Sickmafs a level maths core 1 notes questions pages1to15 by Khalid Ahmad

A Level Maths Core 1 (Edexcel) Revision Notes with Questions FREE BOOK – you can download this book for free. FREE VIDEOS – videos that explain everything in this book. ONLINE COURSE – based on this book. Get all the above at sickmafs.com

Contents 1. Quadratic equations ………………………………………………..… 2. Straight line graphs ……………………………………………………. 3. Indices (Powers) ………………………………………………………... 4. Surds ………………………………………………………………………….. 5. Sketching graphs ……………………………………………………….. 6. Transforming graphs …………………………………………………. 7. Simultaneous equations …………………………………………….. 8. Arithmetic series ……………………………………………………….. 9. Inequalities ……………………………………………………………...... 10.Discriminant ………………………………………………………………. 11.Differentiation …………………………………………………………... 12.Integration …………………………………………………………………. 13.The Answers ………………………………………………………………. 14.Quotes from Buddha, Jesus, Muhammad, …………………. Einstein and Steve Jobs

p1 p3 p5 p7 p9 p11 p13 p15 p17 p19 p21 p23 p25 p27

How to use this book: You shouldn’t need to use a calculator for the questions in this book, unless stated otherwise. I just think calculators can be a distraction so I made questions that didn’t need them.

Each example is explained on video (for free) on my website sickmafs.com. To find the video that explains an example in my book, look for a tag like this Vid 3.1 . “Vid 3.2” means the second video of the third group of videos in that section (e.g. a section like Quadratic Equations in A level > Core 1). The bottom of each page tells you where the section you are reading is on my website.

I don’t give answers to the questions in this book at the back as my experience tells me that students trick themselves into believing they know the answers after looking at the answers. Instead, I’ve I make students put their answers to these or similar questions in the Sick Courses quizzes of my website, if they know the answers, the automated system will mark it right. If you’re stuck on something, ask me your questions by commenting on my videos.

1. Quadratic Equations Simple (HCF) factorising Solve ͸‫ ݔ‬ൌ ͻ‫ ݔ‬ଶ. ͸‫ ݔ‬ൌ ͻ‫ ݔ‬ଶ ֜ ͸‫ ݔ‬െ ͻ‫ ݔ‬ଶ ൌ Ͳ ֜ ͵‫ݔ‬ሺʹ െ ͵‫ݔ‬ሻ ൌ Ͳ

Vid 1

If ܾܽ ൌ Ͳ so ܽ ൌ Ͳ and/or ܾ ൌ Ͳ therefore ͵‫ݔ‬ሺʹ െ ͵‫ݔ‬ሻ ൌ Ͳ so ͵‫ ݔ‬ൌ Ͳ and ʹ െ ͵‫ ݔ‬ൌ Ͳ .

ଷ

Know this pattern.

ܽ‫ ݔ‬ଶ ൅ ܾ‫ ݔ‬൅ ܿ (trinomials) have ͵ bits.

Factorising trinomials i.e. ࢇ࢞૛ ൅ ࢈࢞ ൅ ࢉ ൌ ሺ࢖࢞ ൅ ࢗሻሺ࢓࢞ ൅ ࢔ሻ Using “guess” method to factorise ʹ‫ ݔ‬ଶ െ ͵‫ ݔ‬െ ʹ. Vid 3.1 ଶ

ʹ‫ ݔ‬െ ͵‫ ݔ‬െ ʹ ൌ ሺʹ‫ ݔ‬െ ͳሻሺ‫ ݔ‬൅ ʹሻ

Each (educated) guess at factorising should at least multiply to get the first and last bits of the original expression, in this case ʹ‫ ݔ‬ଶ and ͷ.

െ‫ ݔ‬൅ Ͷ‫ ݔ‬ൌ ͵‫ ݔ‬û

The “double chin” (the curvy lines look like a double chin) multiplies the bits that make the middle bit of the expanded bracket.

ൌ ሺʹ‫ ݔ‬൅ ͳሻሺ‫ ݔ‬െ ʹሻ

‫ ݔ‬െ Ͷ‫ ݔ‬ൌ െ͵‫ ݔ‬ü

Solve ʹ‫ ݔ‬ଶ െ ͳͳ‫ ݔ‬൅ ͳʹ ൌ ͹ by factorising.

This is the correct middle bit (i.e.െͳͳ‫)ݔ‬ which means ሺʹ‫ ݔ‬െ ͳሻሺ‫ ݔ‬െ ͷሻ is ʹ‫ ݔ‬ଶ െ ͳͳ‫ ݔ‬൅ ͷ factorised.

Vid 3.2

Rearrange equation to equal 0 to prepare for trick in next step.

ʹ‫ ݔ‬ଶ െ ͳͳ‫ ݔ‬൅ ͳʹ ൌ ͹ ʹ‫ ݔ‬ଶ െ ͳͳ‫ ݔ‬൅ ͷ ൌ Ͳ ሺʹ‫ ݔ‬െ ͳሻሺ‫ ݔ‬െ ͷሻ ൌ Ͳ ʹ‫ ݔ‬െ ͳ ൌ Ͳ ‫ ݔ‬െ ͷ ൌ Ͳ

Factorise using any method. ሺʹ‫ ݔ‬െ ͳሻሺ‫ ݔ‬െ ͷሻ ൌ Ͳ so ʹ‫ ݔ‬െ ͳ ൌ Ͳ and ‫ ݔ‬െ ͷ ൌ Ͳ. ʹ‫ ݔ‬െ ͳ ൌ Ͳ and ‫ ݔ‬െ ͷ ൌ Ͳ BOTH true for different values of ‫ݔ‬.

ଵ

‫ ݔ‬ൌ ‫ ݔ‬ൌ ͷ ଶ

Completing the square Write ‫ ݔ‬ଶ െ ͺ‫ ݔ‬൅ ͹ in the form ሺ‫ ݔ‬൅ ‫݌‬ሻଶ ൅ ‫ݍ‬ ሺ‫ ݔ‬െ Ͷሻଶ ൌ ‫ ݔ‬ଶ െ ͺ‫ ݔ‬൅ ͳ͸ ሺ‫ ݔ‬െ Ͷሻଶ െ ͻ ൌ ‫ ݔ‬ଶ െ ͺ‫ ݔ‬൅ ͹ ଶ

Means turning ‫ ݔ‬ଶ ൅ ܾ‫ ݔ‬൅ ܿ into ሺ‫ ݔ‬൅ ‫݌‬ሻଶ ൅ ‫ ݍ‬or ܽ‫ ݔ‬ଶ ൅ ܾ‫ ݔ‬൅ ܿ into ܽሺ‫ ݔ‬൅ ‫݌‬ሻଶ ൅ ‫ݍ‬

Vid 4.1

ଶ

Write െʹ‫ ݔ‬െ ʹͲ‫ ݔ‬൅ Ͷ͹ in the form ܽሺ‫ ݔ‬൅ ‫݌‬ሻ ൅ ‫ ݍ‬

Vid 4.2

Solve ‫ ݔ‬ଶ െ ͺ‫ ݔ‬൅ ͹ ൌ ͳ͸ by completing the square.

Vid 4.3

First get ሺ‫ ݔ‬൅ ܾሻଶ ൌ ‫ ݔ‬ଶ െ ͺ‫ ݔ‬൅ ܿ : Using ሺ‫ ݔ‬൅ ܾሻଶ ൌ ‫ ݔ‬ଶ ൅ ʹܾ‫ ݔ‬൅ ܾ ଶ ֜ ሺ‫ ݔ‬െ Ͷሻଶ ൌ ‫ ݔ‬ଶ െ ͺ‫ ݔ‬൅ ͳ͸

െʹ‫ ݔ‬ଶ െ ʹͲ‫ ݔ‬൅ Ͷ͹ ൌ െʹሺ‫ ݔ‬ଶ ൅ ͳͲ‫ݔ‬ሻ ൅ Ͷ͹ ൌ െʹሾሺ‫ ݔ‬൅ ͷሻଶ െ ʹͷሿ ൅ Ͷ͹ ൌ െʹሺ‫ ݔ‬൅ ͷሻଶ ൅ ͷͲ ൅ Ͷ͹ ൌ െʹሺ‫ ݔ‬൅ ͷሻଶ ൅ ͻ͹

Complete the square for ‫ ݔ‬ଶ ൅ ͳͲ‫ݔ‬

‫ ݔ‬ଶ െ ͺ‫ ݔ‬൅ ͹ ൌ ͳ͸ ሺ‫ ݔ‬െ Ͷሻଶ െ ͻ ൌ ͳ͸ ሺ‫ ݔ‬െ Ͷሻଶ ൌ ʹͷ ‫ ݔ‬െ Ͷ ൌ േ ͷ ‫ ݔ‬ൌ േ ͷ ൅ Ͷ ൌ ͻ and െͳ

Quadratic formula

Complete the square. േͷ means ͷ and െͷ

Quadratic formula: if ܽ‫ ݔ‬ଶ ൅ ܾ‫ ݔ‬൅ ܿ ൌ Ͳ Solve ൅ ͷ ൌ ͳͳ‫ݔ‬ ʹ‫ ݔ‬ଶ ൅ ͷ ൌ ͳͳ‫ ݔʹ ֜ ݔ‬ଶ െ ͳͳ‫ ݔ‬൅ ͷ ൌ Ͳ

֜ ‫ ݔ‬ൌ

ି௕ േξ௕ మ ିସ௔௖ ଶ௔

ʹ‫ ݔ‬ଶ െ ͳͳ‫ ݔ‬൅ ͷ ൌ Ͳ ֜ substitute ܽ ൌ ʹǡ ܾ ൌ െͳͳǡ ܿ ൌ ͷ into ‫ ݔ‬ൌ ‫ ݔ‬ൌ

The quadratic formula. It finds the solution to any quadratic equation.

Vid 5

ʹ‫ ݔ‬ଶ

Factorising is opposite of expanding. ͵‫ ݔ‬is highest common factor of ͸‫ ݔ‬and െͻ‫ ʹݔ‬.

͵‫ ݔ‬ൌ Ͳ ʹ െ ͵‫ ݔ‬ൌ Ͳ ଶ ‫ ݔ‬ൌ Ͳ ‫ ݔ‬ൌ

Difference of two squares i.e. ࢇ૛ െ ࢈૛ ൌ ሺࢇ െ ࢈ሻሺࢇ ൅ ࢈ሻ Factorise Ͷ‫ ݐ‬ଶ െ ͻ. Ͷ‫ ݐ‬ଶ െ ͻ ൌ ሺʹ‫ݐ‬ሻଶ െ ͵ʹ ൌ ሺʹ‫ ݐ‬െ ͵ሻሺʹ‫ ݐ‬൅ ͵ሻ Vid 2

Factorising, completing the square and the quadratic formula.

ି௕ േξ௕మ ିସ௔௖ ଶ௔

Quadratic must be arranged in the form ܽ‫ ݔ‬ଶ ൅ ܾ‫ ݔ‬൅ ܿ ൌ Ͳ before using the formula.

ିሺିଵଵሻ േඥሺିଵଵሻమ ିସൈଶൈହ ଵଵ േξଵଶଵ ି ସ଴ ଵଵ േ ଽ ൌ ൌ ଶൈଶ ସ ସ ଵଵ ି ଽ ଵଵ ା ଽ ൌ ൌ ସ ସ ʹ ʹͲ ൌ ൌ Ͷ Ͷ

ൌ ͲǤͷ

sickmafs.com > A level Maths > Core 1

ൌͷ

Exercise ͳ Factorise::

Exercise ͳ݃ Solve::

ͳǤ ͺ‫ ݔ‬ଶ െ ʹͲ‫ݔ‬ ͵Ǥ Ͷ െ ͳ͸‫ݔ‬ ͷǤ Ͷͷ݉‫ ݖݔ‬െ ʹ͹‫ ݔ‬ଶ ൅ ͳͺ‫ݔ‬

ʹǤ ͶǤ ͸Ǥ

͵‫ ݕݔ‬െ ͸‫ݕ‬ 8xy – 12y ‫ ݔ‬ଶ ‫ ݕ‬൅ Ͷ‫ ݔ‬ଶ ൅ ‫ ݔ‬ଶ ‫ ݕ‬ଶ

ͳǤ ͵Ǥ ͷǤ .

