Transcript for:
Understanding Graphing Quadratic Functions

in this video we're going to focus on graphing quadratic functions how to graph it in vertex form standard form how to find the maximum and the minimum values we're going to talk about how to find the axis of symmetry the vertex and also how to write the equation and then we're going to work on a word problem dealing with uh how to find the maximum height the time it takes to get there the range of the object and how long it takes before it hits to ground so let's begin first you need to know the difference between the shapes positive x^2 and Nega X2 positive x^2 is a parabola that opens in the upward Direction because it opens in the upward Direction it has a minimum value the minimum value occurs at the vertex in this problem the vertex is the origin 0 0 the x coordinate of the vertex is the axis of symmetry I'm going to write AOS and it's an equation it's simply the x value of the vertex so you write it as x equal Z the Y value of the vertex is the minimum value now for the graph yal x^2 it opens in a downward Direction and so it has a maximum value at the vertex so now let's work on some examples you need to be familiar with vertex form and standard form this is the vertex form of a quadratic function and a Vertex is H comma K the standard form looks like this ax^2 + BX plus C that's the standard form of a quadratic equation so let's say if we have a function that looks like this y = x - 1^ 2 notice that H is the number that we see here H is one now since there's no number here K is zero so the vertex is 1 comma 0 so this graph it shifts one unit to the right so that's a horizontal shift and the vertex is at this point now if you want to find the next points here's a technique that you can use you can use a table if you want but you don't need to 1 SAR is 1 2^2 is 4 why am I telling you this it turns out that from the vertex if you travel one unit to the right the next point will occur at a a y- value that's one unit higher than a Vertex so travel one unit to the right and up one that's going to give you the next point one unit to the left up one that will give you the point to the left of the vertex now since 2^2 is four if you travel two units to the right you can find the next point if you go up four units starting from the vertex so the next Point occurs at 3 comma 4 and if you travel two units to the left towards NE 1 and if you go if you go up four units you'll get the next point which is uh -14 so that's a quick way that you can graph it now if you prefer to use the table you can do that too but if you do choose to use a table Senter the table around the vertex choose two points to the right of the vertex and two points to the left of the vertex so let's say if we plug in two into the equation so 2 minus 1 is 1 1^ 2 is 1 so we get the 21 if you plug in zero it's going to be the same thing due to the Symmetry around the vertex now if you plug in three into the equation 3 - 1 squared is going to be 2^ squ which is four and -1 should have the same value because these two points are equidistant they have the same y value and -1 and three is equidistant from the the vertex so they share the same y value of four due to the Symmetry around the vertex now what is the axis of symmetry and does this graph have a minimum value or a maximum value so the axis of symmetry is simply the x coordinate of the vertex so therefore it's x = 1 now because it opens upward this graph has a minimum value and that value is the y-coordinate of the vertex so the minimum value is zero now once you have this information you could find the domain and the range the domain for any quadratic function is always going to be negative Infinity to Infinity The Domain represents the allowed X values or the values of x that you can have in this function X could be anything it could be 5 Z8 100 there's no restrictions on the value of x so the domain for a quadratic function will always be negative Infinity to Infinity now the range is going to vary to write the range what is the lowest y value that you see here looking at the values on the y axis the lowest y value is zero and the highest is infinity because it keeps going up towards Infinity so therefore the range is from zero to infinity and since it includes zero you need to use a bracket add zero instead of a parenthesis for Infinity always use parenthesis so that's how you can write the domain and range for this particular function let's try another example let's graph this function y = x^2 + 4 so the number on the outside or that's separate from X that represents a vertical shift it's going to shift up four units so the vertex is going to be 0 comma 4 you can rewrite this function as x - 0^ 2 + 4 so because this is a zero it doesn't shift to the left to the right so the vertex is zero at the x value but we do have a k value of four and so it shifts up four units from the origin now the negative sign in front of the X2 tells us that the graph reflects over the x-axis so it's going to open in a downward Direction but it's going to start at 04 and it's going to point downward now you can make a table at this point if you want just remember to Center the table of values at the vertex so since the x coordinate of the vertex is zero choose two