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
General transformation rules
Composite functions and how to write them
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
Sketching linear functions
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
Gradient of a linear function and the angle formed
How to restrict the domain of a function so that it can be invertible?
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
How to sketch the inverse of a function?
Reflect it about y=x
Finding the horizontal asymptote
Finding the vertical asymptote
Sketching rational functions
Find the vertical and horizontal asymptote and according to them, shift the parent function 1/x
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