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CocoSign's Guide About Finishing Quadratic Functions Explained Form

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How to use The Quadratic Functions Explained Form?

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 board 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 the ground so let's.begin first you need to know the.difference between the shapes positive x.squared and negative x squared positive.x squared 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 a OS and it's an equation it's.simply the x-value of the vertex you.write as x equals 0 the y-value of the.vertex is the minimum value now for the.graph y equals a negative x squared 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 then a vertex is H.comma K the standard form looks like.this ax squared plus 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 is equal to X minus.one squared notice that H is the number.that we see here H is 1 now since.there's no number here K is 0 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 point here's the technique that you.can use you can use the table if you.want but you don't need to 1 squared is.1 2 squared 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 y-value.that's one unit higher than the vertex.so travel one unit to the right an up.one that's going to give you the next.point one unit to the left up one.that'll give you the point to the left.of the vertex.now since 2 squared is 4 if you travel 2.units to the right you can find the next.point if you go up 4 units start from.the vertex so the next point occurs at 3.comma 4 and if you travel 2 units to the.left towards negative 1 if you go up if.you go up 4 units you'll get the next.point which is negative 1/4 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 the.table.Center the table around the vertex.choose 2 points to the right of the.vertex and 2 points to the left of the.vertex so let's say if we plug in 2 into.the equation so 2 minus 1 is 1 1 squared.is 1 so we get the point 2 1 if you plug.in 0 is going to be the same thing due.to the symmetry around the vertex.now if you plug in 3 into the equation 3.minus 1 squared is going to be 2 squared.which is 4 and negative 1 should have.the same value because these two points.are equidistant they have the same.y-value and negative 1 and 3 is.equidistant from the vertex so they.share the same y-value of 4.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 equals 1 now because.it opens upward this graph has a minimum.value and that value is the y-coordinate.the vertex so the minimum value is 0 now.once you have this information you could.find in the domain and range.the domain for any quadratic function is.so it's 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 cubed.anything it could be 5 0 negative 8 100.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 0 and the highest is infinity because.it keeps going up towards infinity so.therefore the range is from 0 to.infinity and since it includes 0 you.need to use a bracket ad zero instead of.a parenthesis for infinity always use.parentheses 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 is equal to negative x.squared plus 4 so the number on the.outside where that's separate from X.that represents a vertical shift it's.going to shift up 4 units so the vertex.is going to be 0 comma 4 you can rewrite.this function as X minus 0 squared plus.4 so because this is a 0 doesn't shift.to the left to the right so the vertex.is 0 at the x value but we do have a K.value of 4 and so it shifts up 4 units.from the origin now the negative sign in.front of the x squared tells us that the.graph reflects over the x axis so it's.going to open in the downward direction.but it's going to start at 0 4 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 0 choose 2 points to.the right and 2 points to the left now.since 1 squared is 1.if we travel 1 to the right we need to.go down 1 unit which will take us to the.point 1 comma 3 then once the left we.also need to go down 1 unit from the.vertex now since 2 squared is 4 as we.travel 2 units from the right or to the.right of the vertex we need to go down 4.units so that will take us to the point.2 0 and if we travel 2 units to the left.down 4 that will take us to the point.negative 2 0.now if you plug in the numbers that we.have in a table into the equation you.should get the same answer so negative 1.squared plus 4 that's negative 1 plus 4.which is 3 if we plug in negative 1.we'll get the same thing now negative 2.squared plus 4 that's negative 4 plus 4.which is 0 if you plug in negative 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 x-axis so 2 0 and.negative 2 0 are the X intercepts the y.intercept is where it touches the y-axis.and that's 0 4 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 squared.it's going to open downward and.therefore it has a maximum value the.vertex is 0 for the maximum value is the.y coordinate of the vertex which is 4.the axis of symmetry is the x-coordinate.the vertex which is 0 now what about the.domain and range of this function.so as we said before the domain.represents its are real numbers X could.be anything however there will be.restriction on the range or the Y values.now what is the lowest Y value you see.in what's the highest the highest Y.value is 4 the lowest is negative.infinity these arrows will keep going.down towards negative infinity so.writing it from left to right or from.low value to high value the range is.going to be negative infinity to 4 but.it includes 4 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 in one or the other way.that's one of those two ways I'll try.this one y is equal to X plus 2 squared.minus 1 so how can you 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 negative 2 comma negative.1 so H is a negative 2 and K is negative.1 so remember to reverse this value but.not this one.so the vertex is at negative two.negative one 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 x squared.function so the next point is going to.be at negative one zero if we travel run.to the left from the vertex and up grun.that will give us another point negative.three zero so those are the x-intercepts.negative 3 0 and negative 1 0 now as we.travel two units to the right let's 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.in 0 into X and then you should get a.y-value of 3 2 squared is 4 minus 1 is 3.to find the x-intercept replace Y with 0.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 one is equal to X plus.two squared and now let's take the.square root of both sides the square.root of one will give you two numbers.plus or minus one the square root of x.plus two squared is just going to be X.plus two so you have two equations X.plus two is equal to a positive one and.X plus two is equal to negative one if.you subtract to one minus two will give.you an x intercept of negative one which.is this one here and negative one minus.two will give you an x intercept of.negative three 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 in the upward.direction so therefore we have a minimum.value the minimum value is the y.coordinate of the vertex so it's.negative one the axis of symmetry.is the equation X is equal to negative.two 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 negative 1 and since it's a.minimum value the lowest Y value is.negative 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 equal.to negative 2 times X minus 1 squared.plus 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 string.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 have the function y.was equal to x squared but now it's.negative 2 x squared so if you plug in 1.into X you should get negative 2 for y.so as you travel run unit to the right.you need to go down 2 units you have to.multiply it by 2 instead of going down 1.unit you need to go down 2 units so.that's going to take us to this point.and if we travel 1 to the left we need.to go down 2 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 y-value of the vertex is stream so.three minus eight will take us to a.y-value of negative 5 so the next point.is going to be 3 negative 5 and if we.travel to to the left it's going to be.negative 1 negative 5 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 0 for y and let's solve for x so.let's subtract 3 from both sides so.negative 3 is equal to negative 2 times.X minus 1 squared so let's divide both.sides by negative 2 so the two negative.signs will cancel and it's going to be 3.over 2 and that's equal to X minus 1.squared so let's take the square root of.both sides.so plus the minus root 3 over root 2.which if you rationalize it that's going.to be root 6 over 2 that's equal to X.minus 1 so if we add 1 to both sides we.get two answers it's 1 plus or minus.root 6 over 2 so this point here is 1.plus root 6 over 2 and this other.x-intercept is 1 minus root 6 over 2 now.since the graph opens downward we have a.maximum value and the max value is the.y-coordinate which is 3 and the axis of.symmetry which is this vertical line.here that's x equals 1 d domain is all.real numbers and what is the range so.notice that the lowest Y value is.negative infinity but the highest.sistering 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 squared plus 2x minus 8 how.would you grab this function so it's.positive x squared we know it's going to.open upward now if you want to find the.vertex you want to use this equation X.is equal to negative B divided by 2a.so this equation is in the form ax.squared plus BX plus C so a is the.number in front of X since we don't see.a number it's a 1 B is 2 C is negative 8.so let's find the x-coordinate of the.vertex so b is 2 a is 1 negative 2.divided by 2 is negative 1 so now to.find the y-coordinate let's replace X.with negative 1 so it's negative 1.squared plus 2 times negative 1 minus 8.negative 1 squared is negative 1 times.negative 1 which is positive 1 and 2.times negative 1 that's minus 2 so 1.minus 2 is negative 1 and negative 1.minus 8 is negative 9 so the.x-coordinate the vertex is negative 1 I.mean the y-coordinate is negative 9 so.but the vertex coordinate is negative 1.negative 9 so now that we have that.let's find the x-intercepts so let's.replace Y with 0 and let's solve for X.so notice that we have a trinomial where.the leading coefficient is 1 so we need.to factor it we need to find two numbers.that multiply to negative 8 but add to.the middle term 2 so this is going to be.4 and negative 2 4 times negative 2 is.negative 8 but 4 plus negative 2 is 2 so.it's a factor it's going to be X plus 4.times X minus 2 now to solve for X we.need to set each factor equal to 0 so.for the first one let's subtract 4 from.both sides so X is equal to negative 4.and for the next one let's add 2 to both.sides so X is equal to positive 2 so.those are the two x-intercepts negative.4 0 and 2 0.now let's find a y-intercept so let's.plug in 0 into the equation so we've got.to replace 0 with X so 0 squared plus 2.times 0 minus 8 is simply negative 8 so.the y-intercept is 0 negative 8 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 negative one negative nine so.the y-intercept is very close to the.vertex is 0 negative 8 so if it's put it.over here.so since the y-intercept is 1 unit to.the right of the vertex one unit to the.left must also share the same y value of.negative 8.now the x-intercepts are negative 4 0.and 2 0 notice that the x-coordinate of.the vertex is the average of the.x-intercepts if you