ሺ‫ ݔ‬െ ʹሻሺ‫ ݔ‬൅ ͵ሻ ൌ Ͳ ሺʹ‫ ݕ‬൅ ͳሻሺ‫ ݕ‬െ ͷሻ ൌ Ͳ ‫݌‬ሺ‫ ݌‬൅ ͹ሻ ൌ Ͳ

ʹǤ ͶǤ ͸Ǥ

ሺ͵‫ ݔ‬൅ Ͷሻሺ‫ ݔ‬െ ͷሻ ൌ Ͳ ሺ݉ െ Ͷሻሺ݉ ൅ ͺሻ ൌ Ͳ ‫ݔ‬ሺͶ‫ ݔ‬െ ͳሻ ൌ Ͳ

‫ ݔ‬ଶ ൅ ͳͲ‫ ݔ‬ൌ Ͳ ͷ െ ʹ‫ ݔ‬ൌ Ͳ ͺ‫ ݔ‬ଶ ൅ ʹͲ‫ ݔ‬ൌ Ͳ ͳʹ‫ ݔ‬ൌ ͸‫ ݔ‬ଶ ͳͲ‫ ݔ‬ଶ ൌ ͳͷ‫ݔ‬

ʹǤ ͶǤ ͸Ǥ ͺǤ ͳͲǤ

‫ ݔ‬െ ͷ‫ ݔ‬ଶ ൌ Ͳ ʹͳ‫ ݔ‬ଶ െ ͹‫ ݔ‬ൌ Ͳ Ͷ‫ ݔ‬ଶ െ ʹ‫ ݔ‬ൌ Ͳ ͺ‫ ݔ‬ଶ ൌ െʹͲ‫ݔ‬ Ͷ‫ ݕ‬ଶ െ Ͷ‫ ݕ‬ൌ Ͳ

ͳǤ ͵Ǥ ͷǤ ͹Ǥ ͻǤ

‫ ݔ‬ଶ ൅ Ͷ‫ ݔ‬൅ ͵ ‫ ݔ‬ଶ ൅ ͸‫ ݔ‬൅ ͷ ‫ ݔ‬ଶ െ ͸‫ ݔ‬൅ ʹͲ Ͷ‫ ݔ‬ଶ െ ͳ͸‫ ݔ‬൅ ͳͺ ʹ‫ ݔ‬ଶ െ ͺ‫ ݔ‬൅ Ͷ

ʹǤ ͶǤ ͸Ǥ ͺǤ ͳͲǤ

‫ ݔ‬ଶ ൅ ͳʹ‫ ݔ‬൅ ͶͲ ‫ ݔ‬ଶ ൅ ʹ‫ ݔ‬൅ ͳ ‫ ݔ‬ଶ െ ͺ‫ ݔ‬൅ ͳ͹ ͷ‫ ݔ‬ଶ െ ʹͲ‫ ݔ‬൅ ͹Ͳ ͵‫ ݔ‬ଶ ൅ ͳͺ‫ ݔ‬െ ͻ

Exercise ͳ Solve by factorising: ͳǤ ͵Ǥ ͷǤ ͹Ǥ ͻǤ

Exercise ͳܿ Factorise:

ͳǤ ͵Ǥ ͷǤ ͹Ǥ ͻǤ ͳͳǤ ͳ͵Ǥ ͳͷǤ

‫ ݔ‬ଶ െ ʹͷ ͳ͸ െ ‫ ݔ‬ଶ ͳ െ ͻ‫ ݕ‬ଶ ͷͲଶ െ Ͷͻଶ ʹͷ݉ଶ െ ͸Ͷ‫ ݐ‬ଶ ͵͸݁ ଶ െ Ͷ‫ ݎ‬ଶ ‫ ݕ‬ସ െ Ͷ݁ ଶ ͳʹܿ ସ െ ͹ͷ݁ ଶ

ʹǤ ͶǤ ͸Ǥ ͺǤ ͳͲǤ ͳʹǤ ͳͶǤ ͳ͸Ǥ

ͻ‫ ݔ‬ଶ െ Ͷ ‫ ݕ‬ଶ െ ͳͲͲ Ͷͻ െ ͳ͸‫ ݔ‬ଶ ͷͲͲଶ െ Ͷͻͻଶ ͳͲͲ݀ଶ െ Ͷͻ‫ ݐ‬ଶ ͳ͸݂ ଶ െ ͻ‫ ݔ‬ଶ ͻ െ ͳ͸‫ ݔ‬ସ ʹ݉ସ െ ͺ‫ݑ‬ଶ

Exercise ͳ݀ Factorise:

Exercise ͳ݄ Write in the form ሺ‫ ݔ‬൅ ܾሻଶ ൅ ܿ ‫ܽ ݎ݋‬ሺ‫ ݔ‬൅ ܾሻଶ ൅ ܿ:

Exercise ͳ݅ Write in the form ሺ‫ ݔ‬൅ ܾሻଶ ൅ ܿ or ܽሺ‫ ݔ‬൅ ܾሻଶ ൅ ܿ, then solve. ͳǤ ͵Ǥ ͷǤ ͹Ǥ ͻǤ ͳͳǤ ͳ͵Ǥ

ʹ‫ ݔ‬ଶ ൅ ͹‫ ݔ‬൅ ͵ ͵‫ ݔ‬ଶ ൅ ͷ‫ ݔ‬൅ ʹ ‫ ݔ‬ଶ ൅ Ͷ‫ ݔ‬൅ ͵ ͷ‫ ݔ‬ଶ െ ͳͶ‫ ݔ‬െ ͵ ʹ‫ ݔ‬ଶ ൅ ͻ‫ ݔ‬൅ Ͷ Ͷ‫ ݔ‬ଶ െ Ͷ‫ ݔ‬െ ͵ ‫ ݔ‬ଶ െ ͸‫ ݔ‬൅ ͷ ͸‫ ݔ‬ଶ ൅ ʹ͵‫ ݔ‬െ Ͷ

ʹǤ ͶǤ ͸Ǥ ͺǤ ͳͲǤ ͳʹǤ ͳͶǤ ͳ͸Ǥ

ʹ‫ ݔ‬ଶ ൅ ͵‫ ݔ‬൅ ͳ ͹‫ ݔ‬ଶ െ ͺ‫ ݔ‬൅ ͳ ‫ ݔ‬ଶ ൅ ͺ‫ ݔ‬൅ ͹ ʹ‫ ݔ‬ଶ െ ͳͳ‫ ݔ‬൅ ͷ ‫ ݔ‬ଶ െ ͵‫ ݔ‬൅ ʹ ʹ‫ ݔ‬ଶ െ ͹‫ ݔ‬െ ͷ ͵‫ ݔ‬ଶ ൅ ͳ͵‫ ݔ‬൅ Ͷ ͺ‫ ݔ‬ଶ ൅ Ͷͻ‫ ݔ‬൅ ͸

ͳǤ ͵Ǥ ͷǤ ͹Ǥ ͻǤ ͳͳǤ ͳ͵Ǥ ͳͷǤ

ͳǤ ͵Ǥ ͷǤ ͹Ǥ ͻǤ

ʹ‫ ݔ‬ଶ ൅ ͳʹ‫ ݔ‬൅ ͳͲ ͸‫ ݔ‬ଶ ൅ ͳͷ‫ ݔ‬൅ ͻ Ͷ‫ ݔ‬ଶ ൅ ͳͶ‫ ݔ‬൅ ͸ ʹͲ‫ ݔ‬ଶ െ ͷͲ‫ ݔ‬െ ͵Ͳ ͳͲ‫ ݔ‬ଶ െ ʹ͸‫ ݔ‬െ ͳʹ

ʹǤ ͶǤ ͸Ǥ ͺǤ ͳͲǤ

ʹ‫ ݔ‬ଶ ൅ ͸‫ ݔ‬൅ Ͷ ͳͲ‫ ݔ‬ଶ ൅ ͳͷ‫ ݔ‬൅ ͷ ͳʹ‫ ݔ‬ଶ െ ͳͺ‫ ݔ‬൅ ͸ ʹͲ‫ ݔ‬ଶ െ ͳͳͲ‫ ݔ‬െ ͷͲ ͸‫ ݔ‬ଶ ൅ ͳͺ‫ ݔ‬൅ ͺ

ͳǤ ͵Ǥ ͷǤ

Exercise ͳ݁ Factorise into the form ሺܽ‫ ݔ‬൅ ܾሻሺܿ‫ ݔ‬൅ ݀ሻ and ܽሺܾ‫ ݔ‬൅ ܿሻሺ݀‫ ݔ‬൅ ݁ሻ

Exercise ͳ݂ Solve by factorising: ʹ‫ ݔ‬ଶ െ ‫ ݔ‬െ ͵ ൌ Ͳ ͵‫ ݔ‬ଶ ൅ ͷ‫ ݔ‬൅ ʹ ൌ Ͳ ͷ‫ ݔ‬ଶ െ ͳ͸‫ ݔ‬൅ ͵ ൌ Ͳ ʹ‫ ݔ‬ଶ െ ͷ‫ ݔ‬ൌ ͵ ʹ‫ ݔ‬ଶ ൅ ͻ‫ ݔ‬ൌ െͶ

ʹǤ ͶǤ ͸Ǥ ͺǤ ͳͲǤ ͳʹǤ ͳͶǤ

‫ ݔ‬ଶ ൅ ͳʹ‫ ݔ‬൅ ʹ͹ ൌ Ͳ ‫ ݔ‬ଶ ൅ ʹ‫ ݔ‬െ ͷ ൌ Ͳ ‫ ݔ‬ଶ ൅ ͺ‫ ݔ‬ൌ Ͳ ‫ ݔ‬ଶ െ ͳʹ‫ ݔ‬ൌ ͳ͵ ‫ ݔ‬ଶ ൌ ͳͲ‫ ݔ‬െ ͻ ͷ‫ ݔ‬ଶ ൌ ʹͲ‫ ݔ‬൅ ͹Ͳ ͵‫ ݔ‬ଶ ൅ ͳͺ‫ ݔ‬െ ͻ ൌ Ͳ

ʹǤ ͶǤ ͸Ǥ ͺǤ ͳͲǤ ͳʹǤ ͳͶǤ ͳ͸Ǥ

͵‫ ݔ‬ଶ െ ͺ‫ ݔ‬൅ Ͷ ൌ Ͳ ʹ‫ ݔ‬ଶ െ ͳͳ‫ ݔ‬൅ ͷ ൌ Ͳ ʹ‫ ݔ‬ଶ െ ͹‫ ݔ‬െ ͷ ൌ Ͳ ‫ ݔ‬ଶ ൅ ͺ‫ ݔ‬൅ ͹ ൌ Ͳ ‫ ݔ‬ଶ ൌ െ͵‫ ݔ‬െ ʹ െ͹‫ ݔ‬െ ͷ ൌ െʹ‫ ݔ‬ଶ ͳ͸ െ ‫ ݔ‬ଶ ൌ Ͳ ݉ଶ െ ʹͷ ൌ Ͳ

Exercise ͳ݆ Solve using quadratic formula:

ͳǤ ͵Ǥ ͷǤ ͹Ǥ ͻǤ ͳͳǤ ͳ͵Ǥ ͳͷǤ

ͳǤ ͵Ǥ ͷǤ ͹Ǥ ͻǤ

‫ ݔ‬ଶ ൅ Ͷ‫ ݔ‬൅ ͵ ൌ Ͳ ‫ ݔ‬ଶ െ ͸‫ ݔ‬൅ ͷ ൌ Ͳ ‫ ݔ‬ଶ െ ͸‫ ݔ‬ൌ Ͳ ‫ ݔ‬ଶ െ ͸‫ ݔ‬ൌ ͹ ‫ ݔ‬ଶ ൌ ͻ െ ͺ‫ݔ‬ Ͷ‫ ݔ‬ଶ െ ͳ͸‫ ݔ‬െ ͺ ൌ Ͳ ʹ‫ ݔ‬ଶ െ ͺ‫ ݔ‬ൌ Ͷ

ʹǤ ͶǤ ͸Ǥ ͺǤ ͳͲǤ

sickmafs.com > A level Maths > Core 1

ʹ‫ ݔ‬ଶ െ ͻ‫ ݔ‬െ ͷ ൌ Ͳ ‫ ݔ‬ଶ െ ͸‫ ݔ‬൅ ͷ ൌ Ͳ ͵‫ ݔ‬ଶ ൅ ͺ‫ ݔ‬൅ Ͷ ൌ Ͳ ͵‫ ݔ‬ଶ ൌ ͺ‫ ݔ‬െ Ͷ െ͵‫ ݔ‬ଶ െ ‫ ݔ‬െ Ͷ ൌ Ͳ

ʹ‫ ݔ‬ଶ ൅ ͹‫ ݔ‬൅ ͵ ൌ Ͳ ͵‫ ݔ‬ଶ െ ͳͳ‫ ݔ‬െ Ͷ ൌ Ͳ ‫ ݔ‬ଶ ൅ Ͷ‫ ݔ‬൅ ͵ ൌ Ͳ ʹ‫ ݔ‬ଶ ൌ ‫ ݔ‬൅ ͵ ͵‫ ݔ‬ଶ ൌ െͳͳ‫ ݔ‬൅ Ͷ ͵‫ ݔ‬ଶ ൌ ͳͳ‫ ݔ‬൅ Ͷ ͻ‫ ݔ‬ଶ െ Ͷ ൌ Ͳ ‫ ݕ‬ଶ ൌ ͳͲͲ

Exercise ͳ݇ Find length of each shape:

A rectangle has a length ͳͷ ܿ݉ less than its width and area ͺͷͲ ܿ݉ଶ . A cylinder has length twice its radius and surface area ͶͲͲ ܿ݉ଶ . A cuboid has a square cross section and its length is twice its height. Its surface area is cuboid is ͺͲͲ ݉ଶ . Find its width, accurate to ͳ decimal place.

ʹǤ ͶǤ ͸Ǥ

A triangle has length ͷͲ ܿ݉ shorter than its perpendicular height and area ͵ͲͲ ܿ݉ଶ .

The surface area of a sphere is ͵Ͳ times its length (diameter). A cuboid has a square cross section and its length is half its height. Its surface area is cuboid is ͸ͲͲ ݉ଶ . Find its volume, accurate to ͳ decimal place.

Facts about straight line graphs.

