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.

Main content

# Examples: Graphing and interpreting quadratics

## Video transcript

we're on problem 36 it says which of the following sentences is true about the graphs of y is equal to three times X minus five squared plus one and y equals 3 times X plus five squared plus one so let's just let's do it something very similar to what we did in the past and if you think about it both of these equations y is going to be 1 or greater let me just you know let's just analyze this a little bit right this term right here let me this term right here since we're squaring is always going to be positive right now even if that what's inside the parentheses becomes negative if we have you know X is minus 10 inside the parentheses becomes negative but when you square it it always becomes positive and you're going to multiply 3 times a positive number so you're going to add a positive number so the lowest value that this could be 0 so the lowest value that Y could be is actually 1 and same thing here the lowest value that you know this number can become very negative but when you square it it's going to become positive so this expression with this with the squared here is going to be positive and you multiply 3 is going to be positive so the lowest value here is always going to be 0 and include this whole term when you include this whole term so similarly the lowest value Y could be is 1 I just want to think about it a little bit just to give you a little bit of an intuition and let's let's think of this in the context of what we talked last time with the shifting so let me draw it in a color that you can see so if that is the y-axis and I'll just draw mainly the positive area so see this is so if I were to just draw Y is equal to x squared plus 1 it would look like this where this is 1 well that's y is equal to 1 and the graph would look something of along this y is equal to x squared plus 1 that's a horrible drawing normally I wouldn't redo it but that was just atrocious y is equal to x squared plus 1 looks something like that it's symmetric you know the you get the idea you've seen these parabolas before this is y is equal to x squared plus 1 x squared plus 1 now if we were to do X X minus 5 squared plus 1 what happens to it well let me think about what is 3 x squared plus 1 well then it just increases a little bit faster so if I were to say equals three x squared plus one it might look something like this it'll just increase a little bit faster three times as fast actually so that would be three x squared plus one right the rate of increase in both direction just goes faster because you have this constant term three out there multiplying the numbers okay now what happens when you shift it when you shift this let's do X minus five so where x equals zero was the minimum point before now when if we step through the five here that'll be our minimum point right because then that whole term becomes zero so this vertex will now be shifted to the right let me do another color so if this is the point five now this would be the graph you just took this graph and you shifted it over to the right by five I won't draw the whole thing that graph right there would be three times X minus five squared plus one and remember the the y shift is always intuitive if you add one you're shifting it up if you subtract one you're shifting it down the X shift isn't we subtracted five X minus 5 we replaced X with X minus 5 but we shifted to the right and the intuition is there is because now plus five makes this expression zero so that's 3x minus five squared and then the same logic three times X plus five squared is going to be here plus one so if we shift it that's going to be shifted to the let me pick a good color to the left this it's going to look something like this going to be this blue graph shifted to the left so this is minus five so this is the graph right here of three times X plus five squared plus one now hopefully you have an intuition so let's read their statements and see which one makes sense which of the following is true their vertices are maximums no that's not true of any of these because the vertices is the is that point right there and they're actually it's a minimum point right a maximum point would look something like that and we know that because you just go positive this term can only be positive if this was a negative three then it would flip it over okay so it's not choice a the graphs have the same shape with different vertices yeah both of these graphs have the shape of three x squared but their one vertices is ten to the left of the other one so I think B is our choice let's read the other ones the graphs have different shapes with different vertices now they have the same shape they definitely of the same shape I mean they both have this 3 x squared shape one graph has a vertex that is a maximum while the other has a growth no that's not right they both are upward-facing so they both have minimum points so it's choice B choice B next problem problem 37 37 let me see what it says what are the x-intercepts let me copy and paste that okay I'll paste it there what are the x-intercepts of the graph of that well the x-intercepts whatever this graph looks like I don't know exactly what it looks like you know I don't know what you know this graphs going to look something like this I actually have no idea what it looks like until I solve it it's going to look something like this when they say x-intercepts they're like where does it intersect the x-axis so that's like there and there I don't know if those are the actual points right and to do that we set the function equal to 0 because this is the point y is equal to 0 you're essentially saying when does this function equal 0 because that's the x-axis when Y is equal to 0 so you set Y is equal to 0 and you get 0 is equal to 12x squared minus 5x minus 2 and whenever I have a coefficient larger than 1 in front of the x squared term I find that very hard to just eyeball and factor so I use a quadratic equation so negative B this is B B is minus 5 so negative negative 5 is plus 5 right negative B plus or minus the square root of B squared negative 5 squared is 25 minus 4 times a which is 12 times C which is minus 2 times minus 2 so let's just make that x plus 2 and put the plus out there right minus times minus is a plus all of that over 2a all that over 24 two times a so that is equal to we see five plus or minus the square root see it was twenty five plus four tot 4 plus 12 four times 12 times 2 right because that was a minus two but we had a minus there before so eight times 12 so 96 96 all of that over 24 what's 25 plus 96 that's 100 once 121 right this is 121 which is 11 squared so this becomes 5 plus or minus 11 over 24 so remember these are the points where these are the X values where that original function will equal 0 it's always important to remember what we're even doing so let's see so if X is equal to 5 plus 11 over 24 that is equal to 16 over 24 which is equal to 2/3 that's one potential intercept so you know maybe that's right here right that's X is equal to 2/3 and Y is equal to 0 and the other value is X is equal to 5 minus 11 over 24 and that's what minus 6 over 24 which is equal to minus 1/4 which could be this point I actually drew the graph not that far off of what it could be so this would be X is equal to minus 1/4 and those are the x-intercepts of that graph to see 2/3 and minus 1/4 is choice C choice C on the test we have time for at least one more see where oh boy they drew us all this graph so which is the graph let me shrink it I want to be able to fit all the graphs so let me copy and paste their graphs so this is one where the clipboard is definitely going to come in useful let me it doesn't okay that's good enough okay so they want to know which is the graph of let me I've never done something this graphical let's see so the graph they say is y is equal to minus 2 times X minus 1 squared plus 1 so that's what we have to find the graph of so immediately when you look at it so you say ok this is like the same thing as Y is equal to minus 2 x squared plus 1 but they shifted the X right they shifted the X to the right by 1 I know it says a minus 1 but think about it when X is equal to positive 1 this is equal to 0 and and if so it's going to be shifted to the right by 1 right plus 1 we know that we know that it's going to be shifted up by 1 right so up plus 1 and then we have to think is it going to be opening upwards or downwards and think of it this way if this was if this was y is equal to 2x squared plus 1 then this term would always be positive and it'll just become more and more positive as you get further and further away from 0 so it would open up but if you put a negative number there if you say Y is equal to minus 2x squared plus 1 then you're going to open downward you're going to get more and more negative as you get away from your vertex right so we're shifted to the right by 1 we're shifted up by 1 and we're going to be opening downwards so if we look at our choices only these two are shifting are opening downwards and both of them are shifted up by one their Y the vertex is at Y is equal to 1 but this is shifted one to the right and this is shifted one to the left and remember we said it was X minus 1 squared so the vertex happens when this whole expression is equal to 0 and this whole expression is equal to 0 when X minus 1 when X is equal to positive 1 when X is equal to positive 1 so that's right here so it's actually choice C and that's probably that's when you're shifting graphs that can be one of them kind of hardest things to agree but I just really encourage you to explore graphs practice grab practice with graphs with a graphing calculator and really try to plot points and try to get a really good grasp of why when you go from minus 2x squared plus 1 to minus 2 times X minus 1 squared Y when you replacing X with an X minus 1 y this shifts the graph to the right by 1 anyway I'll see you in the next video