average 2 and.negative 4 if you add them up and divide.by 2 this will give you the x-coordinate.the vertex which is negative 1 so now we.have enough points to make a graph out.of this equation.negative nine is all the way at the.bottom so let's plot the vertex force.which is negative one negative nine and.then the y-intercept which is zero.negative eight and we have another point.at negative two and negative eight and.then the x intercept at two zero and.negative four zero 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 of the vertex so that's at.x equals negative one and it represents.this line we have a minimum value and a.minimum value is the Y corner of the.vertex so it's negative nine be domain.as always is all real numbers and the.range.notice that the lowest Y value is.negative nine and the highest is.infinity so the range is from negative.nine 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 negative three but.add to two this is going to be positive.three and negative one so it's going to.be X plus three times X minus one.sitting intercepts are negative three.and one so as an ordered pair we can.write the X intercepts as negative three.comma 0 and 1 comma 0 now what is the.midpoint between negative 3 and 1 if we.add these two numbers and divide by 2 if.we average them what is the midpoint so.negative 3 plus 1 is negative 2 and.negative 2 divided by 2 is negative 1 so.this will give us the x coordinate of.the vertex now to prove it you can use.the equation X is equal to negative B.divided by 2a so B is the number in.front of X which is 2 and a is the.number in front of the x squared if you.don't see anything it's a 1 negative 2.divided by 2 is indeed negative 1 so now.let's find the y coordinate of the.vertex so let's plug the negative 1 into.the equation so negative 1 squared plus.2 times negative 1 minus 3 this is 1.minus 2 minus 3 so that's 1 minus 5.which is negative 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.squared so half of 2 is 1 so we need to.add 1 squared now for the right side to.be equal if we add 1 squared to the.right side we must also take away 1.squared from the right side so that we.haven't changed a value at the right.side so we have x squared plus 2x plus 1.minus 3 minus 1 or minus 4 so notice.that 1 minus 4 still equals to the.original value of negative 3 now how can.we factor x squared plus 2x plus 1 two.numbers that multiply to 1 but add to 2.are 1 and 1 so it's going to be X plus 1.times X plus 1 which we can simply write.it as X plus 1 squared minus 4 so that.is the equation in vertex form.and notice that you can get the vertex.from it which is negative 1 negative 4.and we have that here so that's another.way you can find the vertex.next let's find the y-intercept if we.replace X with zero we can see that it's.going to be negative three so we have.the point 0 negative 3 so now we can.organize everything into a table and so.let's start with the vertex which is.negative 1 negative 4 the y-intercept is.1 unit away from the vertex it's to the.right of it so one unit to the left must.also have a y-value of negative 3.now the x-intercepts are 1 0 and.negative 3 0 so as you can see these two.are the same and these two are the same.and makes it a lot easier if you Center.it around the vertex it's very easy to.find the miss importance so now we can.make the graph.now let's start with the vertex which is.negative 1 negative 4 and then we have.the point is 0 negative 3 and negative 2.negative 3 and then after that we have 1.0 and negative 4 0 so the graph looks.something like this you see that it has.a minimum value at negative 4 the axis.of symmetry is x equals negative 1 the.domain is all real numbers and the range.is from negative 4 to infinity so that's.it for this particular function now.let's try this word problem our ball is.thrown upward at a speed of 16 meters.per second from a cliff that is 32.meters higher now we're given the height.function this function tells us the.height at any time seen how long does it.take the ball to reach its maximum.height so let's say if it's at 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.negative 4.9 T squared plus 16 plus 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 negative x.squared which is a downward parabola so.to find the height.we need to find a y-coordinate of the.vertex and to 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 a negative B over 2a so.B is the number in front of T which is.16 and a is the number in front of T.squared which is 90 to 4.9 two times.negative four point nine is negative 9.8.the two negative signs will cancel so T.is going to be positive and 16 divided.by nine point eight will give us a T.value of one point six three so this is.the time it takes to reach the max and.pipe it's one point six three three 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 one point.six three three.so one point six three three squared.times negative four point nine that's.about negative thirteen point zero six.seven and then sixteen times one point.six three three that's twenty six point.one two eight and let's add 32 to it so.the maximum height occurs at positive 45.point zero six one 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 45 point.zero six and the time it takes to get.there is one point six three three so at.this point the ball is that it's 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 at ground level so we get the.fine we need to find the 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.zero is equal to negative four point.nine T squared plus 16 plus 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 negative B plus or minus.square root B squared minus 4ac divided.by 2a so B is 16 B squared or 16 squared.that's 256 minus 4 times a which