2. Straight line graphs Simple straight line graphs (horizontal and vertical ones) ‫ ݕ‬ൌ ܿ graphs are horizontal and pass through ሺͲǡ ܿሻ ‫ ݔ‬ൌ ݇ graphs are vertical and pass through ሺ݇ǡ Ͳሻ

‫ݕ‬

Vid 1

Gradient and ࢟ intercept from graph with equation ࢟ ൌ ࢓࢞ ൅ ࢉ

‫ ݕ‬ൌ ݉‫ ݔ‬൅ ܿ ൝

݉ ൌ ൌ

Vid 2.1

୴ୣ୰୲୧ୡୟ୪ ୢ୧ୱ୲ୟ୬ୡୣ ୠୣ୲୵ୣୣ୬ ୲୵୭ ୮୭୧୬୲ୱ

Find gradient and ‫ ݕ‬intercept for graph ‫ ݕ‬ൌ ͵‫ ݔ‬െ ͳ gradient ൌ ͵, ‫ݕ‬-intercept ൌ െͳ

Vid 2.2

Find gradient and ‫ ݕ‬intercept for graph ͷ‫ ݕ‬൅ ʹ‫ ݔ‬ൌ ͵Ͳ ଶ ଶ ͷ‫ ݕ‬൅ ʹ‫ ݔ‬ൌ ʹͲ ֜ ‫ ݕ‬ൌ െ ‫ ݔ‬൅ Ͷ means gradient = െ , ‫ ݕ‬intercept ൌ ͸ Gradient =

௨௣ ௗ௜௦௧௔௡௖௘

ୟୡ୰୭ୱୱ ୢ୧ୱ୲ୟ୬ୡୣ

ହ

Vid 3.1

௬మ ି௬భ

௫మ ି௫భ

Distance between two points (coordinates) For ሺ‫ݔ‬ଶ ǡ ‫ݕ‬ଶ ሻ and ሺ‫ݔ‬ଶ ǡ ‫ݕ‬ଶ ሻ: Pythagoras ֜ distance ൌ ඥሺ‫ݔ‬ଶ െ ‫ݔ‬ଵ ሻଶ ൅ ሺ‫ݕ‬ଶ െ ‫ݕ‬ଵ ሻଶ Midpoints between two points (coordinates) For ሺ‫ݔ‬ଶ ǡ ‫ݕ‬ଶ ሻ and ሺ‫ݔ‬ଶ ǡ ‫ݕ‬ଶ ሻ:

Midpoint ൌ (middle ‫ݔ‬, middle ‫ )ݕ‬ൌ (average ‫ݔ‬, average ‫ )ݕ‬ൌ ቀ

Vid 3.2

௫మ ା ௫భ ௬మ ା ௬భ

Parallel and perpendicular graphs Vid 4 ௔ ‫ݕ‬ ‫ݕ‬ൌ ‫ݔ‬൅ܿ ௕

௕

‫ ݕ‬ൌ െ ‫ ݔ‬൅ ݀ ௔

ଶ

ቁ

௔ ௕

Vid 3.3

Parallel lines have the same gradient ௔ ௕ Perpendicular lines have gradients and െ (i.e. negative reciprocals) ௕

௔

ଷ

Example 2. ‫ ݕ‬ൌ ‫ ݔ‬൅ ͹ and ‫ ݕ‬ൌȂ ‫ ݔ‬൅ ͶͲͲ are perpendicular. ଷ

ଶ ଵ

Example 3. ‫ ݕ‬ൌȂ ͷ‫ ݔ‬൅ ͳ and ‫ ݕ‬ൌ ‫ݔ‬Ȃ Ͷ are perpendicular. ହ

Finding equation of line from coordinate and gradient Find equation of line that passes through ሺ͵ǡͶሻ and gradient ͷ. Vid 5.1 Substitute gradient ͷ into ‫ ݕ‬ൌ ݉‫ ݔ‬൅ ܿ: ‫ ݕ‬ൌ ͷ‫ ݔ‬൅ ܿ Substitute ሺ͵ǡͶሻ into ‫ ݕ‬ൌ ͷ‫ ݔ‬൅ ܿ: Ͷ ൌ ͷ ൈ ͵ ൅ ܿ so ܿ ൌ െͳͳ ‫ ݕ‬ൌ ͷ‫ݔ‬Ȃ ͳͳ

Find equation of line that passes through ሺ͵ǡͶሻ and gradient ͷ, give answer in the form ܽ‫ ݔ‬൅ ܾ‫ ݕ‬൅ ܿ ൌ Ͳ. Vid 5.2 Substitute gradient ͷ into ‫ ݕ‬െ ‫ݕ‬ଵ ൌ ݉ሺ‫ ݔ‬െ ‫ݔ‬ଵ ሻ: ‫ ݕ‬െ ‫ݕ‬ଵ ൌ ͷሺ‫ ݔ‬െ ‫ݔ‬ଵ ሻ Substitute ሺ͵ǡͶሻ into ‫ ݕ‬െ ‫ݕ‬ଵ ൌ ݉ሺ‫ ݔ‬െ ‫ݔ‬ଵ ሻ: ‫ ݕ‬െ Ͷ ൌ ͷሺ‫ ݔ‬െ ͵ሻ ‫ ݕ‬െ ͷ‫ ݔ‬൅ ͳͳ ൌ Ͳ

sickmafs.com > A level Maths > Core 1

‫ݔ‬ൌͶ

‫ ݕ‬ൌ ͵‫ ݔ‬െ ͳ ͷ

Ȃ ͳ

െʹ

‫ݔ‬

ଶ

‫ݕ‬ൌെ ‫ݔ‬൅Ͷ ହ

Find gradient ݃ between ሺ͵ǡ͸ሻ and ሺ͹ǡͻሻ ‫ݕ‬ ͻ ͵ ͸ Ͷ ‫ݔ‬ ͵ ͹ ଽି଺ ଷ ݃ൌ ൌ ଻ିଷ

ସ

Find distance ݀ between ሺ͵ǡ͸ሻ and ሺ͹ǡͻሻ ‫ݕ‬ ͻ

‫ݔ‬

Find midpoint ݉ between ሺ͵ǡ͸ሻ and ሺ͹ǡͻሻ ‫ݕ‬ ͹Ǥͷ

Example 1. ‫ ݕ‬ൌ ʹ‫ ݔ‬൅ ͸ͷͶ and ‫ ݕ‬ൌ ʹ‫ݔ‬Ȃ ͷ are parallel. ଶ

‫ݔ‬

͵ ͹ ݀ଶ ൌ ሺ͹Ȃ ͵ሻଶ ൅ ሺͻȂ ͸ሻଶ ݀ ൌ ඥሺ͹Ȃ ͵ሻଶ ൅ ሺͻȂ ͸ሻଶ ൌ ͷ

‫ ݕ‬ൌ ‫ݔ‬൅݀ ‫ݔ‬

୦୭୰୧୸୭୬୲ୟ୪ ୢ୧ୱ୲ୟ୬ୡୣ ୠୣ୲୵ୣୣ୬ ୲୵୭ ୮୭୧୬୲ୱ

Gradients between two points (coordinates) For ሺ‫ݔ‬ଵ ǡ ‫ݕ‬ଵ ሻ and ሺ‫ݔ‬ଶ ǡ ‫ݕ‬ଶ ሻ:

െʹ

‫ݕ‬

ܿ ൌ ‫ ݕ‬ሺ ሻ

ହ

െʹ ‫ݕ‬

‫ݕ‬ൌʹ

݉ൌቀ

͸ ͵

ଷ ା ଻ ଺ ା ଽ ଶ

ହ

ଶ

ቁ ൌ ሺͷǡ ͹Ǥͷሻ

‫ݔ‬

ଵ

െͷ ൌ െ ֜ negative reciprocal ൌ െ െ ൌ ଵ

gradient ݉ ൌ

ହ

௬ି௬భ ௫ି௫భ

֜ ‫ ݕ‬െ ‫ݕ‬ଵ ൌ ݉ሺ‫ ݔ‬െ ‫ݔ‬ଵ ሻ

െ‫ ݕ‬൅ ͷ‫ ݔ‬െ ͳͳ ൌ Ͳ would also be correct. 3

ଵ ହ

Exercise ʹܽ Find gradient and ‫ ݕ‬intercept for these equations of graphs then draw them: ͳǤ

͵Ǥ ͷǤ ͹Ǥ ͻǤ ͳͳǤ ͳ͵Ǥ

ଵ

ଶ

ʹǤ

‫ ݕ‬ൌ ‫ ݔ‬െ ͵ ଶ

‫ ݕ‬ൌ ͵‫ ݔ‬൅ ʹ

ͶǤ ͸Ǥ ͺǤ ͳͲǤ ͳʹǤ ͳͶǤ

‫ ݕ‬ൌ ‫ݔ‬൅ʹ

͵‫ ݕ‬ൌ ͻ‫ ݔ‬െ ͳͷ

ʹ‫ ݔ‬൅ ‫ ݕ‬ൌ ͳ

ͺ‫ ݔ‬൅ Ͷ‫ ݕ‬ൌ ͳʹ ହ

ͷ‫ ݕ‬ൌ ‫ ݔ‬െ ͳͲ ଷ

‫ ݕ‬ൌ ‫ ݔ‬െ ͷ ଷ

‫ ݕ‬ൌ െ͵‫ ݔ‬൅ ͷ ‫ ݕ‬ൌ െ‫ ݔ‬െ ͵

͵‫ ݕ‬ൌ െʹ‫ ݔ‬െ ͸ ‫ݔ‬൅‫ݕ‬ൌʹ

ͳͲ‫ ݔ‬െ ͷ‫ ݕ‬െ ͳͷ ൌ Ͳ ଽ

͵‫ ݕ‬ൌ Ȃ ‫ ݔ‬൅ ͸ ଶ

Exercise ʹ݀ Find equations for graphs that pass through the following coordinates and have the following gradients: ͳǤ

͵Ǥ ͷǤ

͹Ǥ

ͻǤ

ʹǤ

Passes through ሺʹǡ ͳሻ with gradient ͷ

Passes through ሺͶǡ ͳሻ with

ͶǤ ͸Ǥ

ହ gradient െ ଷ

Passes through ሺʹǡ െ͵ሻ with ହ gradient ଶ

Passes through ሺͲǡ ͵ሻ with gradient െʹ

ͺǤ

Passes through ሺെʹǡ ͵ሻ with

ͳͲǤ

ହ gradient ଶ

Passes through ሺͳǡ ͳሻ with gradient ͷ. Passes through ሺͳǡ Ͷሻ with gradient

ହ ଶ

Passes through ሺെʹǡ ͵ሻ with gradient

ହ ଶ

Passes through ሺͲǡ Ͳሻ with gradient െʹ

Passes through ሺെʹǡ ͵ሻ with gradient െ

ଶ ଷ

Exercise ʹܾ Between the coordinate pairs below, find the ሺܽሻ gradient ሺܾሻ distance ሺܿሻ midpoint :

Exercise ʹ݁ Find equations, in the form ‫ ݕ‬ൌ ݉‫ ݔ‬൅ ܿ , for graphs that pass through the following pairs of coordinates:

ͳǤ ͵Ǥ ͷǤ ͹Ǥ ͻǤ

ሺ͵ǡ͸ሻ and ሺ͹ǡͻሻ ሺͳǡʹሻ and ሺͷǡͷሻ

ሺെͳǡͲሻ and ሺͶǡͳʹሻ

ሺെ͵ǡ െͶሻ and ሺെ͸ǡ െͺሻ

ሺ͵ǡ Ͷሻ and ሺെ͸ǡ െͺሻ

ʹǤ ͶǤ ͸Ǥ ͺǤ ͳͲǤ

ሺͳǡʹሻ and ሺͳͲǡͳͶሻ

ሺͲǡͶሻ and ሺͳʹǡͻሻ

ሺെ͵ǡ െͶሻ and ሺͲǡͲሻ

ሺെͳǡ െʹሻ and ሺെͳͲǡ െͳͶሻ

ሺെͳǡ ʹሻ and ሺͳͳǡ െ͹ሻ

Exercise ʹܿ For the graphs below, find equations of graphs that are ሺܽሻ parallel with ‫ ݕ‬intercept ͵ ሺܾሻ perpendicular with ‫ ݕ‬intercept െͳ: ͳǤ

͵Ǥ

ͷǤ

͹Ǥ ͻǤ

ଷ

‫ ݕ‬ൌ ‫ ݔ‬െ ʹ ହ

ସ

‫ ݕ‬ൌ െ ‫ ݔ‬൅ ͵ ହ

‫ ݕ‬ൌ ‫ݔ‬൅ʹ

ʹ‫ ݕ‬ൌ െ͹‫ ݔ‬െ ͳ ‫ݕ‬൅‫ݔ‬െͻൌͲ

ଷ

ʹǤ

‫ ݕ‬ൌ ‫ ݔ‬൅ ʹ ହ

‫ ݕ‬ൌ െ ‫ ݔ‬െ ͵

ͶǤ

ହ

͸Ǥ

ͺǤ ͳͲǤ

sickmafs.com > A level Maths > Core 1

ଵ

‫ ݕ‬ൌ െ‫ ݔ‬െ ͵

͵‫ ݕ‬ൌ െͶ‫ ݔ‬െ ͵ͷ ‫ݕ‬൅‫ݔ‬൅͹ൌͲ

ͳǤ ͵Ǥ ͷǤ ͹Ǥ ͻǤ

ሺ͵ǡ͸ሻ and ሺ͹ǡͻሻ

ሺͳǡʹሻ and ሺͷǡͷሻ

ሺെͳǡͲሻ and ሺͶǡͳʹሻ

ሺെ͵ǡ െͶሻ and ሺെ͸ǡ െͺሻ ሺ͵ǡ Ͷሻ and ሺെ͸ǡ െͺሻ

ʹǤ ͶǤ ͸Ǥ ͺǤ ͳͲǤ

ሺͳǡʹሻ and ሺͳͲǡͳͶሻ ሺͲǡͶሻ and ሺͳʹǡͻሻ

ሺെ͵ǡ െͶሻ and ሺͲǡͲሻ

ሺെͳǡ െʹሻ and ሺെͳͲǡ െͳͶሻ ሺെͳǡʹሻ and ሺͳͳǡ െ͹ሻ

Exercise ʹ݂ Find equations, in the form ܽ‫ ݔ‬൅ ܾ‫ ݕ‬൅ ܿ ൌ Ͳ, for graphs that pass through the following pairs of coordinates: ͳǤ ͵Ǥ ͷǤ ͹Ǥ ͻǤ

ሺ͵ǡ͸ሻ and ሺ͹ǡͻሻ

ሺͳǡʹሻ and ሺͷǡͷሻ

ሺെͳǡͲሻ and ሺͶǡͳʹሻ

ሺെ͵ǡ െͶሻ and ሺെ͸ǡ െͺሻ ሺ͵ǡ Ͷሻ and ሺെ͸ǡ െͺሻ

ʹǤ ͶǤ ͸Ǥ ͺǤ ͳͲǤ

ሺͳǡʹሻ and ሺͳͲǡͳͶሻ ሺͲǡͶሻ and ሺͳʹǡͻሻ

ሺെ͵ǡ െͶሻ and ሺͲǡͲሻ

ሺെͳǡ െʹሻ and ሺെͳͲǡ െͳͶሻ ሺെͳǡ ʹሻ and ሺͳͳǡ െ͹ሻ

3. Indices (Powers)

Vid 2.1

Index rules ൈ ܽ ௬ ൌ ܽ ௫ା௬ Example 1. ͹ଶ ൈ ͹ଷ ൌ ͹ଶାଷ ൌ ͹ହ

Vid 2.2

ܽ௫

Vid 1.1

Example 2. ͺ‫ ݔ‬ଶ ‫ ݕ‬ଽ ൈ ʹ‫ ݕ ଺ ݔ‬ସ ൌ ͺ ൈ ʹ ൈ ‫ ݔ‬ଶ ൈ ‫ ଺ ݔ‬ൈ ‫ ݕ‬ଽ ൈ ‫ ݕ‬ସ ൌ ͳ͸‫ݕ ଼ ݔ‬ଵଷ

