Transforming exponential graphs (example 2) | Mathematics III | High School Math | Khan Academy

We're told the graph of y equals 2 to the x is shown below. So that's the graph; it's an exponential function. Which of the following is the graph of y is equal to negative 1 times 2 to the x plus 3 plus 4? They give us 4 choices down here, and before we even look closely at those choices, let's just think about what this would look like if it was transformed into that.

You might notice that what we have here, this y that we want to find the graph of, is a transformation of this original one. How do we transform it? Well, we replaced x with x plus 3, then we multiplied that by negative 1, and then we add 4.

So let's take it step by step. This is y equals 2 to the x. What I want to do next is let's graph y is equal to 2 to the x plus 3 power. Well, if you replace x with x plus 3, you're going to shift the graph to the left, to the left, by three. That might be a little bit counter-intuitive, but when we actually think about some points, it'll hopefully make some sense.

Here, for example, in our original graph when x is equal to zero, y is equal to one. Well, how do we get y equal? How do we get y equal one for our new graph for this thing right over here? Well, to get y equals one here, the exponent here still has to be zero. So that's going to happen at x equals negative three. So that's going to happen at x equals negative 3, y is equal to 1.

So notice we shifted to the left by 3. Likewise, in our original graph, when x is 2, y is 4. Well, how do we get y equals 4 in this thing right over here? Well, for y to be equal to 4, this exponent here needs to be equal to 2. For this exponent to be equal to 2 (because 2 squared is 4), x is going to be equal to negative 1.

So when x is equal to negative 1, y is equal to 4. Notice we shifted to the left by 3. This thing, which isn't our final graph that we're looking for, is going to look something like, it's going to look something like, something like that - shifted.

Now let's figure out what the graph of, now let's multiply this expression times negative 1. Notice we're slowly building up to our goal, so now let's figure out the graph of y is equal to negative one times two to the x plus 3.

Here, when y equals two to the x plus 3, if we multiply that times negative one, whatever y we had, we're gonna have the negative of that. So instead of when x is equal to negative 3 having positive 1, when x equals negative 3, you're going to have negative 1. We multiply by negative 1.

When x is equal to negative 1, instead of having 4, you're going to have negative 4. So our graph is going to be flipped over; it's flipped over the x-axis and it's going to look something, something, something like this. This is not a perfect drawing, but it'll give us a sense of things.

And we can look at which of these graphs match up to that. Finally, we want to add that 4 there, so we want to figure out the graph of y equals negative 1 times 2 to the x plus 3 plus 4. We want to take what we just had and shift it up by four.

So instead of this being a negative one right over here, this is going to be a negative one plus four, which is three. Instead of this being a negative four, negative four plus 4 is 0. Instead of our horizontal asymptote being at y equals 0, our horizontal asymptote is going to be at y equals 4.

Our graph is going to look something like, we're going to look something like, like this; we just shifted that red graph up by four - shifted it up by four - and we have a horizontal asymptote at y equals four.

So let's look at which of these choices match that. So choice A right over here has a horizontal asymptote y equals four, but it is shifted on the horizontal direction inappropriately. In fact, it looks like it might have not been shifted to the left, so we can rule this one out.

This one over here, well, this one just - this one approaches our asymptote as x increases, so that's not right. It should approach our asymptote as x decreases, so we rule that one out. Choice C looks like what we just graphed: horizontal asymptote at y equals four. When x equals negative three, y is equal to three. That's what we got. When x is equal to negative one, y is equal to zero.

So this looks right, and you can even try those points out. We like choice C, and D is clearly off.