is.negative 4.9 times C which is positive.32 divided by 2a or 2 times negative 4.point 9 negative 4 times negative 4.point 9 times 32 that's 600 27.2 and if.you add 256 to that that's 880 3.2 if.you take the square root you get twenty.nine point seven two so we now have is.it's negative 16 plus or minus twenty.nine point seven two which is equal to.this whole thing inside the radical.included in a radical divided by.negative nine point eight.so let's make some space so now we have.two possible answers the first one is.negative 16 plus twenty nine point seven.two divided by negative nine point eight.negative sixteen plus twenty nine point.seven two that's positive thirteen point.seven two if we divided by negative nine.point eight that will give us a negative.time value which is not what we're.looking for now if we try negative.sixteen minus twenty nine point seven.two divided by negative nine point eight.this will give us a positive value which.is going to be positive four point 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 thirty-two.the max height is forty five point oh.six the time it takes to reach the.maximum height one point six three three.but the time it takes to hit the ground.it's going to be four point six seven so.if that represents this answer what is.the other answer now notice that if you.extend the graph this way it's also.going to touch the x-axis at negative.one point four 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 four point six.seven 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 one negative five and you also.know the y-intercept 0 negative 4 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 minus H squared.plus K so H is 1 K is negative 5 so.let's plug it in so this is going to be.a times X minus now let's insert the.value of H which is 1 squared plus K.which is negative 5 so we have the.formula a X minus 1 squared minus 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 negative 4 and X with 0.since x is 0 Y is negative 4 and solve.for a so 0 minus 1 is negative 1.negative 1 squared is just 1 so if we.add 5 to both sides negative 4 plus 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 minus 1 squared.minus 5 in vertex form you'll notice.that since the graph opens upward it's a.positive x squared now sometimes you may.want the answer in standard form to.convert it from vertex form to standard.form you need to foil X minus 1 squared.X minus 1 squared is the same as X minus.1 times X minus 1 so x times X is x.squared x times negative 1 that's.negative x negative 1 times X that's.also negative x and negative 1 times 1.is plus 1 minus the 5 so X minus X is 2x.1 minus 5 is 94 so in standard form it's.x squared minus 2x minus 4 so that's how.you can write the equation in vertex.form and in standard form if you're.given D vertex and a point it could be a.y-intercept the x-intercept whatever the.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.negative 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 3 it's.the midpoint between the two.x-intercepts but we don't know the.y-value so we can't really use them so.if you have the x-intercepts 1 0 and 5 0.here's what you can do you can write x.minus 1 times X minus 5 and it's.factored for 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.negative 10 so let's replace Y with.negative 10 and let's replace the X with.0 this will allow us to solve for a so 0.minus 1 is negative 1 0 minus 5 is.negative 5 and negative 1 times negative.5 is positive 5 so if we divide both.sides by 5 we see that a is equal to.negative 2 so therefore the equation in.factored form is y is equal to negative.2 times X minus 1 times X minus 5 now to.put it in standard form we need to foil.x times X is x squared x times negative.5 that's negative 5 X negative 1 times X.negative x and negative 1 times negative.5 plus 5 and we still have a negative to.any outside so let's combine like terms.negative 5 X minus X that's negative 6x.now the next thing that we need to do is.distribute the negative to so it's going.to be negative 2x squared plus 12x minus.10 so that is the equation in standard.form now 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 run out the GCF but let's.take out the negative two negative two x.squared divided by negative two is x.squared and positive 12x divided by.negative two is negative 6x and then.we're going to leave a space and here's.a negative 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 negative 6 is.negative 3 but make it positive so it's.going to be positive 3 and then squared.now positive 3 squared is 9 now we have.to incorporate the negative 2 which is.negative 18 if you distribute the.negative 2 to the 9 so at this point we.added negative 18 to the right side of.the equation now so that the equation is.balanced we can either add negative 18.to the left side or positive 18 to the.right side so if you add negative 18 and.positive 18 to the right side the value.of the right side is the same remainder.and that's why this technique works so.now here's a 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.squared that's how you can factor it the.easy way now if you're unsure why that.works let's work it out.three squared is 3 times 3 that's 9.negative 10 plus 18 that's positive 8.so let's find two numbers that multiply.to 9 but that add to the middle term.negative 6 so what are those two numbers.negative 3 times negative 3 multiplied.to negative I mean 2 positive 9.a negative 3 plus negative 3 adds to.negative 6 so it's going to be X minus 3.times X minus 3 which we can simply.write as X minus 3 squared 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 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.associate with it so thanks for watching.and have a great day.

