 So a horizontal stretch or compression of an exponential graph changes the base of the exponential function. What about a horizontal translation? In this case, something very interesting happens because we're working with an exponential function. Let's take a look. So let's try to describe the transformations needed to produce the graph of y equals 2 to power x minus 3 from the graph of y equals 2 to the x, and then identify a different transformation that will produce the same graph from y equals 2 to power x. Since this is the graph of 2 to power x minus 3, we might start as follows. We take the graph of y equals 2 to the x and shift it horizontally to the right by 3 units. Now what happens if we apply our rule of exponents? We note that y equals 2 to power x minus 3. So remember the product rule for exponential expressions. This means that 2 to power x minus 3 is the same as 2 to the x times 2 to power minus 3. And we can simplify that to y equals 1 eighth 2 to power x. And so this means we could also take the graph of y equals 2 to power x and apply a vertical stretch by a factor of 1 eighth. And so there's two ways we can produce this graph. We can take a graph of y equals 2 to power x and shift it horizontally to the right by 3 units. Or we can take the graph of y equals 2 to power x and apply a vertical stretch by a factor of 1 eighth. We can also go backwards. So let's say the graph shown is produced by a horizontal transformation of y equals 2 to the x. Let's find the equation and describe the transformation. Now a horizontal translation will produce the graph of y equals 2 to power x minus h. Now we see the graph goes through the point 0, 4. This means that x equals 0, y equals 4 makes the equation true. And that means we can solve it for h giving us h equals negative 2. And so the equation is y equals 2 to power x plus 2. And the graph is produced by a horizontal translation of 2 units to the left. So we can take our graph of y equals 2 to the x and shift it horizontally by 2 units to the left. What if we include a vertical and a horizontal translation? The first useful thing to notice here is we see the asymptote is y equals 7. Since the graph of y equals 3 to power x has asymptote of y equals 0, the graph has been shifted upward by 7 units. And so this is the graph of y equals 3 to the x that has been shifted upward 7 units to produce the graph of y equals 3 to the x plus 7. Now if the graph undergoes a horizontal shift, its equation will be y equals 3 to power x minus h plus 7. Since the graph goes through 5, 10, we know that x equals 5, y equals 10 makes this equation true. So we'll substitute those values in and solve for h getting h equal to 4. And so we know this graph is shifted 4 units to the right to produce the graph of y equals 3 to power x minus 4 plus 7. So we can take the graph of y equals 3 to the x, shift upward by 7 units, then right by 4 units.