Question

In Exercises 48 and 49 , assume that the graph of the exponential function \(f(x)=k \cdot a^{x}\) passes through the two points. Find the values of \(a\) and \(k .\) $$(1,4.5),(-1,0.5)$$

Short answer

The values of \(a\) and \(k\) for the function \(f(x)=k \cdot a^{x}\) that passes through the points \((1,4.5),(-1,0.5)\) are \(a=3\) and \(k=1.5\).

Step-by-step solution

Create Equation from Given Points

First, substituting the two points \((1,4.5)\) and \((-1,0.5)\) into the general equation \(f(x)=k \cdot a^{x}\), we get two equations as follows: \[4.5=k \cdot a\] \[0.5=k \cdot a^{-1}\]

Solve for a

To find the value of \(a\), we can divide the first equation by the second equation. That gives us \[9=a^{2}\], and by taking the square root on both sides, we get \(a= ±3\). However, because \(a\) is the base of the exponential and it must be positive, so \(a=3\).

Solve for k

To find the value of \(k\), we can substitute \(a=3\) into the first equation \(4.5=3k\). Divide both sides by 3, we get \(k=1.5\).