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  1. Ensure you have a efficient internet connection.
  2. Click the document which needs to be electronically signed.
  3. Click to the option of "My Signature” and drag it.
  4. You will be given choice after selecting 'My Signature'. You can choose your drawn signature.
  5. Create your e-signature and drag 'Ok'.
  6. Select "Done".

You have successfully finish the PDF sign . You can access your form and save it. Except for the e-sign choice CocoSign provides features, such as add field, invite to sign, combine documents, etc.

How to create an electronic signature for the Quadratic Functions Explained Form in Chrome

Google Chrome is one of the most welcome browsers around the world, due to the accessibility of a large number of tools and extensions. Understanding the dire need of users, CocoSign is available as an extension to its users. It can be downloaded through the Google Chrome Web Store.

Follow these basic tips to generate an e-signature for your form in Google Chrome:

  1. Direct to the Web Store of Chrome and in the search CocoSign.
  2. In the search result, select the option of 'Add'.
  3. Now, sign in to your registered Google account.
  4. Click the link of the document and drag the option 'Open in e-sign'.
  5. Select the option of 'My Signature'.
  6. Create your signature and put it in the document where you favor.

After adding your e-sign, save your document or share with your team members. Furthermore, CocoSign provides its users the options to merge PDFs and add more than one signee.

How to create an electronic signature for the Quadratic Functions Explained Form in Gmail?

Nowadays, businesses have altered their mode and evolved to being paperless. This involves the completing tasks through emails. You can easily e-sign the Quadratic Functions Explained Form without logging out of your Gmail account.

Follow the tips below:

  1. Download the CocoSign extension from Google Chrome Web store.
  2. Open the document that needs to be e-signed.
  3. Select the "Sign” option and generate your signature.
  4. Select 'Done' and your signed document will be attached to your draft mail produced by the e-signature software of CocoSign.

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How to create an e-signature for the Quadratic Functions Explained Form straight from your smartphone?

Smartphones have substantially replaced the PCs and laptops in the past 10 years. In order to solved problems for you, CocoSign helps finish your task via your personal phone.

A efficient internet connection is all you need on your phone and you can e-sign your Quadratic Functions Explained Form using the tap of your finger. Follow the tips below:

  1. Direct to the website of CocoSign and create an account.
  2. Then, drag and upload the document that you need to get e-signed.
  3. Select the "My signature" option.
  4. Put down and apply your signature to the document.
  5. Take a look at the document and tap 'Done'.

It takes you a short time to add an e-signature to the Quadratic Functions Explained Form from your phone. Get or share your form the way you want.

How to create an e-signature for the Quadratic Functions Explained Form on iOS?

The iOS users would be pleased to know that CocoSign provides an iOS app to help out them. If an iOS user needs to e-sign the Quadratic Functions Explained Form, utilize the CocoSign software with no doubt.

Here's guide add an electronic signature for the Quadratic Functions Explained Form on iOS:

  1. Download the application from Apple Store.
  2. Register for an account either by your email address or via social account of Facebook or Google.
  3. Upload the document that needs to be signed.
  4. Click to the place where you want to sign and select the option 'Insert Signature'.
  5. Write your signature as you prefer and place it in the document.
  6. You can save it or upload the document on the Cloud.

How to create an electronic signature for the Quadratic Functions Explained Form on Android?

The large popularity of Android phones users has given rise to the development of CocoSign for Android. You can download the software for your Android phone from Google Play Store.

You can add an e-signature for Quadratic Functions Explained Form on Android following these tips:

  1. Login to the CocoSign account through email address, Facebook or Google account.
  2. Click your PDF file that needs to be signed electronically by selecting on the "+” icon.
  3. Direct to the place where you need to add your signature and generate it in a pop up window.
  4. Finalize and adjust it by selecting the '✓' symbol.
  5. Save the changes.
  6. Get and share your document, as desired.

Get CocoSign today to help out your business operation and save yourself a great amount of time and energy by signing your Quadratic Functions Explained Form wherever.

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