If you're seeing this message, it means we're having trouble loading external resources on our website.

If you're behind a web filter, please make sure that the domains *.kastatic.org and *.kasandbox.org are unblocked.

### Course: Algebra 2 (Eureka Math/EngageNY)>Unit 3

Lesson 9: Topic C: Lessons 17-22: Graphs of exponential and logarithmic functions

# Graphing exponential functions

Sal graphs y=-2*3ˣ+5 using our interactive graphing widget.

## Want to join the conversation?

• I csnt really understand when should the asymptote be parallel to x axis and when will it be parallel to y axis. This is one of the hardest math lessons i had ever seen
• An asymptote that is parallel to the x-axis is a horizontal asymptote. An asymptote that is parallel to the y-axis is a vertical asymptote.
• I don't understand why will y approach 5 as x gets negative?
• -

We won't know how a graph is shaped until we know how it responds to certain values of x. (This is why it's good to use a table when working on graphs.)
Here we have y = -2 * 3^x + 5.
We can use a table to find the graph's shape, but we can also observe what happens to the graph as x approaches positive or negative infinity.
3 to the power of super negative numbers almost equals zero, so -2 * 3(-oo) will be a number slightly less than zero. Numbers like this, plus the remaining 5, will approach 5.
• but doesn't the ( -2) indicate that the new function must be reflected across the x-axis? i see no reflections at all..
• Yes, the -2 indicates that the new function must be reflected across the x-axis, but what your forgetting is that this is a transformation from the parent graph, in this case, is the exponential graph. If you look at the standard exponential graph, it basically looks like this graph but flipped over the x-axis and a horizontal aymptote of 0 instead of 5. Hope this helps.
• How on earth does -0-1 equal -1 and not +1?
• First, -0 is just 0.
Start at 0 and move 1 unit to the left. You end up on -1, not +1.

Hope this helps.
• Why wasn't it flipped over the x-axis and stretched vertically by a factor of -2? and what does it mean to stretch vertically by a factor of a number? for example in this case it would be stretched vertically over the x-axis by a factor of -2. There are problems in the practice that say that
• Can someone help me understand how to graph a function where y=f^z-x? I understand that I'm supposed to start by rewriting them as y=f^(-(x-z)), but every time I attempt to graph those specific functions, I always seem to get it wrong.
• Ok, so y = f^(-(x-z)) can be graphed quite simply with a few rules to follow. Firstly, the sign in front of the exponent (which is - in this case) means that it is reflected upon the y axis. As for -z, the function is shifted z units towards the right. Remember, you always want the function to be zero, which means if you have z, you move -z units left. If you have -z, you move z units right. I hope this helped!
• I dont understand why is the asymptote on 5 rather than just moving the whole function 5 upwards? It may be a dumb question but please, any help is appreciated.
• Both are true. An unshifted exponential function has an asymptote on y=0. Since the function is shifted up five, its entire graph, including its asymptote, is shifted up five, so the new asymptote is at y=5.
• how do you know what corresponding points to use when trying to graph a function
someone pleeeeease heeeeeelp me because im very stuck on my problems!
• It partially depends on the function itself, this particular video deals with exponential functions. If you think about how functions move g(x) = a f(x-h)+k, life is easier if you think of values around h (but also depending on a to eliminate fractions). Thus, there is not one easy answer, but hints like above will give some of best values.
• Would the range be y>(the asymptote) and also how would we figure out the anchor points to graph?
• Since you have a -2 as a multiplier, it reflects across x, so the range would be y< (asymptote). O and 1 as x values are generally good points unless there is a horizontal shift (due to channging x such as y = -2 (3)^(x-2) which moves equation 2 units ot the right, this would mean x values such as 1, 2, and/or 3 would be good points
• In {practice:Graphs of exponential } there's a point I don't understand : What's the different between 3^5-x and 3^x-5 ?For 3^x-5, in oder to make y equal to one , x need to be equal to 5, so I transform it 5 to the right.But in 3^5-x, in oder to make y equal to one, x also need to be equal to 5!So again I transform it 5 to the right!So isn't that means:
3^5-x = 3^x-5 ?

## Video transcript

- [Voiceover] We're told, use the interactive graph below to sketch a graph of y is equal to negative two, times three to the x, plus five. And so this is clearly an exponential function right over here. Let's think about the behavior as x is, when x is very negative, or when x is very positive. When x is very negative, three to a very negative number like let's say you had three to the negative 3rd power, that would be 1/27th, or three to the negative 4th power, that would be 1/81st. So this is going to get smaller, and smaller, and smaller. It's going to approach zero as x becomes more negative. And since this is approaching zero, this whole thing right over here is going to approach zero. And so this whole expression is, if this first part is approaching zero, then this whole expression is going to approach five. So we're going to have a horizontal asymptote that we're going to approach as we go to the left. As x gets more and more negative, we're going to approach positive five. And then as x gets larger, and larger, and larger, three to the x is growing exponentially. But then we're multiplying it times negative two, so it's going to become more, and more, and more negative. And then we add a five. And so what we have here, well this looks like a line. We want to graph an exponential. So let's go pick the exponential in terms of x. There you have it. And so we can move three things. We can move this point, it doesn't even need to just be the y-intercept, although that's a convenient thing to figure it out. We can move this point here. And we can move the asymptote. And maybe the asymptote's the first interesting thing. We said as x becomes more, and more, and more, and more negative, y is going to approach five. So let me put this up here. So that's our asymptote. It doesn't look like it quite yet, but when we try out some values for x and the corresponding y's, and we move these points accordingly, hopefully our exponential is going to look right. So let's think about, let's pick some convenient x's. So let's think about when x is equal to zero. If x is equal to zero, three to the zeroth power is one. Negative two times one is negative two, plus three is three. So when x is equal to zero, y is three. And let's think about when x is equal to one, and I'm just picking that 'cause it's easy to compute. Three to the first power is three, times negative two is negative six, plus five is negative one. So when x is one, y is negative one. And so let's see, does this, is this consistent with what we just described? When x is very negative, we should be approaching, we should be approaching positive five, and that looks like the case. As we move to the left, we're getting closer, and closer, and closer to five, in fact, it looks like they overlap, but it's really, we're just getting closer, and closer, and closer 'cause this term, this term right over here is getting smaller, and smaller, and smaller as x becomes more, and more, and more negative. But then as x becomes more and more positive, this term becomes really negative 'cause we're multiplying it times a negative two, and we see that it becomes really negative, so I feel pretty good about what we've just graphed. We've graphed the horizontal asymptote, it makes sense, and we've picked two points that sit on this, on the graph of this exponential. So I can check my answer, and we got it right.