ܽ ௫ ൊ ܽ ௬ ൌ ܽ ௫ି௬ Example 1. ͷଵଶ ൊ ͷଷ ൌ ͷଵଶିଷ ൌ ͷଽ ଶ ଽ

Example 2. ͺ‫ ݕ ݔ‬ൊ ʹ‫ ݕ ݔ‬ൌ

ሺܽ ௫ ሻ௬ ൌ ܽ ௫௬

௕ ௫

௕ೣ

ଷ ିଶ

ହ ଶ

ൌቀ ቁ ൌ ଷ

ସ ିଷ

Example 2. Ͷିଷ ൌ ቀ ቁ ଵ

ೣ

೤

ଶ௫ ల ௬ ర

଼

ൌ ൈ ଶ

మ

௫

ଷమ

మ

ଵ ଷ

ଶହ ଽ

ସ

Example 1. ͺయ ൌ ൫ξͺ൯ ൌ Ͷ భ

ൌ

ൌቀ ቁ ൌ

ଶ

య

ହమ

௫ల

Vid 1.5

ଵ

ସయ

ൌ

Vid 1.6

ଵ

Example 2. ͳ͸మ ൌ ൫ξͳ͸൯ ൌ ξͳ͸ ൌ േͶ

Combining index rules మ

௫మ

Vid 1.4

Example 2. ͳ͵଴ ൌ ͳ

ቀ ቁ ൌ ቀ ቁ ൌ ೣ Example 1. ቀ ቁ ௕ ௔ ହ ௔

ܽ೤ ൌ ൫ ξܽ൯

଼௫ మ ௬ వ

Example 2. ሺ͵‫ ݕݔ‬ହ ሻଶ ൌ ͵ଶ ‫ ݔ‬ଶ ሺ‫ ݕ‬ହ ሻଶ ൌ ͻ‫ ݔ‬ଶ ‫ݕ‬ଵ଴

Example 1. ͷ଴ ൌ ͳ

మ

Solve ͺି య . ͺି య ൌ

ଵ

మ

଼ య

ൌ

య

ଵ

మ

൫ ξ଼൯

ൌ

ଵ ସ

Why ܽ ௫ ൊ ܽ ௬ ൌ ܽ ௫ି௬ : ‫ݔ‬ ܽ௫ ൊ ܽ௬ ൌ

௔ൈ௔ൈ௔ǥൈ௔ൈ௔ൈ௔ǥൈ௔

௔ൈ௔ൈ௔ǥൈ௔

‫ ݔ ݕ‬Ȃ ‫ݕ‬

Vid 1.2

ൈ

௬వ ௬ర

Example 1. ሺͳͲଶ ሻଷ ൌ ͳͲଶൈଷ ൌ ͳͲ଺ ൌ ͳͲͲͲͲͲͲ

ܽ଴ ൌ ͳ ௔ ି௫

଺ ସ

Why ܽ ௫ ൈ ܽ ௬ ൌ ܽ ௫ା௬ : ܽ ௫ ൈ ܽ ௬ ൌ ܽ ൈ ܽǤ Ǥൈ ܽ ൈ ܽ ൈ ܽǤ Ǥൈ ܽ ൌ ܽ ௫ା௬ ‫ݔ‬ ‫ݕ‬ ‫ ݔ‬൅ ‫ݕ‬

Vid 2

ଵ

଺ସ

Vid 2.3

ൌ Ͷ‫ݔ‬

Vid 1.3

ିସ ହ

‫ݕ‬

Vid 2.4

ൌ ܽ ௫ି௬

Why ሺܽ ௫ ሻ௬ ൌ ܽ ௫௬ : ‫ݕ‬

‫ݕ‬

ሺܽ ௫ ሻ௬ ൌ ܽ ௫ ൈ ܽ ௫ ǥ ൈ ܽ ௫ ൌ ܽ ௫ା௫ǥା௫ ൌ ܽ ௫௬ Explaining negative powers: ܽଷ ൌ ܽ ൈ ܽ ൈ ܽ ൊܽ ܽଶ ൌ ܽ ൈ ܽ ൊܽ ܽଵ ൌ ܽ ൊܽ ܽ଴ ൌ ͳ ൊܽ ଵ ܽିଵ ൌ ௔ ൊܽ ଵ ܽିଶ ൌ మ ௔ ൊܽ ଵ ܽିଷ ൌ య ௔

ܽି௡ ൌ

ଵ

௔೙

௫ ି௡

௫

֜ቀ ቁ

ൌ

ଵ଺

, if ܽ ൌ

௬

ൌ

ଵ

ೣ ೙ ቀ ቁ ೤

ൌ

௫೙

௬೙

Re-writing in index form Write

ଵ଺

൅

ଷ௫ మ ିଶ௫

௫మ ξ௫ ଵ଺ ଷ௫ మ ିଶ௫ ௫మ

൅

ξ௫

ൌ ͳ͸‫ି ݔ‬ଶ ൅

ଷ௫ మ భ

௫మ

െ

య మ

in the form ‫ ݔܣ‬௠ ൅ ‫ ݔܤ‬௡ ൅ ‫ ݔܥ‬௣ ଶ௫ భ

௫మ

ൌ ͳ͸‫ି ݔ‬ଶ ൅ ͵‫ ݔ‬െ ͵‫ݔ‬

భ మ

An advance trick Solve ͺ ௫ାଶ ൌ Ͷ ௫ Vid 4 ͺ ௫ାଶ ൌ Ͷ ௫ ֜ ሺʹଷ ሻ௫ାଶ ൌ ሺʹଶ ሻ௫ ֜ ʹଷ௫ା଺ ൌ ʹଶ௫ ֜ ͵‫ ݔ‬൅ ͸ ൌ ʹ‫ݔ‬ ‫ ݔ‬ൌ െ͸

sickmafs.com > A level Maths > Core 1

Vid 3

ͳ͸‫ି ݔ‬ଶ ൌ ͳ͸ ൈ ଷ௫ మ ିଶ௫ ξ௫

ଷ௫ మ భ ௫మ

െ

ൌ

ଶ௫ భ ௫మ

ଷ௫ మ ξ௫

െ

ଵ

௫మ

ଶ௫

ξ௫ భ

ൌ

௫మ

ଷ௫ మ భ ௫మ

െ భ

ଶ௫ భ

௫మ

య

భ

ൌ ͵‫ ݔ‬ଶିమ െ ʹ‫ݔ‬ଵିమ ൌ ͵‫ ݔ‬మ െ ʹ‫ ݔ‬మ

Convert everything to same base i.e. 2 Using ሺܽ ௫ ሻ௬ ൌ ܽ ௫௬ ܽ௫ ൌ ܽ௬ ֜ ‫ ݔ‬ൌ ‫ݕ‬

Exercise ͵ Find the value of following (as an integer or fraction): ͳǤ ͵Ǥ ͷǤ ͹Ǥ ͻǤ ͳͳǤ ͳ͵Ǥ ͳͷǤ ͳ͹Ǥ ͳͻǤ ʹͳǤ

ʹ͵Ǥ

ʹͷǤ

ʹ͹Ǥ

͵ସ ʹ଺ ͷ଴ ͳଵ଴ Ͳସ ͳʹଵ ͳଷସ ͶͲ଴ ͷʹଵ ʹିଵ ͷିଶ

ʹǤ ͶǤ ͸Ǥ ͺǤ ͳͲǤ ͳʹǤ ͳͶǤ ͳ͸Ǥ ͳͺǤ ʹͲǤ ʹʹǤ

ହ ଷ

ቀ ቁ

ʹͶǤ

ଷ ହ ିଶ

ቀ ቁ

ʹ͸Ǥ

ଷ ଷ ିଵ

ቀ ቁ

ʹͺǤ

ସ

͸ଷ Ͷଷ ͷସ ͳ଼଴ Ͳଵ ʹ଴ ͳଷ Ͳ଻ ͷଵ ʹିଷ ͳିଷ

଻ ଶ

ቀ ቁ

ଶ ଷ ିଷ

ቀ ቁ

ଶ ଷ ଴

ቀ ቁ ସ

ʹ͵Ǥ

ʹͷǤ ʹ͹Ǥ ʹͻǤ ͵ͳǤ ͵͵Ǥ ͵ͷǤ

͵͹Ǥ ͵ͻǤ ͶͲǤ

ܽଷ ൈ ܽଶ ܾ ଷ ൈ ܾ ିଶ Ͷܽଷ ൈ ʹܽଶ ͸‫ ݔ‬ൈ ‫ି ݔ‬ହ ʹܽିଷ ൈ ܽଶ ൈ ʹܽଽ ͵ܽଷ ܾ ଶ ൈ ʹܾܽ ିଶ ܽହ ൊ ܽଶ ܾ ଷ ൊ ܾ ିଶ ͺܽଷ ൊ െʹܽଶ ͸‫ ݔ‬ൊ ‫ି ݔ‬ହ ͳͶܽିଷ ‫ ݔ‬ଶ ൊ ʹܽଽ ʹ͹ܽଷ ܾ ଶ ͵ܾܽ ିଶ

ͻܽଷ ൈ ͵ܾ ଶ ͵ܽ ൈ ܾ ିଷ

ሺ‫ ݔ‬ଶ ሻଷ ሺͶ‫ ݔ‬ଶ ሻଷ ሺ͹݉ହ ‫ ݔ‬ଷ ሻଶ ʹሺ͹݉ହ ‫ ݔ‬ଷ ሻଶ ሺͶ‫ ݎ‬ଶ ሻଷ ൊ െͶ‫ ݎ‬ଶ ሺ‫ ݎ‬ଶ ሻଷ ൈ ‫ ݎ‬ଶ ‫ݎ‬ସ

ሺʹ‫ ݎ‬ଶ ሻଷ ൈ ‫ ݎ‬ଶ Ͷ‫ି ݎ‬ସ

ሺʹ‫ ݔ‬ଶ ሻସ ൈ ሺ͵‫ ݔ‬ଶ ሻଷ ʹ‫ି ݔ‬ଵ଴ ൈ ሺ͵‫ ݔ‬ଶ ሻଷ

ͳǤ

͵Ǥ

ͷǤ

͹Ǥ

ͻǤ

ͳͳǤ

ͳ͵Ǥ

ͳͷǤ

Exercise ͵ Simplify: ͳǤ ͵Ǥ ͷǤ ͹Ǥ ͻǤ ͳͳǤ ͳ͵Ǥ ͳͷǤ ͳ͹Ǥ ͳͻǤ ʹͳǤ

Exercise 3 c Find equations for graphs with that pass through the following points with the following gradients: మ

ఱ

ͺయ

ʹǤ

భ

ͺయ

ͶǤ

భ

ʹͷమ ଷ଺

ቀ ቁ ସଽ

͸Ǥ

ସଽ

଺ସ

ͺǤ

ቀ ቁ ଶହ

భ

ଷ଺ ି మ

ቀ ቁ

ସ

భ

ఱ

ͺି య

ͳͶǤ

య

మ

ͺି య

ͳ͸Ǥ

ͺయ

భ

ͳ͸ି మ

ͳʹǤ

ͺି య

భ

ቀ ቁ

భ

భ మ

଼ଵ ି మ

ͳͲǤ

ʹͷି మ

భ

ͳʹͷయ

భ మ

భ

͵ʹఱ

Exercise 3 d Find x in the following: ʹǤ ͶǤ ͸Ǥ ͺǤ ͳͲǤ ͳʹǤ ͳͶǤ ͳ͸Ǥ ͳͺǤ ʹͲǤ ʹʹǤ

ʹͶǤ

ʹ͸Ǥ

ʹͺǤ ͵ͲǤ ͵ʹǤ ͵ͶǤ ͵͸Ǥ

͵ͺǤ

ͶͲǤ

ͶʹǤ

sickmafs.com > A level Maths > Core 1

ܾ଺ ൈ ܾ଻ ݊ିଽ ൈ ݊ିଶ ͳͲ‫ ݒ‬ଷ ൈ ʹǤͷ‫଺଺ ݒ‬ ͹݆ ଽ ൈ െ݆ ିସ ͺܿ ଻ ൈ ʹܿ ିଶ ൈ ܿ ିଵ ͵‫ ݔ‬ସ ‫ ݖ‬ൈ ‫ ݖ‬ହ ‫ ݔ‬ସ െ݊ଷ ൊ ଼݊ ‫ି ݔ‬ହ ൊ ‫ି ݔ‬ଶ ݀ଷ ൊ Ͷ݀ଶ ͻ‫ ଼ ݔ‬ൊ െ‫ି ݔ‬ହ ͳʹܽିଷ ‫ ݔ‬ଶ ൊ ͵ܽଽ ͳͺܽଷ ܾ ଶ ͸ܾܽ ିଷ

ͺ଼݉ ൈ ʹ݊ Ͷ݉ ൈ ݊ିହ

െሺ‫ ݔ‬ଶ଴ ሻିଶ ሺʹ݁ሻହ ሺ͵݁ ଻ ݂ ଼ ሻସ ͵ሺ଼݉ ‫ ଻ି ݔ‬ሻ଻ ሺ͵Ͳ‫ ݔ‬ସ ሻିଶ ൊ ͷ‫ݔ‬ ሺ͵‫ݎ‬ሻଷ ൈ ‫ ݎ‬ଽ െ‫ି ݎ‬ଷ

