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writing exponential functions (4, 112/81), (-1, 21/2)

Sagot :

The given points are (4, 112/81) and (-1, 21/2).

To find an exponential function from the given points, we have to use the forms.

[tex]\begin{gathered} y_1=ab^{x_1} \\ y_2=ab^{x_2} \end{gathered}[/tex]

Now, we replace each point in each equation.

[tex]\begin{gathered} \frac{112}{81}=ab^4 \\ \frac{21}{8}=ab^{-1} \end{gathered}[/tex]

We solve this system of equations.

Let's isolate a in the second equation.

[tex]\begin{gathered} \frac{21}{8}=\frac{a}{b} \\ \frac{21b}{8}=a \end{gathered}[/tex]

Then, we replace it in the first equation

[tex]\frac{112}{81}=(\frac{21b}{8})\cdot b^4[/tex]

We solve for b.

[tex]\begin{gathered} \frac{112\cdot8}{81\cdot21}=b\cdot b^4 \\ \frac{896}{1701}=b^5 \\ b=\sqrt[5]{\frac{896}{1701}}=\frac{2\sqrt[5]{4}}{3} \\ b\approx0.88 \end{gathered}[/tex]

Once we have the base of the exponential function, we look for the coefficient a.

[tex]a=\frac{21b}{8}=\frac{21}{8}(\frac{2\sqrt[5]{4}}{3})=\frac{7\sqrt[5]{4}}{4}[/tex]

Therefore, the exponential function is

[tex]y=\frac{7\sqrt[5]{4}}{4}\cdot(\frac{2\sqrt[5]{4}}{3})^x[/tex]

The image below shows the graph of this function.

View image UbaidullahN168828
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