points to the right and two points to the left now since 1^ squ is 1 if we travel one to the right we need to go down one unit which will take us to the point 1 comma 3 and one to the left we also need to go down one unit from the vertex now since 2^2 is four as we travel two units from the right or to the right of the vertex we need to go down four units so that will take us to the Point 20 and if we travel two units to the left down four that will take us to the point -20 now if you plug in the numbers that we have in the table into the equation you should get the same answer so -12 + 4 that's- 1 + 4 which is 3 if we plug in NE 1 we'll get the same thing now -22 + 4 that's -4 + 4 which is z if you plug in -2 you get the same thing so we'll get the same points that we already have in the graph now what are the X and Y intercepts for this particular function the X intercepts are the values of X where the graph touches the xaxis so two 0 and -2 0 are the X intercepts the Y intercept is where it touches the y- axis and that's 04 so we already have them for this particular example now does this function have a maximum value or a minimum value because there's a negative in front of the x s it's going to open downward and therefore it has a maximum value the vertex is 04 the maximum value is the y-coordinate of the vertex which is four the axis of symmetry is the x coordinate vertex which is zero now what about the domain and range of this function so as we said before the domain represents its all real numbers X could be anything however there will be restriction on the Range or the Y values now what is the lowest yv value you see and what's the highest the highest y value is four the lowest is negative Infinity these arrows will keep going down towards netive Infinity so writing it from left to right or from low value to high value the range is going to be negative Infinity to four But it includes four so notice that the range always has the y-coordinate of the vertex because that's going to be the minimum value or the maximum value if it's the maximum value the y-coordinate will be on the right side if it's a minimum value the y coordinate will be on the left side and then Infinity will be on the other side it's always one or the other way it's one of those two ways let's try this one y is = x + 2^ 2 - 1 so how can we graph this particular quadratic function so notice that it's going to shift two units to the left and down one unit so the vertex is going to be -2 comma 1 so H is -2 and K is 1 so remember to reverse this value but not this one so the vertex is at -21 which is somewhere in this region so as we travel one unit to the right we need to go up one since there's a positive in front of the X2 function so the next point is going to be at 1 0 if we travel one to the left from the vertex and up one that will give us another Point -3 0 so those are the X intercepts -3 0 and 1 Z now as we travel two units to the right we need to go up four units all the way to three two units to the left up to three as well so this is the Y intercept which is 0 comma 3 to find the Y intercept you can plug in0 into X and then you should get a yvalue of 3 2^ 2 is 4 - 1 is 3 to find the x intercept replace y with zero and solve for x so let me show you first let's make some space let's put this over here so let's replace zero or Y with zero and then we'll solve for x so we need to add one to both sides so 1 is = x + 2 and now let's take the square root of both sides the sare root of one will give you two numbers plus or minus one the square root of x + 2^ 2 is just going to be x + 2 so you have two equations x + 2 is equal to POS 1 and x + 2 is = 1 if you subtract 2 1 - 2 will give you an xep of1 which is this one here and-1 - 2 will give you an X intercept of -3 which is the other one so that's how you can find the x intercept in vertex form so now let's go ahead and graph the function so we have a parabola that's going to open any upward Direction so therefore we have a minimum value the minimum value is the y-coordinate of the ver so it's 1 the axis of symmetry is the equation x is equal to -2 the x coordinate of the vertex the domain is all real numbers and what about the range what do you think the range is going to be so the range is going to have the y-coordinate of the vertex Nega 1 and since it's a minimum value the lowest y value is Nega 1 the highest notice that it goes up towards positive Infinity so that's the highest y value so the range is from negative 1 to Infinity now let's try this one let's say that Y is = to -2 * x - 1^ 2 + 3 so feel free to pause the video and try this example if you want to so what is the vertex let's start with that notice that it shifts one unit to the right and up three units so the vertex is 1 comma 3 H is 1 K is 3 so it's somewhere over here now we have a negative sign in front of the equation so we know it's going to open in a downward Direction now how can we find the next Point typically what we did before is as we traveled one unit to the right from the vertex we would go down one unit since we had the function y was equal to x^2 but now it's -2X s so if you plug in one into X you should get --2 for y so as you travel one unit to the right you need to go down two units you have to multiply it by two