 and y=27.
Plugging them, the exponential function becomes:

$\displaystyle{27}={a}{{b}}^{{3}}$ (Let this be
EQ1.)

For the second point (5,243), the values of x and y are x=5 and y=243.
Plugging them, the function becomes:

$\displaystyle{243}={a}{{b}}^{{5}}$ (Let this be
EQ2.)

To solve for the values of a and b, apply substitution method of system
of equations. To do so, isolate the a in EQ1.

$\displaystyle{27}={a}{{b}}^{{3}}$

$\displaystyle\frac{{27}}{{{b}}^{{3}}}={a}$

Plug-in this to EQ2.

$\displaystyle{243}={a}{{b}}^{{5}}$

$\displaystyle{243}=\frac{{27}}{{{b}}^{{3}}}\cdot{{b}}^{{5}}$

And, solve for
b.

$\displaystyle{243}={27}{{b}}^{{2}}$

$\displaystyle\frac{{243}}{{27}}={{b}}^{{2}}$

$\displaystyle{9}={{b}}^{{2}}$

$\displaystyle\pm\sqrt{{9}}={b}$

$\displaystyle\pm{3}={b}$

Take note that in the exponential function $\displaystyle{y}={a}{{b}}^{{x}}$ , the b should be greater than zero
$\displaystyle{\left({b}\gt{0}\right)}$ . When $\displaystyle{b}\leq{0}$ , it is no longer an exponential function.

So,
consider only the positive value of b which is 3.

Now that the value of b is
known, plug-in it to EQ1.

$\displaystyle{27}={a}{{b}}^{{3}}$

$\displaystyle{27}={a}{{\left({3}\right)}}^{{3}}$

And, solve for a.

$\displaystyle{27}={27}{a}$

$\displaystyle\frac{{27}}{{27}}={a}$

$\displaystyle{1}={a}$

Then, plug-in a=1 and b=3 to

$\displaystyle{y}={a}{{b}}^{{x}}$

So this becomes

$\displaystyle{y}={1}\cdot{{3}}^{{x}}$

$\displaystyle{y}={{3}}^{{x}}$

Therefore, the exponential function that
passes the given two points is $\displaystyle{y}={{3}}^{{x}}$ .

Notes

Friday, 29 September 2017

$\displaystyle{\left({3},{27}\right)},{\left({5},{243}\right)}$ Write an exponential function $\displaystyle{y}={a}{{b}}^{{x}}$ whose graph passes through the given points.

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In 1984, is Julia a spy? Please provide specific examples from the book. My teacher says that he knows of 17 pieces of evidence which proves that Julia...

Friday, 29 September 2017

Write an exponential function whose graph passes through the given points.

No comments:

Post a Comment

In 1984, is Julia a spy? Please provide specific examples from the book. My teacher says that he knows of 17 pieces of evidence which proves that Julia...

$\displaystyle{\left({3},{27}\right)},{\left({5},{243}\right)}$ Write an exponential function $\displaystyle{y}={a}{{b}}^{{x}}$ whose graph passes through the given points.