Functions

Question 1

Difference of two squares

Answer 1

Question 2

Ways of solving quadratic functions

Answer 2

• Quadratic formula
• Factorisation
• Completing the square

Question 3

Quadratic formula

Answer 3

Question 4

Factorising quadratic functions where a=1

Answer 4

1. Find two numbers which multiply to c and add to the coefficient of b
2. Put the two values into the factorised form:
   (x +/- q)(x +/- r) = 0
3. Due to the zero product property, the first or second factor must equal zero: make both equal to zero and solve for x
4. The two values are your possible x values

Question 5

Factorising quadratic equations when a ≠ 1

Answer 5

1. Find two numbers which multiply to ac and add to the coefficient of b
2. Subsitute them into the equation so that they are now coefficients of x and replace b
3. Separate the first and second half of the equation and factorise each by taking out a numerical or coefficient value: the two bracketed expressions should be the same
4. Collect the expression formed by the values “taken out” and that in the bracket to form a factorised pair of parentheses
5. Make those equal to zero and solve for x, giving your two possible x values

Question 6

Completing the square when a = 1

Answer 6

Remember:
1. if the sign of c is negative, the final formula will also have -c
2. To solve the equation, make it equal to zero and solve using square root

Question 7

Completing the square when a ≠ 1

Answer 7

Question 8

Discriminant of quadratic equation

Answer 8

Question 9

Inequalities and the existence of roots from the discriminant

Answer 9

Question 10

Vieta’s formulae

Answer 10

Question 11

Form quadratic equations given the roots (two methods)

Answer 11

1. Form the factorised forms of quadratics using the roots into two brackets and make it equal to zero (Remember: the sign of the value in the bracket must be opposite the root)
2. Use vieta’s formulae and substitute the roots to find the a,b and c of the equation

Question 12

Answer 12

Question 13

Find the inverse of a function

Answer 13

1. Change f(x) to y, if needed
2. Swap x and y
3. Solve for y

Question 14

Find the intersection between two lines

Answer 14

1. Make both functions equal to y (if needed)
2. Make them equal to each other
3. Solve for x
4. Substitute this x into either of the original equation
5. Solve for y value

Question 15

How to find if two functions are inverses of each other?

Answer 15

Both f(g(x)) and g(f(x)) must equal to x

Question 16

How does the completed square form tell the minimum/maximum of a quadratic function?

Answer 16

• If a>0 then it is a minimum, if a<0 then it is a maximum
• The coordinates for x is the opposite sign (the negative) of the numerical value in the bracket
• The coordinates for y is the same sign (unchanged) of the value outside the bracket

Question 17

Formula for minimum/maximum of a quadratic for x coordinate

Answer 17

• • b/2a

Question 18

Formula for minimum/maximum of a quadratic for y coordinate

Answer 18

• • D/4a

Question 19

Linear transformation
f(x) + a

Answer 19

Shifts/translates the graph by |a| units along the y-axis
* up if a>0
* down if a<0

Question 20

Linear transformation
f(x + a)

Answer 20

Shifts/translates graph by|a|units along the x-axis
* left if a>0
* right if a<0

Question 21

Linear transformation
cf(x)

Answer 21

Stretch or compress vertically (along the y-axis)
* Stretch by c when c is greater than 1 (c>1)
* Compress by c when c is greater than 0 but less than 1 (0<c<1)

Question 22

Linear transformation
f(cx)

Answer 22

Stretch or compress horizontally (along the x-axis)
* Stretch by c when c is greater than 0 but less than 1 (0<c<1)
* Compressed by c when c is greater than 1 (c>1)

Question 23

Linear transformation
-f(x)

Answer 23

Reflection about the x-axis

Question 24

Linear transformation
f(-x)

Answer 24

Reflection about the y-axis

Question 25

General transformation rules

Answer 25

Question 26

Composite functions and how to write them

Answer 26

A composite function is created when one function is substituted into another function **Remember:** * write down the inner function first with the outer function still as f(x) * replace the x(s) in the outer function with the inner function

Question 27

Sketching linear functions

Answer 27

y = mx + b * m is the gradient (**remember:** if m is a whole number, the gradient can be written as m/1 (e.g. m=2, g=2/1) * b is the y-intercept

Question 28

Gradient of a linear function and the angle formed

Answer 28

Question 29

How to restrict the domain of a function so that it can be invertible?

Answer 29

**Remember:** a function is only invertible if it is one-to-one 1. Find the inverse of the function 2. If there are values in the infinite domain which would not give real solutions, restrict them using interval notation or inequalities

Question 30

How to sketch the inverse of a function?

Answer 30

Reflect it about y=x

Question 31

Finding the horizontal asymptote

Answer 31

Question 32

Finding the vertical asymptote

Answer 32

Question 33

Sketching rational functions

Answer 33

Find the vertical and horizontal asymptote and according to them, shift the parent function 1/x

Question 34

Gener