let's get started on your notes over an introduction to quadratic functions so the first thing I'm going to do is go over the parts of a quadratic function and introduce you to some vocabulary terms and a lot of students really struggle with these vocabulary terms in this unit so as you can see on the graph that's on this page the graph of a quadratic function forms a shape of a U and it'll be right side up you or an upside down u but we call this u-shaped graph a parabola a lot of students want to say parabola but it's parabola and as you can see it's perfectly symmetrical okay it's an even function it's perfectly symmetrical and that line that will divide the parabola into two congruent halves is called the axis of symmetry it's the axis of symmetry and as you can see it goes all the way it's a vertical line and it goes straight through this point right here so we call this lowest or highest point the vertex and all of this is just new vocabulary terms and so you'll hear me say it a lot through this unit and the last new vocabulary term or terms that I want to introduce you to as you can see on this graph it crosses R we have our x axis here and our y axis here it crosses our x axis this particular graph crosses it twice okay so what do we call where it crosses our x axis we call it X intercepts well you also might hear the term roots so if you were ever asked what are the x-intercepts of the graph or of the function well the x-intercept is when y is 0 so we'll plug in 0 for y and we'll solve for x you also might hear what are the roots of the function you also might hear what are the zeros of the function you also might hear what are the solutions of the function and all of these terms mean the same thing so let's move on so we talk about standard form and vertex form of a quadratic standard form is written ax squared plus BX plus C so the number in front of the x squared is our a term so in this particular example the value of a is 2 so ax squared plus BX the B value is the number in front of the X in this example your B value is 5 ax squared plus BX plus C C is your constant term and remember the sign in front goes with the numbers so my constant here or C value is negative 3 so you might see quadratic functions written in what we call standard form and they're written in this ax squared plus BX plus C you also might see quadratics written in vertex form where it's a times X minus H squared plus K where your vertex is H K so I want to talk about linear versus quadratic parent function so you've seen a linear function f of x equals x it starts alright goes through the origin and my slope is 1 right so slope is the number in front of the X if nothing is there you can put a 1 there so when I rise 1 and run 1 it forms this perfectly straight line and I'm going to draw it as straight as possible that looks like this positive slope slope is 1 and it's just a line ok so let's say this is the linear parent function our quadratic parent function it's we're gonna take every input value and we're gonna square it so I'm going to zoom in here I'm going to first plug in 0 0 square root of 0 if I plug in positive 1 remember my x-values or my input positive 1 and I square it I get positive 1 if I plug in negative 1 for X and I square that what's that going to be it's also positive 1 so your quadratic parent function is going to take this shape because it doesn't matter if you square a positive number or a negative number your Y value will always be positive so my Y values are always above the x-axis so let's move on to some definitions so our axis of symmetry what is it and let's let's get to know it a little bit more so what is this axis of symmetry it's the vertical line that divides the parabola into two congruent halves so let's write that down the vertical line divides the parabola into two congruent halves and it always passes through thee I said it on the previous page but it passes through the vertex and we have a formula a handy dandy little formula you might when you think of formulas you might think of like what's the formula to find the area of a triangle well the formula to find the axis of symmetry is x equals negative B over 2a so we're going to be working with this formula today so in this example it says graph a quadratic with x equals four as the axis of symmetry okay so x equals four that's a vertical line so I'm going to go on my x axis to where X is four it's right there so I'm looking for here's gonna be my axis of symmetry x equals four I want to graph a quadratic so a u-shaped graph with this is my axis of symmetry so I could just put my vertex anywhere on this line and I could do a u-shape up or are you shaped down and for my example I'm just gonna do a u-shape up kind of looks like a V but it's really a u so that's the axis of symmetry now let's move on to the vertex the vertex is the highest or lowest point or part of the graph so our y-value of our vertex is determined by plugging in the axis of symmetry you 4x so I'm going to write something down that might confuse you but what this means is if my axis of symmetry always goes through my vertex I'm going to show you this example right here right if my axis of symmetry always goes through my vertex then the x value of my vertex is going to be that value okay so that means the x value of my vertex is negative B over 2 times a so how do I get any Y value when given an x value plug in that x value into the function and see what you get for Y and so I'm going to write it like this F of negative B over 2a that's probably really confusing for you but that just means all I'm gonna do is I'm going to plug in negative B over 2a into my function and I'm gonna get my Y value so this is just fancy schmancy function notation if you remember that so let's move on how do we know if the parabola opens up or opens down well it's actually really easy to determine that when the parabola opens up the a value is positive and the vertex is known as a minimum so this is what that might look like let's graph our parent function f of x equals x squared starts at our origin and it goes like this out one up one out two up four because if I Square two I get positive 4 and if I square negative two I also get positive 4 so it's gonna look like this and it kind of looks like a V but it should be a hue so this vertex right here right there that is known as a minimum so I can call it the vertex but I can also call it a minimum and if I call it a minimum then that just lets you know a little bit more information it lets you know that the parabola opens up so as you can see this a value right here remember if nothing is there I can put a 1 if that a value is positive that's how I know that my parabola opens up so how do I know if it opens down well if that AV you is negative their vertex is known as a maximum and that lets me know that it opens down so let's graph this f of X equals negative 1 x squared and as you can see this a value is negative all that means is I'm going to square X if you think about your order of operations I'm gonna apply my exponent first and then I'm going to apply this negative right here if I Square X I'm gonna take