Question

What point do all graphs of the form \(y=a^{x}\) have in common? Is there a point that all graphs of the form \(y=2(a)^{x}\) have in common? If so, name the point.

Short answer

The common point that all graphs of the form \(y=a^{x}\) have is (0,1). When considering the function \(y=2(a)^{x}\), the common point for these graphs is (0,2).

Step-by-step solution

Discuss the type of function

The given function \(y=a^{x}\) is an exponential function with base \(a\). This means that the value of \(y\) increases or decreases rapidly as the value of \(x\) changes. A key feature of graphs of exponential functions is that they all pass through the y-axis at a certain point.

Identify the common point

Regardless of the base \(a\) in the function \(y=a^{x}\), when \(x=0\), \(y\) always equals 1 since any non-zero number to the power of 0 is 1. Therefore, all graphs of form \(y=a^{x}\) pass through the point (0,1).

Apply the constant to the function

Now consider function \(y=2(a)^{x}\). This function is a variation of the original function, with the added constant 2. So for \(x=0\), \(y=2(a)^{0}=2*1=2\). Therefore, all graphs of the form \(y=2(a)^{x}\) pass through the point (0,2).