Question

Write an exponential function y=ab^x for a graph that passes through (2,1) and (3,12)

Answer 1

Solution:

Given the general exponential function expressed as

$y=ab^x\text{ --- equation 1}$

If the function passes through the points (x₁, y₁) and (x₂, y₂), we can defined the exponential function by solving for a and b.

This is done by substituting the values of x and y into the general exponential function as shown below:

$\begin{gathered} y_1=a(b)^{x_1}\text{ ----- equation 2} \\ y_2=a(b)^{x_2}\text{ ------ equation 3} \end{gathered}$

Given that the graph of the exponential function passes through (2,1) and (3,12), this implies that

$\begin{gathered} x_1=2 \\ y_1=1 \\ x_2=3 \\ y_2=12 \end{gathered}$

Thus, substituting the x and y values into equation 1 , as done in equation 2 and 3, we have

$\begin{gathered} 1=a(b)^2\text{ ----- equation 4} \\ 12=a(b)^3\text{ ------ equation 5} \end{gathered}$

Divide equation 5 by equation 4,

$\begin{gathered} \frac{12}{1}=\frac{a(b)^3}{a(b)^2} \\ \Rightarrow12=b \end{gathered}$

Substitute the obtained value of b into either equation 4 or 5.

Substituting into equation 4, we have

$\begin{gathered} 1=a(12)^2\text{ } \\ divide\text{ both sides by }12^2 \\ \Rightarrow\frac{1}{12^2}=\frac{a(12)^2}{12^2} \\ a=\frac{1}{144} \end{gathered}$

Substitute the obtained values of a and b into equation 1.

From equation 1,

$\begin{gathered} y=a(b)^x \\ where \\ a=\frac{1}{144} \\ b=12 \end{gathered}$

Thus, the exponential function is expressed as

$y=\frac{1}{144}(12)^x$