every Y value is actually going to be negative so obviously zero has no doubt it has no value so it can't be positive or negative so 0 0 is a point on my graph but if I square 1 and then I take the negative value of it I get negative 1 and I can do the same thing with 2 and I'll get negative 4 and it opens down like this ok we'll get him to we'll get into graphing quadratics more but this is what your graph would look like if you have a negative a value it will open down and this vertex right here is known as a maximum it's the highest point on your graph it's the maximum value so let's move on and let's get going on some examples it says use the formulas to determine the vertex and axis of symmetry for each quadratic function then draw a rough sketch of the parabola on the graph provided so let's get started and this function right here f of x equals x squared plus 10 X plus 15 the first thing I'm gonna do is identify a B and C so if nothing is in front of that variable I can put a 1 and a value is the number in front of x squared that's 1 so what's the B value it's 10 and what's the C value it's 15 so now let's find our axis of symmetry and we've got this little formula over here to help us determine our axis of symmetry it's negative B over 2a or opposite B over 2a so we're axis of symmetry is the opposite of B which is negative 10 over 2 times a two times one which is what negative five which means our axis of symmetry is x equals negative five so I'm actually going to go ahead and plot that on my graph over here three four five there's where net X is negative 5 and my axis of symmetry is the vertical line that goes through that point look so obviously there's kind of a lot of stuff going on here and if you're confused at this point you can always pause the video rewind it maybe watch something again if you're strip if you're struggling with something but I'm just gonna keep moving forward and if you really get it then maybe you can fast-forward through some things so now my vertex the vertex is I'm gonna write it like this as an ordered pair I'm gonna take this negative five and I'm gonna plug it in for X and I'm gonna figure out what that y-value is so I'm actually going to write it like this F of negative five that's just function notation for what is the y-value when x is negative five so here's what I like to do given this formula right here I'm replacing every x value with negative five so the first thing I'm going to do is replace every x value with parentheses and then in those parentheses I'm going to put what x equals so I'm replacing X with negative five because I'm looking for the Y value when X is negative five so let's do a little bit at a time negative five squared is positive 25 ten times negative five is negative 50 plus 15 and then I've got 25 minus 50 it's negative 25 plus 15 different signs subtract f of negative 5 is what negative 10 and if you struggle with stuff like this I'm sure you're at a point in your year where if you need to use a calculator you can but obviously part of your SAT is going to be non calculator so I always think it's a good idea to keep up with your integer and try this without a calculator that means that my when X is negative five y is negative ten and this is my vertex so let's graph this negative five negative ten that right there is my graph so what else do we know we know that our a value is positive a is positive so the parabola opens which way when a is positive the parabola opens up which means my graph is gonna look something like this and I'm just drawing a rough sketch so don't even worry about the the points being correct we're just finding our vertex and is this a minimum or a maximum value our vertex since it opens up that vertex is a minimum okay so let's move on to number two the first thing I want to do it's written in standard form is identify a B and C so what's my a value negative to be negative eight C is negative fifteen so now using those values I'm gonna find my axis of symmetry axis of symmetry is negative B over 2a so what's the opposite of B eight over two times a that's two times negative 2 which is 8 over negative four which is what negative 2 which means my axis of symmetry is x equals negative 2 so I'm going to go ahead and graph my axis of symmetry how do you remember that x equals a number is a vertical line I teach a little slope cheer for this and x equals negative 2 so that is my axis of symmetry I know that's the vertical line that's going to divide my parabola into two congruent parts or two congruent halves so the next thing I want to do is determine the vertex how do I do this well my vertex is a point it's an ordered pair and since my axis of symmetry goes through my vertex I know that negative 2 is going to be the x-value so I'm going to plug in negative 2 in my function for X and determine what is y when X is negative 2 so the first thing I'm going to do is again replace every x value X variable with parentheses and then in those parentheses I'm going to put what I'm replacing X with so negative 2 and then I'm just going to fall I'm just going to simplify this using my order of operations so the first thing I want to do is negative 2 times negative 2 squared well that's positive 4 negative 8 times negative 2 is positive 16 and then minus 15 and then here obviously need to multiply a negative 2 times 4 first but I can go ahead and simplify 16 and negative 15 to be plus 1 and then negative 8 plus 1 is negative 7 which means the Y value of my vertex is negative 7 which means my vertex is at negative 2 and I'm going to go from the bottom there's 10 9 8 negative 7 so now what else do we know from our function we know that a is negative so which direction does this graph open does it open up or does it open down well since a is negative the parabola opens down so that's what I'm going to do it opens down so that's what it looks like and that's all you're doing today so then what if there is no B value so I want to go over examples that look like this determine if the axis of or determine the axis of symmetry for the following quadratic function then determine the vertex so when I see what what if there is no B value what if there is no B value and really let's make it look like this so right here I know that a is 1 I know B is wait a second there's no b-value there's no X term which means zero is the value for B C is negative five so if I'm determining my axis of symmetry when there's no B value negative B well that's just 0 over 2 times 1 zero divided by anything is just zero and this will happen every time when there's no B value your axis of symmetry is always x equals zero which means your axis of symmetry is the y axis so then my vertex how am I going to determine that well I'm gonna plug in 0 for X what do I get for y when I plug in 0 for X I get negative 5 so these are kind of easy to do but you just need to make sure that you recognize that hey just because this is the second term when it's written doesn't mean that negative 5 is the B value the B value is the number in front of the X so that concludes your notes over an introduction to quadratic functions I hope it was helpful