## College Algebra (11th Edition) The parent function is $f(x)=2^x$ (with red) the given function is $g(x)=2^{x+2}-4$ (with blue). The parent function can be graphed by calculating a few coordinates and connecting them with a smooth curve: $f(-2)=2^{-2}=\frac{1}{4}$ $f(-1)=2^{-1}=\frac{1}{2}$ $f(0)=2^0=1$ $f(1)=2^1=2$ $f(2)=2^2=4$ For every corresponding x-value the following equation is true: $f(x+2)-4=g(x)$ This means that the graph is translated 2 units to the left left and 4 units down ($g(x)$ involves a horizontal shift of 2 to the left and also a vertical shift of 4 downwards.). First, the horizontal shift. We only consider the g(x) function as $g'(x)=2^{x+2}$ For example if $f(0)=1$ in the original $f(x)$, this will be equal to $g'(-2)=f(-2+2)=f(0)=1$. Here,$f(0)=g'(-2)$ also, $f(1)=g'(-1)$ We can see that here, each point in the parent function was moved to the left by 2 units. Second, we translate this $g'(x)$ function to get the originally given function. Now, the following equation is true: $g'(x)-4=g(x)$ Every $g'(x)$ value will be decreased by 4. For example if $g'(-2)=1$, this will be translated as $g'(-2)-4=1-4=-3$. We can see that here, the $g'(x)$ is greater than $g(x)$ for every corresponding x-value by 4.)