Original post by mqb2766
Its a cubic and the roots correspond to the factors, so what do you think it should be?
Are you referring to part a)?
Original post by kit2247
Are you referring to part a)?
Start with a general cubic and use the info ...
Original post by kit2247
Are you referring to part a)?
I thought you were asking about the last part, so how do you get the cubic. Which part are you asking about.
For part a) it is looking for when the gradient of the curve is less than zero, i.e. the gradient is negative! If you look at the graph you can see that between 2 and 6 the gradient is decreasing so is negative. However 2 and 6 are both turning points so at these coordinates the gradient will be zero so you won't include them in your range of values. Therefore, x will belong to the reals such that 2<x<6! 
For b) if the line only intersects at one point, you will need to have a look at the graph. I recommend using a ruler and creating tangents to each point on the graph to visualise how many intersections there are. From this you can deduce that the graph will only intersect once when the value of k which is a constant so will be a horizontal straight line, can be either less than 0 as it will always have one intersection but if above will have two so has to be less or it can be more than 8 which again will only intersect graph once!
For c) since you know this is a cubic graph as it has 3 intersections, the 3 roots will be equal to what x value in the bracket will equal to. Therefore we have an intersection of the x-axis at 0 and 6 but it's important to note 6 is a repeated root since it touches the x-axis. Therefore we know the equation will be in the format y=m(x)(x-6)^2 where m is a constant to be found. If you plug in the value you are given for the maximum point for the equation you will be able to find m, which should be 1/4!
For b) if the line only intersects at one point, you will need to have a look at the graph. I recommend using a ruler and creating tangents to each point on the graph to visualise how many intersections there are. From this you can deduce that the graph will only intersect once when the value of k which is a constant so will be a horizontal straight line, can be either less than 0 as it will always have one intersection but if above will have two so has to be less or it can be more than 8 which again will only intersect graph once!
For c) since you know this is a cubic graph as it has 3 intersections, the 3 roots will be equal to what x value in the bracket will equal to. Therefore we have an intersection of the x-axis at 0 and 6 but it's important to note 6 is a repeated root since it touches the x-axis. Therefore we know the equation will be in the format y=m(x)(x-6)^2 where m is a constant to be found. If you plug in the value you are given for the maximum point for the equation you will be able to find m, which should be 1/4!
Original post by kit2247
Are you referring to part a)?
Original post by bmedic
For part a) it is looking for when the gradient of the curve is less than zero, i.e. the gradient is negative! If you look at the graph you can see that between 2 and 6 the gradient is decreasing so is negative. However 2 and 6 are both turning points so at these coordinates the gradient will be zero so you won't include them in your range of values. Therefore, x will belong to the reals such that 2<x<6!
For b) if the line only intersects at one point, you will need to have a look at the graph. I recommend using a ruler and creating tangents to each point on the graph to visualise how many intersections there are. From this you can deduce that the graph will only intersect once when the value of k which is a constant so will be a horizontal straight line, can be either less than 0 as it will always have one intersection but if above will have two so has to be less or it can be more than 8 which again will only intersect graph once!
For c) since you know this is a cubic graph as it has 3 intersections, the 3 roots will be equal to what x value in the bracket will equal to. Therefore we have an intersection of the x-axis at 0 and 6 but it's important to note 6 is a repeated root since it touches the x-axis. Therefore we know the equation will be in the format y=m(x)(x-6)^2 where m is a constant to be found. If you plug in the value you are given for the maximum point for the equation you will be able to find m, which should be 1/4!
For b) if the line only intersects at one point, you will need to have a look at the graph. I recommend using a ruler and creating tangents to each point on the graph to visualise how many intersections there are. From this you can deduce that the graph will only intersect once when the value of k which is a constant so will be a horizontal straight line, can be either less than 0 as it will always have one intersection but if above will have two so has to be less or it can be more than 8 which again will only intersect graph once!
For c) since you know this is a cubic graph as it has 3 intersections, the 3 roots will be equal to what x value in the bracket will equal to. Therefore we have an intersection of the x-axis at 0 and 6 but it's important to note 6 is a repeated root since it touches the x-axis. Therefore we know the equation will be in the format y=m(x)(x-6)^2 where m is a constant to be found. If you plug in the value you are given for the maximum point for the equation you will be able to find m, which should be 1/4!
Omg tysmmm, your explanation is amazing!
Glad I could help! <3
<3Original post by kit2247
Omg tysmmm, your explanation is amazing!
Original post by bmedic
For part a) it is looking for when the gradient of the curve is less than zero, i.e. the gradient is negative! If you look at the graph you can see that between 2 and 6 the gradient is decreasing so is negative. However 2 and 6 are both turning points so at these coordinates the gradient will be zero so you won't include them in your range of values. Therefore, x will belong to the reals such that 2<x<6!
For b) if the line only intersects at one point, you will need to have a look at the graph. I recommend using a ruler and creating tangents to each point on the graph to visualise how many intersections there are. From this you can deduce that the graph will only intersect once when the value of k which is a constant so will be a horizontal straight line, can be either less than 0 as it will always have one intersection but if above will have two so has to be less or it can be more than 8 which again will only intersect graph once!
For c) since you know this is a cubic graph as it has 3 intersections, the 3 roots will be equal to what x value in the bracket will equal to. Therefore we have an intersection of the x-axis at 0 and 6 but it's important to note 6 is a repeated root since it touches the x-axis. Therefore we know the equation will be in the format y=m(x)(x-6)^2 where m is a constant to be found. If you plug in the value you are given for the maximum point for the equation you will be able to find m, which should be 1/4!
For b) if the line only intersects at one point, you will need to have a look at the graph. I recommend using a ruler and creating tangents to each point on the graph to visualise how many intersections there are. From this you can deduce that the graph will only intersect once when the value of k which is a constant so will be a horizontal straight line, can be either less than 0 as it will always have one intersection but if above will have two so has to be less or it can be more than 8 which again will only intersect graph once!
For c) since you know this is a cubic graph as it has 3 intersections, the 3 roots will be equal to what x value in the bracket will equal to. Therefore we have an intersection of the x-axis at 0 and 6 but it's important to note 6 is a repeated root since it touches the x-axis. Therefore we know the equation will be in the format y=m(x)(x-6)^2 where m is a constant to be found. If you plug in the value you are given for the maximum point for the equation you will be able to find m, which should be 1/4!
why not just x(x-6)^2 ?
This is because there can always be a multiple in any equation. For example, if you were given a quadratic, it may very well be x^2 or it could be 2x^2, the same follows for any equation! So you always check, it could very well not have a multiple, so m would be 1 but they don't want you to assume this. People aren't used to seeing cubics with multiples so will commit this mistake and hence why they put the multiple there to catch you out!
Another example is an exponential equation, y=ae^kx is the general formula as it could have a multiple of a, hope that clears that up!
Another example is an exponential equation, y=ae^kx is the general formula as it could have a multiple of a, hope that clears that up!
Original post by Lldlldkeke
why not just x(x-6)^2 ?
Original post by Lldlldkeke
why not just x(x-6)^2 ?
It doesnt pass through the point (2,8). A cubic has 4 parameters, there are 3 roots which give you three of them and in factorized form, the fourth is the multiplier "a"
a(x-r1)(x-r2)(x-r3)
You calculate "a" such that the cubic passes through the given point.
(edited 10 months ago)
Easy, nice one guys
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