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We can introduce another parameter \$k\$ into the definition of the exponential function, giving us two dials to play with.

If we call this parameter \$k\$, we can write our exponential function \$f\$ as \$\$f(x)=b^.\$\$ You can explore the influence of both parameters \$b\$ and \$k\$ in the following applet. You can change the parameters \$b\$ and \$k\$ by typing new values in the corresponding boxes.    We can still get the full range of functions if we eliminate either \$b\$ or \$k\$. For example, you can see that the function \$f(x)=3^\$ (\$k=2\$, \$b=3\$) is exactly the same as the function \$f(x)=9^x\$ (\$k=1\$, \$b=9\$).To change the value of \$f(0)\$, you can allow scaling of the function by clicking the corresponding checkbox.Then, the function changes to \$f(x)=c b^\$ with an additional parameter \$c\$ that scales (multiplies) the whole function so that \$f(0)=c\$.You can change the value of \$c\$ by dragging the red point.You can change range of the \$x\$ and \$y\$-axes buttons labeled \$x \$, \$x-\$, \$y \$, and \$y-\$.