ሺ͵‫ ݔ‬ଷ ሻଶ ൈ ‫ ݔ‬ଶ ൈ ‫ ݔ‬ହ ͻ‫ି ݔ‬ଵ

െሺ‫ ݔ‬ଶ ሻହ ൈ െሺ͸‫ ݔ‬ସ ሻଷ ͵‫ି ݔ‬ଵ ൈ ሺʹ‫ݔ‬ሻଷ

ͳǤ

͵Ǥ

ͷǤ

͹Ǥ

ͻǤ

ͳͳǤ ͳ͵Ǥ ͳͷǤ

ͳ͹Ǥ

ͳͻǤ ʹͳǤ ʹ͵Ǥ ʹͷǤ

ʹ͹Ǥ

ʹͻǤ

͵ͳǤ

͹ ௫ ൌ Ͷͻ ͹௫ ൌ ଷ ௫

ସଽ

ଶ

ଶ ି௫ ହ

ଵ

ቀ ቁ ൌ ቀ ቁ

ʹǤ

ଵ

ͺ௫ ൌ

ଵ

ଽ

ସ

ൌ

ʹ௫ ൌ

ͶǤ ͸Ǥ

ଶହ

ͺǤ

ସ

ͳͲǤ

଼

ʹ ௫ ൌ ͷͳʹ ͷ௫ ൌ ͷ ͺ௫ ൌ ͳ

ͳʹǤ ͳͶǤ ͳ͸Ǥ

ͳͺǤ

଼

Ͷ ௫ାଶ ൌ ͺ ௫ ʹ͹ ௫ିହ ൌ ͻଶ௫ ʹ ௫ାଶ ൌ ͳ͸ ௫ାଷ ଺

ൌ ‫ ݁ܣ‬௫

௘మ

ଷ௘ିଶ௘ య ξ௘

ଶξ௡ ௡మ

ൌ

͵݁ ௬

ൌ ‫ ݊ܣ‬௫

௬ య ିଶ௬ మ ௬య

൅

ൌ݉െ

ʹͲǤ ʹʹǤ ʹͶǤ ʹ͸Ǥ

‫ ݁ܥ‬௫

݊‫ ݕ‬௫

ʹͺǤ

͵ͲǤ

͵ʹǤ

ͷ ௫ ൌ ͳʹͷ ͵௫ ൌ ସ ௫

ଵ

ଶ଻

ቀ ቁ ൌ ଷ

ଶ଻

ଷ ௫

ଵ଺

ቀ ቁ ൌ ଶ

଺ସ

ଵ

͵௫ ൌ

଼ଵ

ʹ௫ ൌ

ଵ଺

଼ଵ

ʹͷ ௫ ൌ ͷ ͳ͸ ௫ ൌ ͳ ͵Ǥͳ ௫ ൌ ͵Ǥͳ ଵ

ͳ͸ ௫ିଶ ൌ Ͷ ௫ ͳ͸ଶ௫ିଵ ൌ ͺଶ௫ ͷ ௬ାଵ ൌ ʹͷ ௬ାସ ଵ

ξ௧

ൌ ‫ݐ‬௫

ସ௠ା௠య ξ௠

଻ξ௡ ௡షయ ଶ௬

ଶ యξ௬

ൌ ‫݉ܣ‬௬ ൅ ݉ ௫

ൌ ‫ ݊ܣ‬௫

ൌ ‫ ݕܣ‬௘ 6

Vid 1.1

4. Surds

In all these examples, we are simplifying surd expressions, which just means writing them in their shortest form.

Surd rules ଶ

ଶ

ξܽ ൈ ξܽ ൌ ൫ξܽ൯ ൌ ܽ Vid 1.2

Example 1. ൫ξͻ൯ ൌ ͻ

ଶ

Square-rooting and squaring are inverse functions so they undo each other.

Example 2. ξͷ ൈ ξͷ ൌ ൫ξͷ൯ ൌ ͷ

You can split surds into products of the original. One of the surds should have a square number so that it can be simplified.

ξܽଶ ܾ ൌ ξܽଶ ൈ ξܾ ൌ ܽξܾ Example 1. ξͳͺ ൌ ξͻ ൈ ξʹ ൌ ͵ ൈ ξʹ ൌ ͵ξʹ Vid 1.3

௔

ට௕ ൌ

Example 2. ξ͹ͷ ൌ ξʹͷ ൈ ξ͵ ൌ ͷξ͵ ଽ

ξ௔ ξ௕

ସ

Vid 1.4

Example 2. ට

ଵଶ ଶହ

Prime factor trees can help find factors that are square numbers e.g. Using prime factor tree: ͳ͵ͷ ൌ ͵ ൈ ͵ ൈ ͵ ൈ ͷ ֜ ͻ is square factor of ͳ͵ͷ so ξͳ͵ͷ ൌ ξͻ ൈ ξͳͷ ൌ ͵ξͳͷ

ଷ

Example 1. ට ൌ

ଶ

ൌ

ξସൈξଷ ହ

ൌ

ଶξଷ ହ

Just like ͵‫ ݔ‬൅ ͻ‫ ݔ‬ൌ ͳʹ‫ݔ‬.

ܽξ‫ ݔ‬൅ ܾξ‫ ݔ‬ൌ ሺܽ ൅ ܾሻξ‫ ݔ‬Example 1. ͵ξͷ ൅ ͻξͷ ൌ ͳʹξͷ Vid 2.1

Example 2. ξͳͶ͹ െ ξʹ͹ ൌ ͹ξ͵ െ ͵ξ͵ ൌ Ͷξ͵

ܽξ‫ ݔ‬ൈ ܾξ‫ ݔ‬ൌ ܾܽ‫ݔ‬

Example 1. ͵ξͷ ൈ ͻξͷ ൌ ͵ ൈ ͻ ൈ ξͷ ൈ ξͷ ൌ ʹ͹ ൈ ͷ ൌ ͳ͵ͷ

Vid 2.2

Example 2.

௔ξ௫ ௕ξ௫

௔

ൌ

Example 1.

௕

Vid 2.3

Example 2.

൫͹ ൅ ξ͵൯൫ͳ െ ξ͵൯ ൌ ͹ െ ͹ξ͵ ൅ ξ͵ െ ͵ ൌ Ͷ െ ͸ξ͵ ଷξଶ ହξଶ

ସξଶ ξ଼

ൌ

ଷ ହ

ସξଶ ଶξଶ

ൌʹ

Rationalise denominators: Basic method: ௞

ൌ

ξ௔

௞

ξ௔

ൈ

ξ௔ ξ௔

ൌ

௞ξ௔

Example 1.

௔

Example 2. Advance method: ௞

ξ௔േξ௕

ൌ

௞

ൈ

ξ௔‫ט‬ξ௕

ହ

ൈ

ξ଻ିξଶ

ξ௔േξ௕

Example 1. ହ

ξ଻ାξଶ

ൌ

ξ଻ାξଶ

Example 2. ଵ

ହିξଷ

ൌ

ଵ

ହିξଷ

ൈ

ξ௔‫ט‬ξ௕

ξ଻ିξଶ

ହାξଷ ହାξଷ

ൌ

ൌ ൌ

௞൫ξ௔‫ט‬ξ௕൯

ଶହିଷ

ସ

ൌ

ହξଶ

ହ

ξଷ

ൌ

ൈ

ସ

ହξଶ

ξଷ ξଷ

ൈ

ൌ

ξଶ ξଶ

ହξଷ Vid 3.2 ଷ

ൌ

ସξଶ ଷ

Vid 3.3

଻ିଶ

ൌ

ହ൫ξ଻ିξଶ൯ ହ

Vid 3.1

ξͳͶ͹ ൌ ξͶͻ ൈ ξ͵ ൌ ͹ξ͵ and ξʹ͹ ൌ ξͻ ൈ ξ͵ ൌ ͵ξ͵. Simplify surds into multiples of a surd, in this case ξ͵ , so that you can add or subtract them.

Cancel by ξʹ.

Irrational numbers are numbers like π that go on for an infinite number of decimal places and do not follow a pattern. More precisely, irrational numbers can’t be put into fractions. Since square roots of numbers (except that of square numbers) are irrational, rationalising the denominator just means getting rid of the square root from the bottom of the fraction. Using ξܽ ൈ ξܽ ൌ ሺξܽሻଶ ൌ ܽ.

Simplifying with surds means rationalising the denominator and simplifying the fraction.

‫ ט‬is the opposite of േ i.e. ܽ േ ܾ ‫ ܿ ט‬means ܽ ൅ ܾ െ ܿ or ܽ െ ܾ ൅ ܿ

௔ି௕

ହ൫ξ଻ିξଶ൯

ହାξଷ

ହ

ξଷ

Surds are things with square roots in them e.g. ξͺ. This topic is just about doing tricks with square roots.

Using the difference of two squares. ൌ ξ͹ െ ξʹ

Vid 3.4

ହାξଷ ଶଶ

sickmafs.com > A level Maths > Core 1

Exercise Ͷܽ Find the value of the following (as a fraction or decimal):

Exercise Ͷ݀ Simplify:

ͳǤ

͵Ǥ ͷǤ ͹Ǥ ͻǤ ͳͳǤ ͳ͵Ǥ ͳͷǤ ͳ͹Ǥ

ሺξͳ͸ሻଶ ଶ

൫ξͻ൯ ሺξͷሻଶ ሺξ͹ሻଶ ξ͹ ൈ ξ͹ ξͷ ൈ ξͷ ξ͵ ൈ ξ͵ ሺʹξ͹ሻଶ ሺͲξʹሻଶ

ͳ͵Ǥ

ͳͷǤ

ͳ͹Ǥ ͳͻǤ ʹͳǤ ʹ͵Ǥ

ξͳͺ ൌ ܽξʹ ξͷͲͲ ൌ ܽξͷ ξͶͷ ൌ ܽξͷ ξʹ͹ ൌ ܽξ͵ ξͻͺ ൌ ܽξʹ ξͺͲͲ ൌ ܽξʹ ଵ଺

ට

ଶହ

ଷ଺

ට

ଽ

ൌ

௔ ହ

଺

௔

ʹͷǤ ʹ͹Ǥ ʹͻǤ ͵ͳǤ ͵͵Ǥ ͵ͷǤ

ͳǤ ͵Ǥ ͷǤ ͹Ǥ

ͷξ͵ ൅ ʹξͶͺ ξͳͺ ൅ ξʹͲͲ ξ͸Ͳ െ ξͳ͵ͷ ξͳͺͲ െ ξʹͲ ൅ ξʹͷ

ሺξ͵͸ሻଶ ሺξͶሻଶ ሺξͳ͹ሻଶ ሺξ͸ሻଶ ξͳʹ ൈ ξͳʹ ξͺ ൈ ξͺ ξͻ ൈ ξͻ ሺ͵ξͶሻଶ ሺͷξͺሻଶ

ʹǤ ͶǤ ͸Ǥ ͺǤ ͳͲǤ ͳʹǤ

ͳͶǤ

ͳ͸Ǥ

ͳͺǤ ʹͲǤ ʹʹǤ ʹͶǤ

ʹ͸Ǥ ʹͺǤ ͵ͲǤ ͵ʹǤ ͵ͶǤ ͵͸Ǥ

ξ͹ʹ ൌ ܽξʹ ξͺͲͲ ൌ ܽξʹ ξʹͺ ൌ ܽξ͹ ξͳʹ ൌ ܽξ͵ ξ͵ʹ ൌ ܽξʹ ξʹͲͲ ൌ ܽξʹ ଷ଺

ට

ସଽ

଼ଵ

ට

ସ

ൌ ൌ

௔ ଻

ଽ

௔

͸ξʹ ൅ ξʹ ൌ ܽξʹ ͸ξ͵ െ ξ͵ ൌ ܽξ͵ ξ͵ʹ െ ξͺ ൌ ܽξʹ ͵ξͳʹ ൅ ͸ξ͵ െ ξʹ͹ ൌ ܽξ͵ ʹξͷ ൌ ξܽ ʹξ͹ ൌ ξܽ ͳͲξ͵ ൌ ξܽ ሺξʹ ൅ ͵ሻሺξʹ െ ͷሻ ሺͳ ൅ ξͷሻሺͶ െ ξͷሻ ሺξ͵ ൅ ͷሻሺξ͵ ൅ ͷሻ

Exercise Ͷܿ Simplify:

͵Ǥ ͷǤ

଺ξହ

ͳͲǤ

ହξଷ

ͻǤ

ͳͳǤ

ͳ͵Ǥ

ʹǤ

ଶξଷ

͹Ǥ

ͷξʹ ൈ ʹξʹ ξ͵ ൈ ξͶͷ ξͷͲ ൈ ͵ξʹ ସξହ ξଵଶ ξଷ

ξଵ଼ ξହ଴

ଵ଴ξ଺ ξଷ଴଴

ʹξ͵ ൈ ͹ξ͵ ξͷ ൈ ξͶͷ ξʹͲ ൈ ʹξͷ

ͶǤ ͸Ǥ

଼ξ଻

ͺǤ

ͳʹǤ

ͳͶǤ

ͳ͸Ǥ

଺ξ଻ ଷξଶ ଽξଶ

ξଷଶ ξଶ ξ଼

ξହ଴

ଶ଴ξ଼ ξସ଴଴

Exercise Ͷ݁ Rationalise the denominator:

͵ξͷ ൅ ͻξͷ ൌ ܽξͷ ͵ξ͵ െ ͳ͹ξ͵ ൌ ܽξ͵ ξ͹ͷ െ ξʹ͹ ൌ ܽξ͵ ͵ξʹ ൅ ͸ξ͵ʹ െ ξͳͺ ൌ ܽξʹ ͹ξʹ ൌ ξܽ ͵ξ͹ ൌ ξܽ ʹξʹ ൌ ξܽ ሺξʹ ൅ ͵ሻሺξʹ െ ͷሻ ሺξ͵ ൅ ͷሻሺξ͵ െ ͷሻ ሺξ͵ ൅ ξͷሻሺξ͵ ൅ ξͷሻ

