32: The function 32: The function © Christine Crisp “Teach A Level Maths” Vol. 2: A2 Core Modules

The function "Certain images and/or photos on this presentation are the copyrighted property of JupiterImages and are being used with permission under license. These images and/or photos may not be copied or downloaded without permission from JupiterImages" Module C3 Edexcel Module C4 AQA MEI/OCROCR

The function If you have either Autograph or a graphical calculator, draw the graph of You will have the following graph: On a calculator choose and

The function It is clearly the shape of a sin or cos function but it has been transformed. Can you describe, giving approximate values, the transformations from that give this curve? a stretch of s.f. 5 parallel to the y -axis, and a translation of approx. ANS:

The function The equation of the curve is approximately ( As the cosine curve keeps repeating we could translate much further, for example but there’s no point doing this. )

The function We need an exact method of finding constants R and  so that Using the addition formula the r.h.s. of (1) becomes Substitute in (1): ( R > 0 )

The function Since this is an identity, the l.h.s. and the r.h.s. must be exactly the same. So, the term must be the same on both sides

The function Since this is an identity, the l.h.s. and the r.h.s. must be exactly the same. So, the term must be the same on both sides

The function Since this is an identity, the l.h.s. and the r.h.s. must be exactly the same. So, the term must be the same on both sides and the term on both sides must be the same.

The function Since this is an identity, the l.h.s. and the r.h.s. must be exactly the same. So, the term must be the same on both sides and the term on both sides must be the same.

The function Since this is an identity, the l.h.s. and the r.h.s. must be exactly the same. So, we can equate the coefficients So, the term must be the same on both sides Coef. of : and the term on both sides must be the same.

The function Since this is an identity, the l.h.s. and the r.h.s. must be exactly the same. Coef. of : So, we can equate the coefficients So, the term must be the same on both sides Coef. of : and the term on both sides must be the same.

The function Since this is an identity, the l.h.s. and the r.h.s. must be exactly the same. Coef. of : So, we can equate the coefficients So, the term must be the same on both sides Be careful to check the signs Coef. of : and the term on both sides must be the same.

The function Since this is an identity, the l.h.s. and the r.h.s. must be exactly the same. We can now solve to find R and  Coef. of : So, we can equate the coefficients So, the term must be the same on both sides Coef. of : and the term on both sides must be the same.

The function Coef. of : The easiest way to find  is unusual. Divide (2) by (1) : This equation has an infinite number of solutions, but  gives us the translation of the cosine curve. We can take the principal value which will translate the curve by the least amount. ( 3 s.f. ) To find R we can square (1) and (2) and add them. Why does this give R ?

The function Coef. of : ( 3 s.f. ) R is positive because it gives the stretch from So,

The function Coef. of : ( 3 s.f. ) So,

The function Coef. of : ( 3 s.f. ) So,

The function SUMMARY To express in the form : Expand using the addition formula Write Equate the coefficients of and Divide the equations to find and solve for  Square and add the equations to get (or get this directly from the given expression) Choose the value of R > 0.

The function where a and b are constants and can be positive or negative. We’ve used as an example of the more general function Suppose we have. Let By choosing this addition formula, we have matched the signs on the l.h.s. and the r.h.s. By choosing this addition formula, we have matched the signs on the l.h.s. and the r.h.s. So, and are both positive and  is an acute angle. The translation from is less than

The function where a and b are constants and can be positive or negative. We’ve used as an example of the more general function Suppose we have. Can you complete this to find  correct to 3 d.p. and R ? Let

The function Coef. of : So, Tip: If you have a graphical calculator, check by drawing both forms. They should give one curve.

The function To express as a single trig ratio, we’ve used or Since we can translate instead of to get curves of the same form, we can also use If you can choose which form to use, it’s better to choose the version which, when expanded, gives the same signs for the corresponding terms as the original expression. Just look in the formula book to see which of the 4 addition formulae match the expression. and

The function Exercise For each of questions (1) to (4), select the expression that would be easiest to use from the 4 below (1) For choose (2) For choose (3) For choose (4) For choose For (3) and (4) the terms could be switched so that either or could be used.

The function One reason for expressing in one of the forms or e.g. so the max. is 13 and the min is  13. is that the stretch from or is obvious.

The function Exercise For the following (i)express in the given form where R > 0 and, giving  correct to 1 d.p. (ii) write down the minimum and maximum value of 1. in the form 2. in the form 3. in the form

The function Solution: Coef. of : So, The max. is 2 and the min is  in the form

The function Solution: Coef. of : So, The max. is 5 and the min is  in the form

The function Solution: So, The max. is 25 and the min is  in the form Coef. of :

The function

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The function SUMMARY To express in the form : Expand using the addition formula Write Equate the coefficients of and Divide the equations to find and solve for  Square and add the equations to get (or get this directly from the given expression) Choose the value of R > 0.

The function Coef. of : ( 3 s.f. ) So, ( 3 s.f. ) e.g. Express in the form Solution:

The function where a and b are constants and can be positive or negative. We’ve used as an example of the more general function Suppose we have. It is easier to solve for  if and are positive so we choose to use

The function Coef. of : So, Tip: If you have a graphical calculator, check by drawing both forms. They should give one curve.

The function To express as a single trig ratio, we’ve used or Since we can translate instead of to get curves of the same form, we can also use If can choose which form to use, it’s better to choose the version which, when expanded, gives the same signs for the corresponding terms as the original expression. Just look in the formula book to see which of the 4 addition formulae match the expression. and

The function One reason for expressing in one of the forms or e.g. so the max. is 13 and the min is  13. is that the stretch from or is obvious.