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 \(k .\) $$(1,1.5),(-1,6)$$

Short answer

The values for \(a\) and \(k\) are 3 and 0.5 respectively.

Step-by-step solution

Formulate Equations from Points

The function is given by \(f(x) = k \cdot a^{x}\). We know that the graph of the function passes through the points (1,1.5) and (-1,6). We can create two equations from these two points.\nFrom the first point, substitute \(x = 1\) and \(f(x) = 1.5\) into the function equation to get \(1.5 = k \cdot a\).\nFrom the second point, substitute \(x = -1\) and \(f(x) = 6\) into the function equation to get \(6 = k \cdot a^{-1} = \frac{k}{a}\).

Solve for a

We have a system of two equations, \(1.5 = k \cdot a\) and \(6 = \frac{k}{a}\), and two unknowns, \(a\) and \(k\). \nMultiply the two equations to eliminate \(k\). The result is \(1.5 \cdot 6 = a \cdot \frac{k}{a} \cdot k \cdot a\), which simplifies to \(9 = a^{2}\). Take the square root on both sides to find \(a = \pm3\). Because \(a > 0\) in an exponential function, we can discard the negative value. So, \(a = 3\).

Solve for k

Now, substitute \(a = 3\) back into the first equation to find \(k\). This gives \(1.5 = k \cdot 3 \Rightarrow k = \frac{1.5}{3} = 0.5\).