ʹǤ ͶǤ ͸Ǥ ͺǤ ͳͲǤ ͳʹǤ ͳͶǤ ͳ͸Ǥ ͳͺǤ

ͳͷǤ

Exercise Ͷܾ Find ܽ: ͳǤ ͵Ǥ ͷǤ ͹Ǥ ͻǤ ͳͳǤ

ଶ

ͳǤ

ξହ ଻

͵Ǥ ͷǤ

ξଷ ξଶ

ξହ

ξ଺

͹Ǥ

ଶξହ

ଷξ଺

ͻǤ

ͳͳǤ

ͳ͵Ǥ

ͳͷǤ

ͳ͹Ǥ

ͳͻǤ

ʹͳǤ

ଶξହ

ଷξ଺

ଶξଵ଼

ଷିξ଺ ξ଻ ଻

ଷିξ଺ ξ଻

ଷିξ଺ ξ଼

ξ଻ାξଶ ି଺

ξ଺ ା ଷ

ଵ

ʹǤ

ξଷ ଵ

ͶǤ ͸Ǥ

ξହ

ξଶ ξସ

ξଷ

ͺǤ

ͳͲǤ

ͳʹǤ

ͳͶǤ

ͳ͸Ǥ

଼ξହ ହξ଻ ଻ξହ

ξଶ ହξଵ଼

ξ଺ Ȃ ଷ ξ଻ ଻

ξ଺ Ȃ ଷ ξ଻

ͳͺǤ

ଷ ା ξ଺

ʹͲǤ

ξ଺ାଶ

ʹͲǤ

ξଷ

ξ଻ିξଶ ξ଼

Exercise Ͷ݂ Simplify: ʹǤ ͶǤ ͸Ǥ ͺǤ

sickmafs.com > A level Maths > Core 1

Ͷξͷ ൅ ʹξͶͷ ξͳ͸ʹ ൅ ξͺ ξͳʹ െ ξ͹ͷ ξͳͺ െ ξͷͲ ൅ ξͺ

ͳǤ

͵Ǥ

ͷǤ

ଵ଴

ξହ ξ଼

ξ଻ିξହ ξଶ

ξ଼ାଷ

ͳǤ

͵Ǥ

ͷǤ

଺

ξଷ ξ଼

ଷାξହ ξଵଶ

ଷିξହ

5. Simultaneous equations Simultaneous equations by elimination Solve the simultaneous equations ʹ‫ ݕ‬൅ ‫ ݔ‬ൌ ͻ ‫ݕ‬൅‫ ݔ‬ൌͶ ʹ‫ ݕ‬൅ ‫ ݔ‬ൌ ͹ ͳ െ ‫ ݕ‬൅ ‫ ݔ‬ൌ Ͷ ʹ ‫ ݕ‬ൌ ͵ Substitute ‫ ݕ‬ൌ ͵ into ʹ : ሺ͵ሻ ൅ ‫ ݔ‬ൌ Ͷ ‫ݔ‬ൌͳ

Vid 1 Vid 2.1

Solve the simultaneous equations ݂ െ ʹ݃ ൌ ͳ ʹ݂ ൅ ʹ݃ ൌ ͳͶ ݂ െ ʹ݃ ൌ ͳ ͳ ൅ ʹ݂ ൅ ʹ݃ ൌ ͳͶ ʹ ͵݂ ൌ ͳͷ ݂ ൌ ͷ Substitute ݂ ൌ ͷ into ͳ : ሺͷሻ െ ʹ݃ ൌ ͳ ݃ൌʹ Solve the simultaneous equations ͵ܽ െ ܾ ൌ ͸ ܽ ൅ ʹܾ ൌ ʹ ͵ܽ െ ܾ ൌ ͸ ͳ ܽ ൅ ʹܾ ൌ ʹ ʹ ͳ ൈ ʹ ֜ ͸ܽ െ ʹܾ ൌ ͳʹ ͵ ൅ ܽ ൅ ʹܾ ൌ ʹ ʹ ͹ܽ ൌ ͳͶ ܽ ൌ ʹ Substitute ܽ ൌ ʹ into ʹ : ሺʹሻ ൅ ʹܾ ൌ ʹ ܾൌͲ

Vid 2.2

Vid 2.3

Simultaneous equations by substitution Solve the simultaneous equations ‫ ݕ‬െ ‫ ݔ‬ൌ ͳ Vid 3 ‫ ݕ‬ଶ ൅ ‫ ݔ‬ଶ ൌ ʹͷ ‫ ݕ‬െ ‫ ݔ‬ൌ ͳ ͳ ֜ ‫ ݕ‬ൌ ͳ ൅ ‫͵ ݔ‬ ‫ ݕ‬ଶ ൅ ‫ ݔ‬ଶ ൌ ʹͷ ʹ Substitute ͵ into ʹ : ሺͳ ൅ ‫ݔ‬ሻଶ ൅ ‫ ݔ‬ଶ ൌ ʹͷ ͳ ൅ ʹ‫ ݔ‬൅ ‫ ݔ‬ଶ ൅ ‫ ݔ‬ଶ ൌ ʹͷ ʹ‫ ݔ‬ଶ ൅ ʹ‫ ݔ‬െ ʹͶ ൌ Ͳ ‫ ݔ‬ଶ ൅ ‫ ݔ‬െ ͳʹ ൌ Ͳ ሺ‫ ݔ‬൅ Ͷሻሺ‫ ݔ‬െ ͵ሻ ൌ Ͳ ‫ ݔ‬ൌ െͶ ‫ ݔ‬ൌ ͵ Substitute ‫ ݔ‬ൌ െͶ into ͵ ֜ ‫ ݕ‬ൌ െ͵ Substitute ‫ ݔ‬ൌ ͵ into ͵ ֜ ‫ ݕ‬ൌ Ͷ Solving simultaneous equations for graphs Find coordinates where graphs ‫ ݕ‬ൌ Ͷ‫ ݔ‬൅ ͷ and ‫ ݕ‬ൌ ‫ ݔ‬ଶ intersect. Vid 4 ‫ ݕ‬ൌ Ͷ‫ ݔ‬൅ ͷ ͳ and ‫ ݕ‬ൌ ‫ ݔ‬ଶ ʹ Substitute ʹ into ͳ : ‫ ݔ‬ଶ ൌ Ͷ‫ ݔ‬൅ ͷ ‫ ݔ‬ଶ െ Ͷ‫ ݔ‬െ ͷ ൌ Ͳ ሺ‫ ݔ‬൅ ͳሻሺ‫ ݔ‬െ ͷሻ ൌ Ͳ ‫ ݔ‬ൌ െͳ ‫ ݔ‬ൌ ͷ Substitute ‫ ݔ‬ൌ െͳ into ʹ ֜ ‫ ݕ‬ൌ ͳ Substitute ‫ ݔ‬ൌ ͷ into ʹ ֜ ‫ ݕ‬ൌ ʹͷ ֜ ሺെͳǡͳሻ ֜ ሺͷǡʹͷሻ

Using more than one equation together to solve more than one variable (letter). For every variable you want to solve, you will need an equation, so to solve ͵ variables you will need ͵ equations solved simultaneously. To understand this, if you roll two die and get numbers ‫ ݔ‬and ‫ ݕ‬and then you’re told ‫ ݔ‬൅ ‫ ݕ‬ൌ ͺ, that wouldn’t be enough to know ‫ ݔ‬and ‫ݕ‬. If you’re then told ‫ ݕݔ‬ൌ ͳͷ, then you would know ‫ ݔ‬ൌ ͵ and ‫ ݕ‬ൌ ͷ. Hence, you needed two equations to find ‫ ݔ‬and ‫ݕ‬. Subtract equations when you have the same thing in both, in this case ‫ ݔ‬.

This works because subtracting the left sides of two equations should always be same as subtracting right sides of those equations. Add equations when you have the opposite thing in both, in this case െʹ݃ and ൅ʹ݃.

This works because the sum of the right sides of two equations should equal the sum of the left sides of those equations. ͵ܽ െ ܾ ൌ ͸ ֜ ʹሺ͵ܽ െ ܾሻ ൌ ʹ ൈ ͸ ֜ ͸ܽ െ ʹܾ ൌ ͳʹ Double equation ͳ to get equation ͵ . Now we can add the equations since െʹܾ and ʹܾ are opposites.

Solve linear and quadratic simultaneous equations mostly by substitution method. Rearrange the linear equation to make something the subject, in this case ‫ݕ‬, which means ‫ ݕ‬can be replaced with ͳ ൅ ‫ ݔ‬in the quadratic equation. Use ሺܽ ൅ ܾሻଶ ൌ ܽଶ ൅ ʹܾܽ ൅ ܾ ଶ to expand ሺͳ ൅ ‫ݔ‬ሻଶ

Simultaneous equations with a linear and a quadratic will produce two pairs of answers, in this case ‫ ݔ‬ൌ െͶ, ‫ ݕ‬ൌ െ͵ and ‫ ݔ‬ൌ ͵, ‫ ݕ‬ൌ Ͷ.

You can use either elimination or substitution method, in this case we’ve use substitution. Solving the same equations graphically simply means reading coordinates of intersection: ‫ݕ‬ ‫ ݕ‬ൌ ‫ݔ‬ଶ ‫ ݕ‬ൌ Ͷ‫ ݔ‬൅ ͷ

ሺȂ ͳǡͳሻ

sickmafs.com > A level Maths > Core 1

ሺͷǡʹͷሻ

‫ݔ‬ i.e. intersection ሺͷǡ ʹͷሻ ֜ ‫ ݔ‬ൌ ͷ when ‫ ݕ‬ൌ ʹͷ and intersection ሺെͳǡͳሻ ֜ ‫ ݔ‬ൌ െͳ when ‫ ݕ‬ൌ ͳ

Exercise ͷܽ Solve these simultaneous equations using elimination method to 1d.p:

Exercise ͷܿ Find coordinates of where the following graphs intersect:

ͳǤ

ʹ‫ ݕ‬൅ ‫ ݔ‬ൌ ͳͲ ‫ݕ‬൅‫ ݔ‬ൌ͸

͵Ǥ

ͷǤ

͹Ǥ

ͻǤ

ͳͳǤ

ͳ͵Ǥ

ͳͷǤ

ͳ͹Ǥ

ͳͻǤ

ʹͳǤ

ʹ͵Ǥ

ʹͷǤ

െʹ‫ ݐ‬൅ ʹ݇ ൌ ͻͲ ʹ‫ ݐ‬൅ ݇ ൌ ͵Ͳ ܿ ൅ ͵݀ ൌ ͵͵ ܿ ൅ ݀ ൌ ʹ͹ ʹܽ െ ܾ ൌ ͺ ʹܽ ൅ ʹܾ ൌ ʹ ʹܽ െ ܾ ൌ ͲǤͷ ܽ െ ܾ ൌ ͲǤͶ ͻܿ ൅ ͵݀ ൌ ͵͵ ܿ ൅ ݀ ൌ ͹ ܿ െ ݉ ൌ ͸ ݉ ൅ ͵ܿ ൌ ͳͲ ܽ െ ܾ ൌ ͺ ʹܽ ൅ ʹܾ ൌ ͳʹ ͷ݀ ൅ Ͷ݁ ൌ ʹͲ ͵݀ ൅ ͵݁ ൌ ͳʹ ͷܽ െ ͵ܾ െ ͳͶ ൌ Ͳ ʹܽ ൅ ͵ܾ ൌ Ͳ ‫ ݕ‬ൌ ͹ ൅ ‫ݔ‬ Ͷ‫ ݕ‬ൌ ʹͷ ൅ ‫ ݔ‬ ʹ‫ ݕ‬൅ ‫ ݔ‬ଶ ൌ ͳͲ ‫ ݕ‬൅ ‫ݔ‬ଶ ൌ ͸

ͻܿ ൅ ͵݀ଶ ൌ ͵͵ ܿ ൅ ݀ଶ ൌ ͹

ʹǤ

ʹ‫ ݕ‬െ ‫ ݔ‬ൌ ͳͲ ‫ ݕ‬െ ‫ ݔ‬ൌ ͸

ͶǤ

͸Ǥ

ͺǤ

ͳͲǤ

ͳʹǤ

ͳͶǤ

ͳ͸Ǥ

ͳͺǤ

ʹͲǤ

ʹʹǤ

ʹͶǤ

ʹ͸Ǥ

െʹ‫ ݕ‬൅ ͸‫ ݔ‬ൌ ͺ െʹ‫ ݕ‬൅ ‫ ݔ‬ൌ ͳ͵ ݇ ൅ ͵݉ ൌ ͳͲ െ͵݉ ൅ ʹ݇ ൌ ʹͲ ʹܽ െ ʹܾ ൌ ͳʹ ܽ ൅ ܾ ൌ Ͷ ͵ܽ ൅ ʹܾ ൌ ͳǤ͹ ܽ െ ʹܾ ൌ ͲǤ͵

ͷܽ െ ܾ ൌ ͳ͹ ʹܽ ൅ ͵ܾ ൌ Ͳ ͵݀ ൅ ܿ ൌ ʹ ʹܿ ൅ ͵݀ ൌ ͹ ͵‫ ݔ‬൅ ‫ ݕ‬ൌ ͳͲ ʹ‫ ݔ‬൅ ͵‫ ݕ‬ൌ ͻ ͷ݀ ൅ ͸݁ ൌ െͳ͸ ͵݀ ൅ ͺ݁ ൌ െͳͶ ͷܽ െ Ͷܾ െ ͳͶ ൌ ͳ͸ ʹܽ ൅ ͵ܾ ൌ ͵ͷ ‫ ݔ‬ൌ ͳͳ െ ʹ‫ݕ‬ ͵‫ ݕ‬ൌ ͳ͸ െ ‫ݔ‬ ʹ‫ ݕ‬ଶ ൅ ͸‫ ݔ‬ൌ ͺ ʹ‫ ݕ‬ଶ ൅ ‫ ݔ‬ൌ ͳ͵ ݇ ଶ ൅ ͵݉ ൌ ͳͲ െ͵݉ ൅ ʹ݇ ଶ ൌ ʹͲ

͵Ǥ

ͷǤ

͹Ǥ

ͻǤ

ͳͳǤ

ͳ͵Ǥ

ͳͷǤ

ͳ͹Ǥ

ͳͻǤ

ʹͳǤ

ʹ‫ ݕ‬൅ ‫ ݔ‬ൌ ͻ ‫ ݕ‬൅ ‫ ݔ‬ൌ Ͷ ͵݉ െ ‫ ݔ‬ൌ ͳʹ ݉ െ ‫ ݔ‬ൌ Ͷ ʹ݇ െ ʹ‫ ݐ‬ൌ ͻ ݇ െ ʹ‫ ݐ‬ൌ Ͷ െܾ ൅ ʹܽ ൌ ͹ ܽ െ ܾ ൌ ͸ െʹ݀ ൅ Ͷ݁ ൌ ʹͺ ʹ݀ ൅ ͵݁ ൌ ͳͶ ͻܿ ൅ ͵݀ ൌ ͵͵ ܿ ൅ ݀ ൌ ͹ ͵‫ ݔ‬൅ ʹ‫ ݕ‬ൌ ͳ͵ ʹ‫ ݔ‬൅ ͵‫ ݕ‬ൌ ͹ ͸‫ ݔ‬െ ͵‫ ݕ‬ൌ ͵ ͷ‫ ݔ‬െ ʹ‫ ݕ‬ൌ Ͷ ݄ ൌ ͳ͵ െ Ͷ݅ ͵݅ ൅ ʹ݄ ൌ ͳ͸ ͳͳ‫ ݔ‬െ ͵‫ ݕ‬ൌ ͺ ͻ‫ ݔ‬൅ Ͷ‫ ݕ‬ൌ ͳ͵ ݉ ൌ݊൅͹ ݉ ൅ ͳ ൌ ʹ݊

