in this lesson we're going to focus on graphing exponential functions so what we need to do is pick two points let's use zero and one anything raised to the zero power is always one and two to the first power is itself two you can think of this as two to the x plus zero this number here is the horizontal asymptote so if you don't see a number there the horizontal asymptote is the x-axis so y is equal to zero that's the horizontal asymptote so now let's go ahead and graph it so the first point is 0 1 that is the y-intercept and the next point is 1 comma 2. so we're going to start from the horizontal asymptote which is the x-axis and then it's going to increase exponential functions they increase at an increase in rate and that's how you can graph it now what is the domain of the function and what is the range so the domain has to do with the x values the lowest x value is negative infinity and the highest is infinity so for all exponential functions the domain will be all real numbers now the range is limited the range has to deal with the horizontal asymptote the lowest y value is zero the highest y value is infinity so the range is from zero to infinity but it does not include zero the graph never touches the horizontal asymptote it gets very close to it but never touches it there's no value that you can plug in for x that will give you a y value of zero it just doesn't happen here's the next example graph this one 1 over 3 raised to the x so if you had a fraction like this what should we do the best thing to do is to rewrite it if we move the three to the top positive x will change to negative x and so that's it now let's set negative x equal to two things zero and one if you solve and find the value of x x will be zero and negative one those are the values that you want to plug in so if we plug in zero three to the negative zero is simply one and if we plug in negative 1 into this equation 3 raised to the negative negative 1 that's going to be 3 to the 1 which is equal to 3. now the horizontal asymptote is this number so it's still y equals zero which means it's the x axis so if we plot it we have the point zero one which is the y intercept and negative one three so the graph is going to start from the x-axis and then it's going to follow those two points let's do that again so that's the graph the domain is going to be all real numbers unlike the other one the range the y values vary from 0 to infinity so that's going to be the range let's try this example three raised to the x minus two plus one so what we're going to do is set x minus two equal to two things zero and one and let's solve the value of x when x minus two is zero is going to be two if we add two to both sides one plus two is three so those are the two x values that we're gonna use so what is the value of y when x is two so three raised to the two minus two plus one two minus two is zero and three to the zero is one one plus one is two now let's plug in three three minus two is one three to the first power is three three plus one is four so those are the points of interest the horizontal asymptote is based on this number so its y is equal to one now let's go ahead and graph it so we have a horizontal asymptote at y equals one and we have the points two comma two and three comma four so here's two comma two and this is the point three four so it's going to start from the horizontal asymptote and then it's gonna increase towards those points and so that's how you can graph it as always the domain is all real numbers and as for the range it starts from the horizontal asymptote of one and it goes to infinity the lowest y value is when the highest is infinity so the range is from one to infinity and so that's it for this one let's try one more example five minus two raised to the three minus x so to find the two points that we need let's set three minus x equal to zero therefore that means x is equal to three and if we set it equal to one three minus two is one so x is two now let's make a table and let's use the points two and three so what is y when x is two three minus two is one and two to the first power is two so five minus two is equal to three now what is y when x is three three minus three is zero two to the zero is one five minus one is four now what is the horizontal asymptote the horizontal asymptote is the constant in the equation in this case it's y is equal to 5. now we can graph it so let's start with the horizontal asymptote f5 so we have the point two comma three and also three comma four so the graph is going to start from the horizontal asymptote and then it's gonna follow the two points so it's decreasing in this case due to the negative sign in front and that's how you can graph it the domain is all real numbers negative infinity to infinity now what is the range notice that the lowest y value is negative infinity the highest is positive five so the range is from negative infinity to five now let's review four generic shapes of exponential functions the first one we've considered 2 raised to the x so this function increases towards quadrant one the next one is two raised to the negative x and this graph reflects over the y axis so it increases towards quadrant two next we have negative two raised to the positive x this graph reflects over the x-axis and so it travels towards quadrant form and then finally negative two raised to the negative x this graph reflects across the origin relative to the first one and so it travels towards quadrant three so here's what i like to do to keep this in mind this is the way i remember it we have positive x and positive y positive x travels towards the right positive y travels upward so therefore this graph is going to go towards quadrant one now the next one which goes towards quadrant two notice that x is negative and y is positive x is negative towards the left y is positive going up so you're heading towards quadrant two in the third example we're going towards quadrant four x is positive y is negative x is positive towards the right y is negative as you go down so we're going towards quadrant four and the last one which is towards quadrant three both x and y are negative x is negative towards left y is negative as you go down so we're heading towards quadrant dream