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CCSS.Math: , , , ,

the function M is given in three equivalent forms which form most quickly reveals the y-intercept so let's just remind ourselves if I have a function the graph y is equal to M of X and these are all equivalent forms they tell us that the function M is given in three equivalent forms I should be able to algebraically manipulate any one of these to get any of the other ones and so if I wanted to graph and Y is equal to M of X and let's say it looks something like this and I actually know it's a downward-opening parabola because I can look at this form right over here and see he look the coefficient on the second degree term here is a negative so it's going to be a downward-opening parabola that's a messy drawing of it and so if we're going out the y-intercept we're saying hey where does it intersect the y axis so what is the y-value when x is equal to 0 so it boils down to how quickly can we evaluate M of 0 what is M of X when M is when X is equal to 0 so how quickly can you evaluate M of 0 well in this top one I can substitute 0 for X and so it'll be negative 2 times negative 3 times negative 9 so it's not too hard to figure out but there's going to be some calculation in my head similarly in the second choice for x equals 0 I can get negative 6 squared which is positive 36 times negative 2 which is negative 72 and then I have to add that to positive 18 I can do that but it's a little bit of computation but here for this last one and this is known as standard form if I say x equals 0 that term disappears that terms disappears and I'm just left with M of 0 is equal to negative 54 so standard form this is standard form right over here was by far the easiest so we know the y intercept is 0 comma negative 64 now one rule of caution sometimes you might look at what is called vertex form and as we'll see this is the easiest one where it is to identify the vertex but when you see this little plus 18 hanging out it looks a lot like this negative 54 that was hanging out it's okay when x equals 0 maybe I can just set out the same way that I cross these terms out and be very very careful there because if X equals zero this whole thing does not equal zero when X is equal to zero as I just said you have negative 6 squared which is 36 times negative 3 this is equal to negative 72 so M of 0 is definitely not 18 so be very very very careful but we can see that the best choice is this one standard form not vertex form or factored form factors forms you can meet as you can imagine is very good for figuring out the zeros let's do another example and actually this is this is the same M of X but we're going to ask something else so it's given in those same three forms which for most quickly reveals the vertex well I just called this vertex form before but what's valuable about vertex form is you can really say ok this is going to be this is going to achieve its vertex when this thing over here is equal to 0 how do I know that well once you get used to vertex form you'll just become a bit of second nature but if this is a downward-opening parabola the vertex is when you hit that maximum point and as you can see here X -6 squared is always going to be non-negative you multiply that times the negative 2 it's always going to be non positive it's either going to be 0 or a negative value so this is always going to take away from this 18 and so if you want to find the vertex the maximum point here it would be the x value that makes this thing equal to 0 because because for any other x value this thing is going to be negative it's going to take away from that 18 and so you can see by inspection well what x value will make this equal to 0 well if X is equal to 6 6 minus 6 is equal to 0 0 squared is 0 times negative 2 is 0 and so mo6 is equal to 18 so this lets us know very quickly that the vertex is going to happen at X is equal to 6 and then the Y value there or the M of 6 is going to be equal to 18 you can do it with these other ones the hardest one would be standard form standard form you could complete the square or do some other techniques or you could try to get into factored form factored form you can find the zeros and then you'd know that the x-coordinate of the vertex is halfway between the x-coordinate of our two x-intercepts and then you can figure out the y value there but this one is definitely the easiest revert X form and what is the vertex what's going to happen at the point 6 comma 18 let's do one last example so this is a different function the function f is given in three equivalent forms which for most quickly reveals the zeros or roots of the function so once again when we're talking about zeros or roots if we have let's say that's the x-axis and if you have a parabola that looks like that the roots are or the zeros are the x-values that make that function equal to zero or there did the x-values of the x-intercepts you could say and so what X values make or which one is easy to figure out when this function is equal to zero which of these forms because they're all equivalent you can expand out these first two and you should get this last one in standard form which one is easy to identify the zeros well in factored form I could just say well what makes either this thing zero or that thing zero because an X that makes this first one zero or the second one zero is going to make this whole expression zero so you can quickly say well if X is equal to negative one this is going to be zero or if X is equal to negative 11 this is going to be zero so this is a very fast way to find out the zeros this one here a lot harder you would have to solve 3 times X plus 6 squared minus 75 is equal to zero do some algebraic manipulation and you would eventually get to these answers so I would rule the vertex form right over here and this form standard form the first step I would do is try to get it into factored form and then from factored form I will I would find the zeros and so once again this is definitely more work than if you already have it in factored form so factored form is definitely what you want when you're trying to find the zeros and here it says write one of the zeros I could write x equals negative one or I could have written x equals negative 11