ʹǤ

‫ ݕ‬െ ‫ ݔ‬ൌ ͺ ʹ‫ ݕ‬െ ‫ ݔ‬ൌ ͳͺ Ͷ݂ ൅ ݃ ൌ Ͷͷ ݂ ൅ ݃ ൌ ͻ ʹܽ െ ͵ܾ ൌ ͹ ʹܽ െ ʹܾ ൌ ͺ ͵ܿ െ ݉ ൌ ͳͲ ܿ െ ݉ ൌ Ͷ

ͶǤ

͸Ǥ

ͺǤ

ͳͲǤ ͵݄ ൅ ʹܽ ൌ ͺ െ͵݄ ൅ ܽ ൌ Ͷ ͳʹǤ ͷܾ ൅ ͸ܿ ൌ ͵Ͷ ܾ െ ʹܿ ൌ െ͸ ͳͶǤ ͷ‫ ݌‬൅ Ͷ‫ ݍ‬ൌ ͵Ͷ ʹ‫ ݌‬൅ ʹ‫ ݍ‬ൌ ͳͶ

ͳ͸Ǥ

ͳͺǤ

ʹͲǤ

ʹʹǤ

ͷ‫ ݔ‬െ ͵‫ ݕ‬ൌ ͳ Ͷ‫ ݔ‬൅ ʹ‫ ݕ‬ൌ ͳͶ ʹ݉ ൅ ͵݊ ൌ െͺ ͵݉ ൅ ʹ݊ ൌ െͳʹ ‫ ݕ‬൅ ͳ ൌ ʹ‫ݔ‬ ‫ ݕ‬ൌ‫ݔ‬൅ʹ ͷܽ െ Ͷ ൌ Ͷܾ ʹܾ ൅ ʹ ൌ ܽ

Exercise ͷܾ Solve these simultaneous equations using substitution method:

Exercise ͷ݀ Find coordinates of where the following graphs intersect:

ͳǤ

͵Ǥ

ͷǤ

͹Ǥ

‫ ݕ‬െ ‫ ݔ‬ൌ ͹ ‫ ݕ‬ଶ ൅ ‫ ݔ‬ଶ ൌ ͳ͸ͻ ʹ‫ ݔ‬൅ ‫ ݕ‬ൌ Ͷ ‫ ݕ‬ଶ ൅ ‫ ݔ‬ଶ ൌ ͷ ݉ ൌ݊െ͵ ݉ଶ ൅ ݊ଶ ൌ ͳ͹

ʹ‫ ݕ‬െ ‫ ݔ‬ൌ Ͷ ‫ ݕ‬ଶ ൅ ʹ‫ ݔ‬ଶ ൌ ͳ͹

ʹǤ

ͶǤ

͸Ǥ

ͺǤ

sickmafs.com > A level Maths > Core 1

݉ ൌ ݊ ൅ ʹ ݉ଶ ൅ ݊ଶ ൌ ͹͵ ݉ ൅ ݊ ൌ ͷ ݉ଶ ൅ ݊ଶ ൌ ͳ͵ ‫ ݕ‬൅ ‫ ݔ‬ൌ ͳͲ ‫ ݔݕ‬ൌ ʹͶ ‫ ݕ‬ൌ ʹ‫ ݔ‬൅ ͵ ‫ ݕ‬ଶ ൅ ‫ ݔ‬ଶ ൌ ͻͲ

͵Ǥ

ͷǤ

͹Ǥ

‫ ݕ‬ൌ ͵‫ ݔ‬െ ʹ ‫ ݕ‬ൌ ‫ ݔ‬ଶ

‫ ݕ‬ൌ െͶ‫ ݔ‬െ ͵ ‫ ݕ‬ൌ ‫ ݔ‬ଶ ‫ ݕ‬ൌ ‫ݔ‬൅͵ ‫ ݕ‬ൌ ʹ‫ ݔ‬ଶ ʹ‫ ݔ‬൅ ‫ ݕ‬ൌ Ͷ ‫ ݕ‬ଶ ൅ ‫ ݔ‬ଶ ൌ ͷ

ʹǤ

ͶǤ

͸Ǥ

ͺǤ

‫ ݕ‬ൌ ͳͳ‫ ݔ‬൅ ͷ ‫ ݕ‬ൌ ʹ‫ ݔ‬ଶ ‫ ݕ‬ൌ ͸‫ ݔ‬െ ͷ ‫ ݕ‬ൌ ‫ݔ‬ଶ

‫ ݕ‬ൌ ͹‫ ݔ‬൅ ͷ ‫ ݕ‬ൌ ʹ‫ ݔ‬ଶ

‫ ݕ‬ൌ ʹ‫ ݔ‬൅ ͵ ‫ ݕ‬ଶ ൅ ‫ ݔ‬ଶ ൌ ͻͲ

6. Sketching graphs

Vid 1.1

Shapes of important graphs Quadratic graphs: Vid 2.1 ‫ݕ‬ Quadratic graphs have: ‫ݔ‬ Cubic graphs: ‫ݕ‬

ଶ

Equation ‫ ݕ‬ൌ ܽ‫ ݔ‬൅ ܾ‫ ݔ‬൅ ܿǡ ܽ ് Ͳ One vertex Vertical line of symmetry through vertex ‫ ݕ‬ൌ ‫ ݔ‬ଶ (see dotted graph) is most used quadratic graph with vertex at ሺͲǡͲሻ.

· · · ·

Vid 2.2

‫ݔ‬

Reciprocal graphs: ‫ݕ‬ ‫ݕ‬ൌ‫ݔ‬ ‫ݔ‬

‫ ݕ‬ൌ െ‫ݔ‬

Cubic graphs have: · Equation ‫ ݕ‬ൌ ܽ‫ ݔ‬ଷ ൅ ܾ‫ ݔ‬ଶ ൅ ܿ‫ ݔ‬൅ ݀ǡ ܽ ് Ͳ · Two vertices except for graphs like ‫ ݕ‬ൌ ܽ‫ ݔ‬ଷ , ܽ ് Ͳ (see dotted graph) which flatten out at ሺͲǡͲሻ.

௫

Asymptotes (lines which it gets closer to but never reaches) at the ‫ ݔ‬and ‫ ݕ‬axes. Symmetrical about ‫ ݕ‬ൌ ‫ ݔ‬and ‫ ݕ‬ൌ െ‫ݔ‬.

· ·

݂ሺ‫ݔ‬ሻ ൌ ‫ ݔ‬൅ ‫ ݔ‬ଷ . Find in terms of ‫ݔ‬: (a) ݂ሺͷሻ (b) ͵݂ሺ‫ ݔ‬൅ ͳሻ (a) ݂ሺͷሻ ൌ ͷ ൅ ͷଷ ൌ ͳ͵Ͳ (b) ͵݂ሺ‫ ݔ‬൅ ͳሻ ൌ ͵ሾሺ‫ ݔ‬൅ ͳሻ ൅ ሺ‫ ݔ‬൅ ͳሻଷ ሿ

݂ሺ‫ݔ‬ሻ ൌ ‫ ݔ‬ଶ . Find in terms of ݂ሺ‫ݔ‬ሻ: (a) ͷሺ‫ ݔ‬൅ ͳሻଶ (b) ‫ ݔ‬ଶ െ ͳͲ‫ ݔ‬൅ ʹ͸ (a) ͷሺ‫ ݔ‬൅ ͳሻଶ ൌ ͷ݂ሺ‫ ݔ‬൅ ͳሻ (b) ‫ ݔ‬ଶ െ ͳͲ‫ ݔ‬൅ ʹ͸ ൌ ሺ‫ ݔ‬െ ͷሻଶ ൅ ͳ ൌ ݂ሺ‫ ݔ‬െ ͷሻ ൅ ͳ Finding where graphs cross or touch the coordinate axes Vid 4.1 ‫ݕ‬ ‫ ݕ‬ൌ ݂ሺ‫ݔ‬ሻ For graphs of ‫ ݕ‬ൌ ݂ሺ‫ݔ‬ሻ: · ‫ ݕ‬ൌ ݂ሺ‫ݔ‬ሻ crosses y axis when ‫ ݕ‬ൌ ݂ሺͲሻ ‫ݔ‬ ‫ ݕ‬ൌ ݂ሺ‫ݔ‬ሻ crosses x axis when Ͳ ൌ ݂ሺ‫ݔ‬ሻ · ·

Graph touches x axis at duplicate solutions for Ͳ ൌ ݂ሺ‫ݔ‬ሻ

Sketch graph of ‫ ݕ‬ൌ ሺ‫ ݔ‬െ ʹሻሺ‫ ݔ‬൅ ͵ሻଶ . Vid 4.1 Substitute ‫ ݕ‬ൌ Ͳ into ‫ ݕ‬ൌ ሺ‫ ݔ‬െ ʹሻሺ‫ ݔ‬൅ ͵ሻଶ ֜ ሺ‫ ݔ‬െ ʹሻሺ‫ ݔ‬൅ ͵ሻଶ ൌ Ͳ ֜ ‫ ݔ‬ൌ ʹ, െ͵, െ͵ i.e. crosses ‫ ݔ‬axis at ʹ, touches ‫ ݔ‬axis at െ͵. Substitute ‫ ݔ‬ൌ Ͳ into ‫ ݕ‬ൌ ሺ‫ ݔ‬െ ʹሻሺ‫ ݔ‬൅ ͵ሻଶ ֜ ሺͲ െ ʹሻሺͲ ൅ ͵ሻଶ ൌ െͳͺ i.e. graph crosses ‫ ݕ‬axis at െͳͺ: െ͵

‫ ݕ‬ൌ ሺ‫ ݔ‬െ ʹሻሺ‫ ݔ‬൅ ͵ሻଶ

െͳͺ

e.g. ‫ ݕ‬ൌ ͵‫ ݔ‬ଶ ൅ Ͳ‫ ݔ‬െ ͻ

A vertex is a point on a graph where its gradient changes from positive to negative (or vice versa).

e.g. ‫ ݕ‬ൌ ͳ‫ ݔ‬ଷ െ ʹ‫ ݔ‬ଶ ൅ Ͳ‫ ݔ‬൅ ͹

Vid 2.3

Reciprocal graphs have: ௞ · Equation ‫ ݕ‬ൌ , ݇ ് Ͳ

What are functions? Vid 3 Functions are algebraic expressions with a name e.g. · ݂ሺ‫ݔ‬ሻ is a function of x i.e. an expression in ‫ݔ‬. The function is called ݂ · ݃ሺ‫ݐ‬ሻ is a function of t i.e. an expression in ‫ݐ‬. The function is called ݃

‫ݕ‬

Sketching graphs means drawing graphs roughly, taking into account their vertex, asymptotes and where they cross the axis. You also have to be aware how graphs transform as their equations change.

‫ݔ‬

sickmafs.com > A level Maths > Core 1

e.g. ‫ ݕ‬ൌ

ଵ

௫

Functions usually have one letter names. ݂ is the most popular name for a function. (a) Replace ‫ ݔ‬with ͷ. (b) Replace ‫ ݔ‬with ‫ ݔ‬൅ ͳ then multiply entire function by ͵.

“‫ ݕ‬ൌ ݂ሺ‫ݔ‬ሻ” is often used to say “some graph with an equation in terms of ‫ݔ‬.” Duplicate solutions technically means graph crosses ‫ ݔ‬axis twice at the same point which practically means graph is touching x axis at that point. In module Core 1 graphs only touch the ‫ ݔ‬axis, not ‫ ݕ‬axis. Solve by factorising: ሺ‫ ݔ‬െ ʹሻሺ‫ ݔ‬൅ ͵ሻଶ ൌ Ͳ ֜ ‫ ݔ‬െ ʹ ൌ Ͳ and ‫ ݔ‬൅ ͵ ൌ Ͳ twice. As െ͵ is a duplicate solution, so it touches, rather than crosses, the x axis.

Duplicate solutions to ݂ሺ‫ݔ‬ሻ ൌ Ͳ doesn’t tell you if graph touches ‫ ݔ‬axis from above or below but when you attempt to sketch the graph you will find only one way to do it.

Exercise ͸ܽ Copy and complete following tables for given equations then plot and draw the graphs accordingly (ideally, use paper with 5mm squares).

ͳǤ ʹǤ ͵Ǥ

ͶǤ

ͷǤ ͸Ǥ ͹Ǥ

‫ ݕ‬ൌ ‫ݔ‬ଶ ‫ ݔ‬Ȃ Ͷ Ȃ ͵ Ȃ ʹ Ȃ ͳ Ͳ ͳ ʹ ͵ Ͷ ‫ݕ‬

͹Ǥ ͻǤ

ଵ

When plotting this graph, use Ͷ squares per unit on the ‫ ݔ‬axis, Ͷ squares per unit on the ‫ ݕ‬axis. ଶ

௫

ଵ

‫ ݔ‬െ ͵ െ ʹ െ ͳ െ െ െ Ͳ ͳ ʹ ͵ Ͷ ଶ ଷ ସ ସ ଷ ଶ ‫ݕ‬ When plotting this graph, use Ͷ squares per unit on the ‫ ݔ‬axis, Ͷ squares per unit on the ‫ ݕ‬axis. ‫ ݕ‬ൌ െ

ଵ

௫

ଵ

‫ ݔ‬െ ͵ െ ʹ െ ͳ െ െ െ Ͳ ͳ ʹ ͵ Ͷ ଶ ଷ ସ ସ ଷ ଶ ‫ݕ‬ When plotting this graph, use Ͷ squares per unit on the ‫ ݔ‬axis, Ͷ squares per unit on the ‫ ݕ‬axis.

