Functions Flashcards
Difference of two squares
Ways of solving quadratic functions
- Quadratic formula
- Factorisation
- Completing the square
Quadratic formula
Factorising quadratic functions where a=1
- Find two numbers which multiply to c and add to the coefficient of b
- Put the two values into the factorised form:
(x +/- q)(x +/- r) = 0 - Due to the zero product property, the first or second factor must equal zero: make both equal to zero and solve for x
- The two values are your possible x values
Factorising quadratic equations when a ≠ 1
- Find two numbers which multiply to ac and add to the coefficient of b
- Subsitute them into the equation so that they are now coefficients of x and replace b
- 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
- Collect the expression formed by the values “taken out” and that in the bracket to form a factorised pair of parentheses
- Make those equal to zero and solve for x, giving your two possible x values
Completing the square when a = 1
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
Completing the square when a ≠ 1
Discriminant of quadratic equation
Inequalities and the existence of roots from the discriminant
Vieta’s formulae
Form quadratic equations given the roots (two methods)
- 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)
- Use vieta’s formulae and substitute the roots to find the a,b and c of the equation
Find the inverse of a function
- Change f(x) to y, if needed
- Swap x and y
- Solve for y
Find the intersection between two lines
- Make both functions equal to y (if needed)
- Make them equal to each other
- Solve for x
- Substitute this x into either of the original equation
- Solve for y value
How to find if two functions are inverses of each other?
Both f(g(x)) and g(f(x)) must equal to x
How does the completed square form tell the minimum/maximum of a quadratic function?
- 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
Formula for minimum/maximum of a quadratic for x coordinate
- b/2a
Formula for minimum/maximum of a quadratic for y coordinate
- D/4a
Linear transformation
f(x) + a
Shifts/translates the graph by |a| units along the y-axis
* up if a>0
* down if a<0
Linear transformation
f(x + a)
Shifts/translates graph by|a|units along the x-axis
* left if a>0
* right if a<0
Linear transformation
cf(x)
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)
Linear transformation
f(cx)
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)
Linear transformation
-f(x)
Reflection about the x-axis
Linear transformation
f(-x)
Reflection about the y-axis
