### Home > PC3 > Chapter 13 > Lesson 13.3.3 > Problem13-133

13-133.

An exponential function has an asymptote at $y = 10$. The function passes through the points $\left(4, 6\right)$ and $\left(7, 8\right)$. Determine the $y$-intercept for the function exactly and then approximate it to three decimal places.

Write the exponential form:

$y = \left(a\right)b^{x} + 10$

Substitute each point in for $\left(x, y\right)$ forming two equations and two unknowns.

$6 = \left(a\right)\left(b\right)^{4} + 10$
$8 = \left(a\right)\left(b\right)^{7} + 10$

Solve for '$a$' and '$b$' and substitute the contants into the function.

Using the new function, substitute '$0$' for '$x$' and simplify. This is the $y$-intercept.