௫

ͳͳǤ

‫ ݔ‬െ ͵ െ ʹ െ ͳ െ െ െ Ͳ ͳ ʹ ͵ Ͷ ଶ ଷ ସ ସ ଷ ଶ ‫ݕ‬

‫ݕ‬ൌ

‫ ݔ‬ଶ ൅ Ͷ‫ ݔ‬൅ ͳͲ

When plotting this graph, use Ͷ squares per unit on the ‫ ݔ‬axis, Ͷ squares per unit on the ‫ ݕ‬axis. ଵ

‫ ݔ‬ଶ ൅ ͸‫ ݔ‬൅ ͳͳ

‫ ݕ‬ൌ ‫ ݔ‬ଷ െ ʹ‫ ݔ‬൅ ͵ ‫ ݔ‬െ Ͷ െ ͵ െ ʹ െ ͳ Ͳ ͳ ʹ ͵ Ͷ ‫ݕ‬

ଵ

ሺ‫ ݔ‬൅ ͵ሻଶ ൅ ͷ

When plotting above graph, use Ͷ squares per unit on the ‫ݕ‬ axis, ͳ square per unit on the ‫ ݔ‬axis.

ଵ

ͷሺ‫ ݔ‬െ ͺሻହ

ͷǤ

‫ݕ‬ൌ ‫ ݔ‬Ȃ Ͷ Ȃ ͵ Ȃ ʹ Ȃ ͳ Ͳ ͳ ʹ ͵ Ͷ ‫ݕ‬

௫

͵Ǥ

‫ݔ‬ଷ

‫ݕ‬ൌ

ሺ‫ ݔ‬൅ ͺሻଶ

ͳǤ

‫ ݕ‬ൌ ‫ݔ‬ଶ െ ‫ ݔ‬൅ ʹ ‫ ݔ‬Ȃ Ͷ Ȃ ͵ Ȃ ʹ Ȃ ͳ Ͳ ͳ ʹ ͵ Ͷ ‫ݕ‬

ଵ

Exercise ͸ܾ ଵ For functions (a) ݂ሺ‫ݔ‬ሻ ൌ ‫ ݔ‬ଶ (b) ݂ሺ‫ݔ‬ሻ ൌ ‫ݔ‬ ଶ (c) ݂ሺ‫ݔ‬ሻ ൌ ሺ͵ െ ‫ݔ‬ሻଶ , find the following in terms of ݂ሺ‫ݔ‬ሻ:

ଷ

ሺ‫ ݔ‬െ Ͷሻଷ

ͶǤ

െሺ‫ ݔ‬െ ͹ሻଶ

͸Ǥ

‫ ݔ‬ଶ ൅ ͳͲ‫ ݔ‬൅ ʹͳ ͺǤ ͳʹǤ ‫ ݔ‬ଶ െ ͺ‫ ݔ‬൅ ͸Ͳ

ͳͶǤ

ʹ‫ ݔ‬൅ ͳ

Exercise ͸ܿ ଵ For functions (a) ݂ሺ‫ݔ‬ሻ ൌ ‫ ݔ‬ଶ (b) ݂ሺ‫ݔ‬ሻ ൌ ‫ݔ‬ ଶ (c) ݂ሺ‫ݔ‬ሻ ൌ ሺ͵ െ ‫ݔ‬ሻଶ , find the following in terms of ‫ݔ‬: ͳǤ

ʹ݂ሺ‫ݔ‬ሻ

ʹǤ

݂ሺ͵‫ݔ‬ሻ

͵Ǥ

ͷǤ

͹Ǥ ͻǤ ͳͳǤ

݂ሺ‫ ݔ‬െ ͷሻ ݂ሺെ‫ݔ‬ሻ

ͶǤ

͸Ǥ

ଶ ଷ

݂ሺ‫ݔ‬ሻ

݂ሺെͷ‫ݔ‬ሻ

݂ሺͶ െ ‫ݔ‬ሻ

െ݂ሺ‫ݔ‬ሻ ͺǤ ͳʹǤ ͵ െ ݂ሺͷ‫ݔ‬ሻ

݂ሺ͵‫ݔ‬ሻ ൅ ͳ

݂ሺെ‫ݔ‬ሻ ൅ Ͷ

ͳͶǤ

݂൫െξ‫ݔ‬൯

Exercise ͸݀ Find where the following equations of graphs cross (a) the y axis (b) the x axis (c) if and where it touches the x axis (d) sketch the graphs showing point of intersection with axis. ͳǤ ʹǤ ͵Ǥ ͶǤ ͷǤ ͸Ǥ ͹Ǥ ͺǤ

‫ ݕ‬ൌ ʹ‫ ݔ‬െ ͷ ‫ ݕ‬ൌ ሺ‫ ݔ‬െ ͷሻሺ‫ ݔ‬൅ ͹ሻ ‫ ݕ‬ൌ ሺ‫ ݔ‬െ ͳሻሺ‫ ݔ‬൅ ʹሻሺ‫ ݔ‬െ Ͷሻ ‫ ݕ‬ൌ ሺʹ‫ ݔ‬െ ͷሻሺ‫ ݔ‬൅ ͹ሻ ‫ ݕ‬ൌ ሺ‫ ݔ‬െ ͳሻሺͷ‫ ݔ‬൅ ʹሻሺͶ െ ‫ݔ‬ሻ ‫ ݕ‬ൌ ‫ݔ‬ሺ‫ ݔ‬൅ ʹሻሺ‫ ݔ‬െ Ͷሻ ‫ ݕ‬ൌ ʹ‫ ݔ‬ଶ ൅ ͵‫ ݔ‬൅ ͳ ‫ ݕ‬ൌ ‫ ݔ‬ଶ െ ͷ

ͻǤ

‫ݕ‬ൌ

ͳʹǤ ͳ͵Ǥ ͳͶǤ ͳͷǤ ͳ͸Ǥ ͳ͹Ǥ ͳͺǤ

଻

௫ Ȃ ଶ ଶ

െͷ

‫ݕ‬ൌ െͷ

ͳͲǤ ͳͳǤ

sickmafs.com > A level Maths > Core 1

ሺͺ ൅ ‫ݔ‬ሻଶ

ʹǤ

‫ݕ‬ൌ

௫

଻

௫ Ȃ ଷ

‫ ݕ‬ൌ ሺ‫ ݔ‬െ ͺሻሺ‫ ݔ‬൅ ͹ሻଶ ‫ ݕ‬ൌ ሺ‫ ݔ‬൅ Ͷሻଶ ሺ‫ ݔ‬െ ͳሻ ‫ ݕ‬ൌ ‫ ݔ‬ଶ ሺ‫ ݔ‬െ ͷሻ ‫ ݕ‬ൌ ሺ‫ ݔ‬െ ʹሻ‫ ݔ‬ଶ ‫ ݕ‬ൌ ‫ݔ‬ሺ‫ ݔ‬൅ ͵ሻଶ ‫ ݕ‬ൌ ሺ‫ ݔ‬൅ ͳʹሻଷ ‫ ݕ‬ൌ ሺ‫ ݔ‬െ ͳሻଷ

Graphs can be transformed by sliding and stretching/squeezing.

7. Transforming graphs

Vid 2

Summary of transformations table: Transforming graphs Vid 1.1 Here are the different ways to transform graphs: Vid 1.2 Vid 1.3

move ‫ ݕ‬direction ‫ݕ‬

slide

Vid 1.4

‫ ݕ‬ൌ ݂ሺ‫ݔ‬ሻ

stretch/ squeeze

Vid 1.5

Vid 1.6

reflect

൅ܽ

‫ ݕ‬ൌ ݂ሺ‫ݔ‬ሻ ൅ ܽ

‫ ݕ‬ൌ ݂ሺ‫ݔ‬ሻ ‫ ݕ‬ൌ ݂ܽሺ‫ݔ‬ሻ

‫ݔ‬

‫ݕ‬ ‫ݕ‬

‫ ݕ‬ൌ െ݂ሺ‫ݔ‬ሻ

‫ݔ‬

‫ ݕ‬ൌ ݂ሺ‫ݔ‬ሻ

‫ݔ‬

ൈܽ

move ‫ ݔ‬direction ‫ݕ‬ െܽ ‫ ݕ‬ൌ ݂ሺ‫ ݔ‬൅ ܽሻ

ൊܽ ‫ݕ‬

‫ݔ‬ ‫ ݕ‬ൌ ݂ሺ‫ݔ‬ሻ ‫ݔ‬

‫ ݕ‬ൌ ݂ሺ‫ݔ‬ሻ ‫ ݕ‬ൌ ݂ሺܽ‫ݔ‬ሻ

‫ݕ‬

‫ ݕ‬ൌ ݂ሺ‫ݔ‬ሻ

‫ݔ‬

‫ ݕ‬ൌ ݂ሺെ‫ݔ‬ሻ

Combining transformations of graphs Vid 3 Find transformations of graph from of ‫ ݕ‬ൌ ݂ሺ‫ݔ‬ሻ to ‫ ݕ‬ൌ ʹ݂ሺ‫ݔ‬ሻ ൅ ͳ. Ͳ Enlarges graph by scale factor ʹ from ‫ ݔ‬axis then translates graph ቀ ቁ. ͳ

Vid 1.7

Find transformations of graph from of ‫ ݕ‬ൌ ‫ ݔ‬ଷ to ‫ ݕ‬ൌ െሺʹ‫ݔ‬ሻଷ . ‫ ݕ‬ൌ ‫ ݔ‬ଷ to ‫ ݕ‬ൌ െሺʹ‫ݔ‬ሻଷ is same as ‫ ݕ‬ൌ ݂ሺ‫ݔ‬ሻ to ‫ ݕ‬ൌ െ݂ሺʹ‫ݔ‬ሻ ଵ i.e. enlarges graph by scale factor from ‫ ݕ‬axis then reflects graph about ‫ݔ‬ ଶ axis. Completing the square to sketch quadratic graphs Vid 4 Sketch graph of ‫ ݕ‬ൌ ‫ ݔ‬ଶ ൅ Ͷ‫ ݔ‬൅ ͳͲ showing coordinate of vertex. ‫ ݕ‬ൌ ‫ ݔ‬ଶ ൅ Ͷ‫ ݔ‬൅ ͳͲ ൌ ሺ‫ ݔ‬൅ ʹሻଶ ൅ ͸ െʹ translates graph ቀ ቁ ͸ For axes intercepts: ‫ ݕ‬ൌ Ͳ ֜ no solution, ‫ ݔ‬ൌ Ͳ ֜ ‫ ݕ‬ൌ ͳͲ: ‫ݕ‬

ሺȂ ʹǡ͸ሻ

ͳͲ

‫ ݕ‬ൌ ‫ ݔ‬ଶ ൅ Ͷ‫ ݔ‬൅ ͳͲ ‫ݔ‬

sickmafs.com > A level Maths > Core 1

For any graph, these changes in function lead to these change in coordinates: Change in function Change in coordinates from ‫ ݕ‬ൌ ݂ሺ‫ݔ‬ሻ to: from ሺ‫ݔ‬ǡ ‫ݕ‬ሻ to: ‫ ݕ‬ൌ ݂ሺ‫ݔ‬ሻ ൅ ܽ ሺ‫ݔ‬ǡ ‫ ݕ‬൅ ܽሻ ‫ ݕ‬ൌ ݂ሺ‫ ݔ‬൅ ܽሻ ሺ‫ ݔ‬െ ܽǡ ‫ݕ‬ሻ ‫ ݕ‬ൌ ݂ܽሺ‫ݔ‬ሻ ሺ‫ݔ‬ǡ ܽ‫ݕ‬ሻ ௫ ‫ ݕ‬ൌ ݂ሺܽ‫ݔ‬ሻ ሺ ǡ ‫ݕ‬ሻ ௔ ‫ ݕ‬ൌ െ݂ሺ‫ݔ‬ሻ ሺ‫ݔ‬ǡ െ‫ݕ‬ሻ ‫ ݕ‬ൌ ݂ሺെ‫ݔ‬ሻ ሺെ‫ݔ‬ǡ ‫ݕ‬ሻ Summary: change outside function moves graph vertically in the same way i.e. ݂ሺ‫ݔ‬ሻ to ݂ሺ‫ݔ‬ሻ ൅ ܽ moves graph vertically ൅ܽ. Change inside function moves graph horizontally in the opposite way i.e. ݂ሺ‫ݔ‬ሻ to ݂ሺ‫ ݔ‬൅ ܽሻ moves graph horizontally Ȃ ܽ. ‫ ݕ‬ൌ ݂ሺ‫ݔ‬ሻ to ‫ ݕ‬ൌ ʹ݂ሺ‫ݔ‬ሻ enlarges graph by scale factor ʹ from ‫ ݔ‬axis. ‫ ݕ‬ൌ ʹ݂ሺ‫ݔ‬ሻ to ‫ ݕ‬ൌ ʹ݂ሺ‫ݔ‬ሻ ൅ ͳ translates Ͳ graph ቀ ቁ. ͳ ‫ ݕ‬ൌ ݂ሺ‫ݔ‬ሻ to ‫ ݕ‬ൌ ݂ሺʹ‫ݔ‬ሻ enlarges graph by scale factor ʹ from ‫ ݕ‬axis. ‫ ݕ‬ൌ ݂ሺʹ‫ݔ‬ሻ to ‫ ݕ‬ൌ െ݂ሺʹ‫ݔ‬ሻ reflects graph about ‫ ݔ‬axis. Complete square to compare graph to ‫ ݕ‬ൌ ‫ ݔ‬ଶ so that you can find vertex. ‫ ݕ‬ൌ ‫ ݔ‬ଶ to ‫ ݕ‬ൌ ሺ‫ ݔ‬൅ ʹሻଶ ൅ ͸ same as ‫ ݕ‬ൌ ݂ሺ‫ݔ‬ሻ to ‫ ݕ‬ൌ ݂ሺ‫ ݔ‬൅ ʹሻ ൅ ͸

‫ ݕ‬ൌ ‫ ݔ‬ଶ to ‫ ݕ‬ൌ ሺ‫ ݔ‬൅ ʹሻଶ adds ʹ inside the function i.e. equivalent to ݂ሺ‫ݔ‬ሻ to ݂ሺ‫ ݔ‬൅ ʹሻ. ‫ ݕ‬ൌ ሺ‫ ݔ‬൅ ʹሻଶ to ‫ ݕ‬ൌ ሺ‫ ݔ‬൅ ʹሻଶ ൅ ͸ adds ͸ outside the function i.e. equivalent to ݂ሺ‫ ݔ‬൅ ʹሻ to ݂ሺ‫ ݔ‬൅ ʹሻ ൅ ͸. No solution for ‫ ݕ‬ൌ Ͳ means graph doesn’t cross ‫ ݔ‬axis.