instead of going down one unit you need to go down two units so that's going to take us to this point and if we travel one to the left we need to go down two units as well so that will take us to the point 0 1 which is the y intercept now if we travel two units to the right typically we would go down by four units but we got to multiply that by two so we need to go down eight units So currently the yvalue of the vertex is three so 3 - 8 will take us to a yvalue of5 so the next point is going to be 35 and if we travel two to the left it's going to be -15 so the graph looks like this now we can't clearly see what the X intercepts are so let's solve it let's replace zero for y and let's solve for x so let's subtract 3 from both sides so -3 is equal to -2 * x - 1^ 2 so let's divide both sides by -2 so the two negative signs will cancel and it's going to be 3 over2 and that's equal to x -1 s so let's take the square root of both sides so plus or minus < tk3 over tk2 which if you rationalize it that's going to beunk 6 over 2 that's equal to x - one so if we add one to both Sid sides we get two answers it's 1 plus orus < TK 6/ 2 so this point here is 1 + root 6 over2 and this other x intercept is 1 minus otk 6 over2 now since the graph opens downward we have a maximum value and the max value is the y-coordinate which is three and the axis of symmetry which is this vertical line here that's x = 1 the domain is all real numbers and what is the range so notice that the lowest y value is negative Infinity but the highest is three so it's from negative Infinity to 3 and that's all we could do for this particular quadratic function now what if the function is in standard form so let's say if we have the equation Y is equal to x^2 + 2x - 8 how would you graph this function so it's positive x^2 we know it's going to open upward now if you want to find a Vertex you want to use this equation x is equal to B / 2 a so this equation is in the form ax^2 Plus + b x + C so a is the number in front of X since we don't see a number it's a one B is 2 C is8 so let's find the x coordinate of the vertex so B is 2 a is 1 -2 / 2 is -1 so now to find the y-coordinate let's replace X with1 so it's -1 2 + 2 * -1 - 8 - 1^ 2 is-1 * 1 which is POS 1 and 2 * 1 that's -2 so 1 - 2 is1 And1 - 8 is9 so the x coordinate of the vertex is1 I mean the y coordinate is9 so but the vertex coordinate is1 9 so now that we have that let's find the X intercepts so let's replace y with zero and let's solve for x so notice that we have a trinomial where the leading coefficient is one so we need to factor it we need to find two numbers that multiply to8 but add to the middle term two so this is going to be 4 and -2 4 * -2 is8 but 4 + -2 is 2 so if to factor it's going to be X + 4 * x - 2 now to solve for x we need to set each factor equal to 0 so for the first one let's subtract four from both sides so X is equal to4 and for the next one let's add two to both sides so X is equal to pos2 so those are the two x intercepts -40 and two 0 now let's find a y intercept so let's plug in zero into the equation so we got to replace 0 with X so 0^ 2 + 2 * 0 - 8 is simply 8 so the Y intercept is 08 so at this point let's organize the data that we have in a table and let's Center it based on the vertex which is -19 so the Y intercept is very close to the vertex is 08 so if it's let me put it over here so since the Y intercept is one unit to the right of the vertex one unit to the left must also share the same y value of8 now the X intercepts are -4 0 and 2 notice that the x coordinate of the vertex is the average of the X intercepts if you average two and4 if you add them up and divide by two this will give you the x coordinate vertex which is1 so now we have enough points to make a graph out of this equation A9 is all the way at the bottom so let's plot the vertex first which is -19 and then the Y intercept which is 08 and we have another point at -28 and then the x intercept at 2 0 and4 0 so the graph is going to look something like okay that side was messed up let's do that again it's going to look something like that so we can see that the axis of symmetry is the x coordinate the vertex so that's at x = -1 and it represents this line we have a minimum value and the minimum value is the y coordinate the vertex so it's 9 the domain as always is all real numbers and the range notice that the lowest y value is9 and the highest is infinity so the range is from9 to infinity and so that is it for this particular example let's try one more example like that last problem go ahead and try this one so find the vertex the X intercepts the Y intercepts and then go ahead and graph it so let's start with the X intercepts what two numbers multiply to -3 but add to two this is going to be positive 3 and negative 1 so it's going to be x + 3 * x -1 so the intercepts are -3 and 1 so as an ordered pair we can write the X intercepts as -3 comma 0 and 1 comma 0 now what is the midpoint between -3 and 1 if we add these two numbers and divide by two if we average them what is the midpoint so -3 + 1 is -2 and -2 / 2 is 1 so this will give us the x coordinate of the vertex now to prove it you can use the equation xal to B / 2 a so B is the number in front of X which is two and a is the number in front of the X squ if you don't see anything it's a one ne2 ID 2 is indeed negative 1 so now let's find the y-coordinate of the vertex so let's plug in1 into the equation so - 1^2 + 2 * -1 - 3 this is 1 - 2 - 3 so that's 1 - 5 which is -4 now there's another way in which you could find the coordinates of the vertex you can use the complete the square method and convert the equation from standard form back into vertex form so let's separate the first two terms from the last term so to complete the square we need to find the perfect square that will complete this trinomial and that number is going to be half of this number that you see here in front of X and then Square it so half of two is one so we need to add one squar now for the right side to be equal if we add one squ to the right side we must also take away one squ from the right side so that we haven't changed the value of the right side so we have x^2 + 2x + 1 - 3 - 1 or - 4 so notice that 1 - 4 still equals to the original value of -3 now how can we Factor x^2 + 2 x + 1 two numbers that multiply to one but add to two are 1 and one so it's going to be x + 1 * x + 1 which we can simply write it as x + 1^ 2 - 4 so that is the equation in vertex form and notice that you can get the vertex from it which is -14 and we have that here so that's another way you can find a Vertex next let's find the Y intercept if we replace x with Z we could see that it's going to be -3 so we have the point 03 so now we can organize everything into a table and so let's start with the vertex which is-14 the Y intercept is one unit away from the vertex it's to the right of it so one unit to the left must also have a y value of3 now the X intercepts are 1 Z and -3 0 so as you can see these two are the same and these two are the same it makes it a lot easier if you Center it around the vertex it's very easy to find the missing points so now we can make the graph now let's start with the vertex which is-14 and then we have the point is0 -3 and 2-3 and then after that we have 1 0 and -40 so the graph looks something like this we can see that it has a minimum value at-4 the axis of symmetry is x = -1 the domain is all R numbers and the range is from -4 to Infinity so that's it for this particular function now let's try this word problem a ball is thrown Upward at a speed of 16 m/s from a cliff that is 32 M high now we're given the height function this function tells us the height at any time team how long does it take the ball to reach its maximum height so let's say if it's out a cliff here's the ground level and here's the ball so it's thrown upward it reaches its Max height and then it falls down now if we were to plot the height equation which if we rearrange it it's -4.9 t^2 plus 16t + 32 we can see that the Y intercept is 32 so that's basically the height of the cliff it goes up and then it falls back down this graph is similar to netive x^2 which is a downward Parabola so to find the height we need to find the y-coordinate of the vertex and to find find the time it takes to reach the height that's the x coordinate of the vertex X is associated with t y is associated with h so let's go ahead and do that so T is equal to B over 2 a so B is the number in front of T which is 16 and a is the number in front of t^2 which is 4.9 2 * 4.9 is 9.8 the two negative signs will cancel so T is going to be positive and 16 ID 9.8 will give us a t value of 1.63 so this is the time it takes to reach the maximum height it's one . 633 now for Part B we need to find the maximum height so we have to plug in the T value into the equation we need to see what the height is when the time is 1.6 33 so 1. 633 s time 4.9 that's about -3.0 67 and then 16 * 1. 633 that's 26128 and let's add 32 to it so the maximum height occurs at positive 4.61 so that's the answer to Part B so if we consider the graph again we know it starts at a height of 32 it goes up and then it goes back down so at the maximum height the Y value is 4.06 and the time it takes to get there is 1. 633 so at this point the ball is at its maximum now for part C we want to find out how long it's going to take for it to hit the ground so what is the time value when it's that ground level so we get to find we need to find a time at that point where the Y value is zero so what we're going to do is we're going to replace H with zero and solve for T So 0 is equal to -4.9 T ^2 + 16t + 32 so basically we're finding the x intercept but it's going to be very difficult to factor this expression so if you can't Factor it the best thing to do is to use the quadratic equation so T is equal to B plus orus < TK b^ 2us 4 a c / 2 a so B is 16 B ^2 or 16 s that's 256 - 4 * a which is -4.9 time C which is POS 32 / 2 a or 2 * 4.9 -4 * 4.9 * 32 that's 627.736 if you take the square root you get 29.72f thing inside the radical including the radical / 9.8 so let's make some space so now we have two possible answers the first one is -16 plus 29.72f if we divide it by 9.8 that will give us a negative time value which is not what we're looking for now if we try -16 minus 29.72f 4.67 so now let's make sense of the information that we have here so if we make a graph we know the initial height is 32 the max height is 45.0 6 the time it takes to reach the maximum height 1. 