in this video we're going to focus on graphing exponential functions including functions with base e but we'll cover that later in a video so let's start with a simple function y is equal to 2 ra to X power how can we graph this particular function what would you do to graph it the best thing to do is to make a table and you want to plug in the right points personally I would pick a zero and one now this function has a horizontal ASM toote the horizontal ASM toote is the x-axis it's y equal 0 now if you see a number like plus three the horizontal ASM toote will shift three units up so it's now y equal 3 so if the function was 2 to the x minus let's say 4 the horizontal ASM toote is y4 but if you don't see that number it's then it's y equal 0 it's the xaxis now if we plug in zero into the function 2 to the 0 power is 1 anything raised to Z power is one two to the first power is two and then simply plot the points so we have the0 01 and 1 comma 2 now start from the xaxis that is the horizontal ASM toote and connect a graph to the two points so that's the graph of the function that's all you got to do now what are the domain and range of this function let's start with the domain how can we write it in interval notation the domain of any function represents the X values that the function can have for exponential functions it's always going to be negative Infinity to Infinity the lowest x value that we see on the left side is negative infinity and the function can keep going to the right there's no restrictions on the value of x we can plug in any value for x now what about the range the range is the Y values that the function can have the lowest yvalue is zero it starts from the horizontal asmo the highest y value it goes up to Infinity so it's from zero to Infinity but it does not include zero there's no value of x that you can plug in that will make y equal to Zer so this is the range let's try another example let's say that Y is equal to 3 x + 1 - 2 so the minus 2 tells you that the graph has been shifted down two units so the horizontal ASM toote is y is equal to -2 now x + 1 that's tells you that the graph has been shifted one unit to the left the standard X values that we should typically choose are zero and one now if it's been shifted one unit to the left then instead of plugging in zero we're going to plug in 1 and instead of plug in one we're going to plug in zero to find those X values you can set x + one equal to Z and equal to one our original values and then if you solve for x you should get negative 1 and zero the new X values that you should plug in so now let's go ahead and make a graph so let's plot the horizontal ASM toote at y = -2 and let's plug in the numbers so let's start with negative 1 3 rais to1 + 1 - 2-1 + 1 is 0 3 0 is 1 1 - 2 is 1 now what is the value of y if we replace x with 0 0 + 1 is 1 3 to the first power is 3 3 - 2 is 1 so now let's graph it so we have the point1 negative 1 which is here and 01 which is there so to graph it start from the horizontal ASM toote and connect to the two points that we see here so it's going to go in that direction let me see if I can do that better there we go so now what are the domain and range of the function as we said before the domain of the function is the same it's all row numbers negative Infinity to Infinity now for the range the lowest y value begins at the horizontal ASM toote -2 and the function goes up towards positive Infinity so the range is -2 to Infinity let's try this one let's say Y is equal to e to the xus 1 now even if you have the base e the rules are still the same e is just a number e is about 2.718 but let's use 2.7 just to keep things simple now just like before we're going to plug in two values into x0 and one when X is zero y will be equal to z e to 0 is 1 1 - 1 is 0 now when X is one what is the value of y e to the first Power which is e that's 2.7 2.7 - 1 is 1.7 and the last thing we need is the horizontal asmil which is y isal to1 so we have plenty of information to graph the function at this point so let's plot the horizontal ASM toote which is not there it's over here the first point is 0 0 and the next point is 1 comma 1.7 which is approximately there so as before we're going to start from the horizontal ASM toote and connect to the two points and so that's a rough sketch of the curve that we have the domain of the function is negative Infinity to infinity and the range is from the horizontal ASM toote up to positive Infinity so the range is from1 to Infinity but not including 1 try this one let's say Y is equal to 3 - e to the x - 2 so what values should we plug into the table if you're not sure set the exponent in this case x - 2 equal to two numbers 0 and 1 therefore the X values that we should choose are two and three the graph has been shifted two units to the right so instead of choosing zero and one add two to both of those numbers now what's the horizontal Asm toote the horizontal ASM toote is the constant that you see in front of the function which is in this case three you can rewrite the function like this if you want now notice that we have a negative sign in front of the exponential term that negative sign is going to cause it to reflect over the horizontal ASM toote so the horizontal ASM toote is y is equal to 3 so before the graph was above the horizontal aste now it's going to be below it now let's plug in these values into the equation and let's get the Y values that correspond to them let's start with 2 2 - 2 is 0 e^ 0 is 1 and 3 - 1 is two now if we plug in three 3 - 2 is 1 e to the 1 is e which is 2.7 3 - 2.7 is.3 so let's plot the horizontal ASM toote which is y is equal to 3 and we have the point 2 comma 2 which is right here and we have the point 3 comma 3 which is here in this uh General vicinity so always start from the horizontal asmo and follow the two points so the graph looks something like that so the domain is going to be the same negative Infinity to Infinity but the range is a little different starting from the bottom to the top the lowest y value is negative Infinity the highest in this case is the horizontal ASM toote of three so the range is from negative Infinity to three but not including three