Maths - Graphing Techniques

bonkysleuth SG

Need help with part (V)

The curve has equation y = (x^2 + ax + 4)/(x+b)

It is given that C has a vert. asymptote x = -1 and a stationary point at x =2.

Some answers for the guiding questions.

Values of a and b are -4 and 1 respectively. The equation of the other asymptote is y = x-5.

Now, deduce the no. of real roots of the equation (4-x^2)(x+1)^2 = (x^2 - 4x +4)^2

Of course we were told to sketch the graph of y = (x^2 -4x+ 4)/(x+1). After manipulating with the above formula, the graph becomes

(x^2 -4x+ 4)^2/(x+1)^2 = (4-x^2)

How do I relate the first equation to that given in the final one, which will allow me to find the no. of real roots?

Thanks

Mikethm SG

sketch y = 4-x^2

y= (x^2-4x+4)^2/(x+1)^2-------------(1)

y= 4-x^2--------------(2)

Look at the number of intersections. This is what you learnt in EM in graphical solution in Sec 3.

This is just a guess as to what the concept/idea being tested as I do not track the A level syllabus... so I might be wrong.

Oppsss didn't realise that the sketch was of y = (x^2-4x+4)/(x+1) :P

Forbiddensinner SG

Need help with part (V)

The curve has equation y = (x^2 + ax + 4)/(x+b)

It is given that C has a vert. asymptote x = -1 and a stationary point at x =2.

Some answers for the guiding questions.

Values of a and b are -4 and 1 respectively. The equation of the other asymptote is y = x-5.

Now, deduce the no. of real roots of the equation (4-x^2)(x+1)^2 = (x^2 - 4x +4)^2

Of course we were told to sketch the graph of y = (x^2 -4x+ 4)/(x+1). After manipulating with the above formula, the graph becomes

(x^2 -4x+ 4)^2/(x+1)^2 = (4-x^2)

How do I relate the first equation to that given in the final one, which will allow me to find the no. of real roots?

Thanks

Sketch sqrt (4-x^2) and - sqrt (4-x^2).

The intersections with your sketched graph will be the answer.

Oh, you only need to sketch from x= -2 to x= 2 for the two above graphs btw.

wee_ws SG

Hi,

Given an equation like

(4-x^2)(x+1)^2 = (x^2 - 4x +4)^2,

the first step is to rewrite it in such a way that one part looks like the expression of a graph we are already familiar with, i.e.

4-x^2 = (x^2 - 4x +4)^2 / (x+1)^2.

The next step is to determine the modification(s) one needs to make on the work of the earlier parts. In this case, a possibility may be:

1. Sketch y^2 = f(x), where f(x) is (x^2 -4x+ 4)/(x+1), based on the graph of y = f(x) which has already been sketched. [ This has been corrected below. ]

2. Insert the graph of y = 4-x^2 on the same diagram.

Finally, one determines the number of intersections between the two graphs and arrives at the number of roots for the equation one started with.

Thanks.

Cheers,
Wen Shih

anpanman SG

Hi,

1. Sketch y^2 = f(x), where f(x) is (x^2 -4x+ 4)/(x+1), based on the graph of y = f(x) which has already been sketched.

2. Insert the graph of y = 4-x^2 on the same diagram

Hi everyone, I have questions for bonky's question

I do have some questions regarding Mr Wee's method. When we draw the graph of y^2 =(x^2 -4x+ 4)/(x+1), is it the same as y = (x^2 -4x+ 4)/(x+1)? Or is this a totally diff. graph that requires to sketch another one with the GC? I dont understand why it is y^2 for (x^2 -4x+ 4)/(x+1) and only y^1 for 4 - x^2.

Another question is,

When we draw a parametric graph, if the range is not given, how do we know if the GC hides any part of the graph? Is there a way to observe such a trend?

Last question.

We were told to draw the graph for y = (x-1)^2 / (x+2) and x^2 + y^2 = 16. At the end, we were asked:

Hence show that the equation 2x^4 - 6x^2 - 68x - 63 =0 has exactly 2 real roots. Been looking at this question for hours, still no idea how to manipulate with the 2 equations I have.

Thanks for helpin'

wee_ws SG

Hi,

Sorry, I made a mistake. Allow me to clarify.

Starting with

(4-x^2)(x+1)^2 = (x^2 - 4x +4)^2,

we may rewrite it as

4-x^2 = (x^2 - 4x +4)^2 / (x+1)^2 = y^2, where

y = (x^2 -4x+ 4)/(x+1).

Finally, we obtain

x^2 + y^2 = 4.

Now, we just need to insert additionally, the graph of a circle given by

x^2 + y^2 = 4.

The number of roots will be determined by the number of intersections between the original curve that was sketched and the circle.

The range of a parameter is usually given. If not, consider the largest possible range, such that both x and y are well-defined.

Could

2x^4 - 6x^2 - 68x - 63 = 0

be the result of substituting

y = (x-1)^2 / (x+2)

into

x^2 + y^2 = 16,

and expanding it?

Thanks.

Cheers,
Wen Shih

anpanman SG

Anyway, is bonky in the same sch as me? your questions are the same as mine. lol

anpanman SG

Hi,

Sorry, I made a mistake. Allow me to clarify.

Starting with

(4-x^2)(x+1)^2 = (x^2 - 4x +4)^2,

we may rewrite it as

4-x^2 = (x^2 - 4x +4)^2 / (x+1)^2 = y^2, where

y = (x^2 -4x+ 4)/(x+1).

Finally, we obtain

x^2 + y^2 = 4.

Now, we just need to insert additionally, the graph of a circle given by

x^2 + y^2 = 4.

The number of roots will be determined by the number of intersections between the original curve that was sketched and the circle.

The range of a parameter is usually given. If not, consider the largest possible range, such that both x and y are well-defined.

Could

2x^4 - 6x^2 - 68x - 63 = 0

be the result of substituting

y = (x-1)^2 / (x+2)

into

x^2 + y^2 = 16,

and expanding it?

Thanks.

Cheers,
Wen Shih

thanks!