633 but the time it takes to hit the ground it's going to be 4.67 so if that represents this answer what is the other answer now notice that if you extend a graph this way it's also going to touch the x-axis at 1.4 but for a real life situation time won't be negative so the answer for part C the time that it takes to hit the ground is 4.67 seconds now let's talk about how to write the equation if you're given the graph so let's say this is the graph and it looks something like this let's say you're given two points you know the vertex which is 1 5 and you also know the Y intercept 04 if you have two points you can find the equation so if you have the vertex it's easier if you use the vertex form of the equation which is a x - h^2 + K so H is 1 K is5 so let's plug it in so this is going to be a * x minus now let's insert the value of H which is 1 squar plus K which is5 so we have the formula a x - 1 - 5 now the only thing that we need is to find the value of a once we do that then we have the equation in vertex form so that's where the second Point comes in replace y with -4 and x with zero since x is 0 Y is4 and solve for a so 0 - 1 is -1 -1 2 is just 1 so if we add five to both sides -4 + 5 is 1 so in this case a is 1 so now at this point to write the equation simply replace a with what it is so the equation is x - 1^ 2 - 5 in vertex form and notice that since the graph opens upward it's positive x^2 now sometimes you may want the answer in standard form to convert it from vertex form to standard form you need to foil x -1 2 x -1 2 is the same as x -1 * x -1 so x * X is x^2 x * -1 that's X1 * X that's also Negative X and 1 * 1 is + one minus the 5 so x - x is 2x 1 - 5 is4 so in standard form it's x^2 - 2x - 4 so that's how you can write the equation in vertex form and in standard form if you're given the vertex and a point it could be a y intercept the x intercept whatever that other point is plug in X and Y and solve for a let's try one more example so let's say if you're given the X intercepts which are 1 and 5 and also you know the Y intercept which is -10 what can you do to write the equation of this graph now we don't have the coordinates of the vertex but we do know that the the x coordinate is three it's the midpoint between the two x intercepts but we don't know the Y value so we can't really use that so if you have the X intercepts 1 Z and 5 0 here's what you can do you can write x -1 * x - 5 and it's factored form because once you factor it these will be the X intercepts now you still need another point to find the value of a and that's where the Y intercept comes the Y intercept is 0-10 so let's replace y with -10 and let's replace x with zero this will allow us to solve for a so 0 minus 1 is 1 0 - 5 is5 and -1 * 5 is POS 5 so if we divide both sides by five we see that a is equal to -2 so therefore the equation in factored form is y is = to -2 * x -1 * x - 5 now to put it in standard form we need to foil x * X is x^2 x * -5 that's 5x1 * XX and-1 * 5 + 5 and we still have a -2 on the outside so let's combine like terms -5x - x that's -6x now the next thing that we need to do is distribute the -2 so it's going to be -2X ^2 + 12x - 10 so that is the equation in standard form now how how can we write the equation in vertex form what you could do at this point is you can complete the square so to complete the square let's focus on the first two terms let's factor out the GCF which is well not the GCF but let's take out the -2 2x^2 / -2 is x^2 and POS 12x / -2 is -6x and then we're going to leave a space and here's a -10 so to complete the square we need a number to add here such that when we Factor this trinomial it's going to be a perfect square trinomial how can we do that so we need to find half of this number half of -6 is -3 but make it positive so it's going to be posi 3 and then Square now POS 3^ 2 is 9 now we have to incorporate the -2 which is 8 if you distribute the -2 to the 9 so at this point we added 8 to the right side of the equation now so that the equation is balanced we can either add 18 to the left side or positive 18 to the right side so if you add 18 and positive 18 to the right side the value of the right side is the same it remains zero and that's why this technique works so now here's the shortcut way to factor this perfect square trinomial it's going to be this letter X and then this sign so minus and then what you see here 3^ squ that's that's how you can Factor it the easy way now if you're unsure why that works let's work it out 3^ 2 is 3 * 3 that's 9 -10 + 18 that's postive 8 so let's find two numbers that multiply to 9 but that add to the middle term -6 so what are those two numbers -3 * -3 multiplies to negative I mean to positive 9 and -3 + -3 adds to -6 so it's going to be x - 3 * x - 3 which we can simply write as x - 3^ 2 so the shortcut technique does work it'll save you a lot of time so this is the equation in vertex form so you can always use the completing the square method to convert it from standard form to vertex form so that is it for this video now you've uh mastered quadratic functions you now to find the domain range you now to find the X and Y intercepts axis of symmetry maximum and minimum values and you also know how to solve word problems associated with it so thanks for watching and and have a great day