struggle with graphs and functions (1 Viewer)

[toonkev667]

anyone know answer for question 4?

 

[cossine]

q4. There will be an intersection along the x-axis at x=a, b, c.

The behaviour will depend on the powers.

Suppose n is even, then the graph will touch the x-axis at x=a. If n is not even then the graph will cut the x-axis. This assumes a != b or a !=c.

Similar things can be said about m, p.

If you learn about composite graphs e.g. multliplication of two functions this should make sense.

 

[Vaibhav123456]

a,b and c are the x intercepts of the graph (three x-intercepts). If n,m or p are 1 then their corresponding x-intercepts will just be normal x intercepts (cut or cross the x axis like a linear graph). If n,m or p are even their x-intercepts will be turning points (note that the turning pts will become flatter as n/p/m increase). If n/p/m are odd then the x-intercepts will be points of inflexion (which will again become flatter as the power n/m/p increases).

 

[cossine]

q4. There will be an intersection along the x-axis at x=a, b, c.

The behaviour will depend on the powers.

Suppose n is even, then the graph will touch the x-axis at x=a. If n is not even then the graph will cut the x-axis. This assumes a != b or a !=c.

Similar things can be said about m, p.

If you learn about composite graphs e.g. multliplication of two functions this should make sense.

 

[CM_Tutor]

Can you clarify what aspect(s) of question 4 are troubling you?

The question, IMO, reads as being much broader than I think is intended. The requirement that "" covers not only the cases where they are positive integers (which is what I suspect was intended) but extends to rational values and negative values, making the investigation much broader. Rational powers can produce roots with vertical tangents. Negative powers will lead to vertical asymptotes. Non-integer powers can also lead to a non-continuous domain.

My advice is to take as small, non-zero integers and to take . Then, the graph of will be the graph of a polynomial with roots at , , and . Your investigations / experiments with different values should show:

• When the power is 1, the root at will be a single root.
  • The graph of will simply crossing the -axis at .
  • The sign of can be changing from positive to negative or negative to positive, and so will resemble one of the roots of a parabola which crosses the -axis twice.
  • To be more precise, the appearance of the root will match that of a parabola with the corresponding sign change and where the concavity of the parabola (up or down) matches that of the root of in the vicinity of
  • That is, if is concave up near , a single root will resemble a parabola like looks near its roots... depending on the sign change of the graph of , it will be similar in appearance near to the parabola near either or .
  • If is concave down near , a single root will resemble a concave down parabola like looks near its roots... depending on the sign change of the graph of , it will be similar in appearance near to the parabola near either or .
• When the power is 2, the root at will be a double root.
  • The graph of will touch but not crossing the -axis.
  • The -axis will be a tangent to the curve at the point .
  • The sign of will not change: it will either be positive both before and after and zero at , or be negative both before and after and zero at .
  • In the region around , the curve will resemble either or near their double roots at the origin.
  • Which of these two parabolas resembles can be determined either by considering the concavity at (concave up resembles , whereas concave down resembles ) or by looking at the sign of around the root (if follows the pattern negative - zero - negative around , it resembles ; if follows the pattern positive - zero - positive around , it resembles )
• When the power is 3, the root at will be a triple root.
  • The graph of will have the -axis as a tangent at and it will cross the -axis.
  • The point will be a horizontal point of inflection on the curve .
  • The sign of will change around .
  • In the region around , the curve will resemble either or in the region around the origin.
  • Which cubic resembles can be determined either by examining the sign change around , or by looking at the concavity change: negative and concave down - zero - positive and concave up will resemble ; positive and concave up - zero - negative and concave down will resemble .
• When the power is 4, the root at will be a quadruple root.
  • The graph of will touch but not crossing the -axis.
  • The -axis will be a tangent to the curve at the point .
  • The sign of will not change: it will either be positive both before and after and zero at , or be negative both before and after and zero at .
  • In the region around , the curve will resemble either or near their quadruple roots at the origin.
  • Which of these two quartics resembles can be determined either by considering the concavity at (concave up resembles , whereas concave down resembles ) or by looking at the sign of around the root (if follows the pattern negative - zero - negative around , it resembles ; if follows the pattern positive - zero - positive around , it resembles )
• The same reasoning / approach applies to the values of , governing the behaviour around , and to the behaviour around that occurs with different values of .

So, for example, a curve like (which is certainly more complicated than would be expected in Advanced Maths) will have:

• a triple root at
• a triple root at ​
• a quadruple root at ​
• a single root at ​
• a double root at ​
• a triple root at ​

Since the function will have a positive value for sufficiently large (as every factor will be positive), it will be

• positive for
• zero at , and, since this a triple root and sign changes at odd-power roots
• negative for
• zero at , and, since this a double root and sign does not change at even-power roots
• negative for
• zero at , and, since this a single root and sign changes at odd-power roots
• positive for
• zero at , and, since this a quadruple root and sign does not change at even-power roots
• positive for - note, at , so the -intercept is positive, as expected
• zero at , and, since this a triple root and sign changes at odd-power roots
• negative for
• zero at , and, since this a triple root and sign changes at odd-power roots
• positive for

In appearance,

• the root at resembles the root of at the origin
• the root at resembles the root of at the origin
• the root at resembles the root of at (in that, in the immediate region around the root, is positive for -values below the root and negative for for -values above the root), or even more similarly (in that it also matches the concavity being down at the root ), the root of at
• the root at resembles the root of at the origin
• the root at resembles the root of at the origin
• the root at resembles the root of at the origin

Now, if you try the Desmos Calculator at https://www.desmos.com/calculator, you can explore these concepts.

As an example, attached is a graph of



The factor of 250,000 is to make the -scale reasonable. You can see the distinctive horizontal point of inflexion shapes of the triple roots at . You can see that the quadruple root at has a much broader / flatter shape around the root than does the double root at . You can see the single roots at and look like the roots / intercepts of a parabola.